Abstract
The aim of this paper is to deal with the anisotropic doubly critical equation
where H is in some cases called Finsler norm, \(H^\circ \) is the dual norm, \(1<p<N\), \(0 \le \gamma < \left( (N-p)/p\right) ^p\) and \(p^*=Np/(N-p)\). In particular, we provide a complete asymptotic analysis of \(u \in \mathcal {D}^{1,p}({\mathbb {R}}^N)\) near the origin and at infinity, showing that this solution has the same features of its euclidean counterpart. Some of the techniques used in the proofs are new even in the Euclidean framework.
Similar content being viewed by others
1 Introduction and main results
This work is devoted to the study of the following anisotropic doubly critical problem
where \(1<p<N\), \(p^*:=Np/(N-p)\) is the Sobolev critical exponent, \(\displaystyle 0 \le \gamma < C_H:=\left( (N-p)/p\right) ^p\) is the Hardy constant and
where \(\Delta _p^H\) is the so-called anisotropic p-Laplacian or Finsler p-Laplacian. We point out that H is a Finsler type norm and \(H^\circ \) is its the dual norm (H satisfies assumptions \((h_H)\), see Sect. 2 for further details). In particular, when \(H(\xi )=|\xi |=H^\circ (\xi )\) the Finsler type p-Laplacian coincides with the classical p-Laplacian, and, hence it is singular when \(1<p<2\) and degenerate when \(p>2\). Then, according with standard regularity theory [13, 27] and the regularity results in the anisotropic framework [2, 6], we say that any solution of (\({\mathcal {P}}_H\)) has to be understood in the weak distributional meaning, i.e. \(u \in D^{1,p}({\mathbb {R}}^N)\) satisfies the following integral equality
The literature about critical problems is really huge. Going back to the Euclidean framework, i.e. when we consider \(H(\xi )=|\xi |=H^\circ (\xi )\) in (\({\mathcal {P}}_H\)), we deal with
In the seminal paper [5], Caffarelli, Gidas and Spruck classified any positive solution to (1.3) with \(p=2\), \(N \ge 3\), and \(\gamma =0\). We point out that a first result, under stronger assumption on the decay of solutions, was obtained by Gidas, Ni and Nirenberg in [14]. Moreover, in this setting a complete answer in the subcritical case was done in the celebrasted work of Gidas and Spruck [15], where the authors proved Liouville-type theorems.
In the quasilinear framework, the situation is much more involved due to the nonlinear nature of the operator. Recently, a classification result of positive solutions to (1.3) with \(p > 2\), \(\gamma =0\) and \(u \in {\mathcal {D}}^{1,p}({\mathbb {R}}^N):=\{u \in L^{p^*}({\mathbb {R}}^N) \ | \ \nabla u \in L^p({\mathbb {R}}^N)\}\) was obtained in [24]. The proof of this result is based on a refined version of the well-known moving plane method of Alexandrov-Serrin [1, 23] and on some a priori estimates of the solutions and their gradients, proved in [29]. To be more precise, we note that the classification result of positive solution to the Sobolev critical quasilinear equation with finite energy started in [12] in the case, and then was extended in [29] for every \(1<p<2\). Subsequently the full case was obtained in [24]. Recently, we refer to the papers [7, 21, 30] for new partial results on the classification of positive solutions without a priori assumption on the energy of solutions. In the anisotropic setting, Ciraolo, Figalli and Roncoroni [9], obtained a complete classification result for positive solution to (\({\mathcal {P}}_H\)) with \(\gamma =0\) using different techniques that do not require the use of the moving plane method, which could not be used in the anisotropic context due to the lack of invariance.
When \(\gamma \ne 0\) the situation is really different. In the seminal paper of Terracini [26], it was proved for the first time the classification result for positive solutions to (1.3) in the case \(p=2\). The author firstly showed the existence of solutions to this problem with a minimization argument based on the concentration and compactness principle. Subsequently, she proved that any solution to this problem is radial and radially decreasing about the origin combining the moving-plane technique and the use of the Kelvin transformation, in the same spirit of [5]. The case \(p \ne 2\) and \(\gamma \ne 0\) is much more involved and it is available in [20], where the techniques used are mainly based on a fine asymptotic analysis at infinity and refined versions of the moving plane procedure, and also on some asymptotic estimates proved in [31, 32].
Our aim is to prove some decay estimates for positive weak solutions to (\({\mathcal {P}}_H\)) in the anisotropic framework \(1<p<N\) and \(\gamma \ne 0\). More precisely, our first main result is the following:
Theorem 1.1
Let \(u \in {\mathcal {D}}^{1,p}({\mathbb {R}}^N)\) be a weak solution of (\({\mathcal {P}}_H\)) with \(1<p<N\), \(0 \le \gamma < C_H\). Then there exist positive constants \(0< R_1< 1 < R_2\) depending on \(N, p, \gamma \) and u, such that
and
where \(\mu _1\), \(\mu _2\) are the solutions of
\(C_1, C_2\) are positive constants depending on \(N,p, \gamma , H\) and u, \(c_1\) is a positive constant depending on \(N,p, \gamma , H, R_1, \mu _1\) and u, \(c_2\) is a positive constant depending on \(N,p, \gamma , H, R_2, \mu _2\) and u.
Remark 1.2
In the following we shall assume that \(\mu _1<\mu _2\) and it is easy to see that
furthermore \({\mathcal {B}}_R^{H^\circ }\) is the dual anisotropic ball also known as Frank diagram (see Sect. 2 for further details).
In the proof we will also exploit some clever ideas from [31] facing the difficulties of the anisotropic issue. A different approach is in fact needed for the study of the asymptotic behaviour of the gradient. In particular, the fact that the moving plane plane technique cannot be applied, a crucial point is given by the following classification result:
Theorem 1.3
Let \(v\in C^{1,\alpha }_{loc}({\mathbb {R}}^N\setminus \{0\})\) be a positive weak solution of the equation
where \(0 \le \gamma < C_H\). Assume that there exist two positive constants C and c such that
where \(\mu _i\) (\(i=1,2\)) are the roots of (1.6) and suppose that there exists a positive constant \({{\hat{C}}}\) such that
then
for some \({{\bar{c}}}>0\).
Theorem 1.3 is new and interesting in itself. The proof is very much different than the ones available in the euclidean case \(H(\xi )=|\xi |\). Here we shall exploit it to deduce the precise asymptotic estimates for the gradient. more precisely we have the following:
Theorem 1.4
Let \(u \in {\mathcal {D}}^{1,p}({\mathbb {R}}^N)\) be a weak solution of (\({\mathcal {P}}_H\)) with \(1<p<N\), and \(0 \le \gamma < C_H\). Then there exist positive constants \({{\tilde{c}}}\), \({{\tilde{C}}}\) depending on \(N,p, \gamma , H\) and u such that
and
where \(\mu _1, \mu _2\) are roots of (1.6) as in Theorem 1.1, and \(0< R_1< 1 < R_2\) are constants depending on \(N, p, \gamma \) and u.
The paper is structured as follows:
-
In Sect. 2 we recall some notions about Finsler type anisotropic geometry, and we prove some technical lemmas that will be crucial in the proof of the main results.
-
In Sect. 3 we prove some preliminary estimates, elliptic estimates and weak comparison principles in bounded and exterior domains that will be essential in the proof of Theorem 1.1.
-
In Sect. 4 we give the proof of decay estimates of solutions to (\({\mathcal {P}}_H\)) near the origin and at infinity, i.e. we prove Theorem 1.1. The, using this result we also prove decay estimates for the gradient of positive weak solutions to (\({\mathcal {P}}_H\)) near the origin and at infinity, i.e. we prove Theorem 1.4.
-
Although the existence of solutions can be easy deduced in the radial-anisotropic setting, in the “Appendix A” we show that problem (\({\mathcal {P}}_H\)) admits at least a positive solution \(u \in D^{1,p}({\mathbb {R}}^N)\) that minimizes the Hardy-Sobolev anisotropic inequality. This result follows using classical arguments (see also [26]) that we decide to add for the readers’ convenience.
2 Preliminaries
Notation. Generic fixed and numerical constants will be denoted by C (with subscript in some case) and they will be allowed to vary within a single line or formula. By |A| we will denote the Lebesgue measure of a measurable set A.
The aim of this section is to recall some properties and geometrical tools about the anisotropic elliptic operator defined above. For \(a, b \in {\mathbb {R}}^N\) we denote by \(a \otimes b\) the matrix whose entries are \((a \otimes b)_{ij}=a_ib_j\). We remark that for any \(v,w \in {\mathbb {R}}^N\) it holds that:
Now, we recall the definition of anisotropic norm.
- \((h_H)\):
-
Let \(H \in C^2({\mathbb {R}}^N \setminus \{0\})\). In all the paper we assume that H is a anisotropic norm if it satisfies the following set of assumptions:
-
(i)
\(H(\xi )>0 \quad \forall \xi \in {\mathbb {R}}^N \setminus \{0\}\);
-
(ii)
\(H(s \xi ) = |s| H(\xi ) \quad \forall \xi \in {\mathbb {R}}^N \setminus \{0\}, \, \forall s \in {\mathbb {R}}\);
-
(iii)
H is uniformly elliptic, that means the set \({\mathcal {B}}_1^H:=\{\xi \in {\mathbb {R}}^N: H(\xi ) < 1\}\) is uniformly convex
$$\begin{aligned} \exists \Lambda > 0: \quad \langle D^2H(\xi )v, v \rangle \ge \Lambda |v|^2 \quad \forall \xi \in \partial {\mathcal {B}}_1^H, \; \forall v \in \nabla H(\xi )^\bot . \end{aligned}$$(2.1)
A set is said uniformly convex if the principal curvatures of its boundary are all strictly positive. Moreover, assumption (iii) is equivalent to assume that \(D^2 (H^2)\) is definite positive.
The dual norm \(H^\circ : {\mathbb {R}}^N \rightarrow [0,+\infty )\) is defined as:
It is possible to show that \(H^\circ \) is also a Finsler norm and it has the same regularity properties of H. Moreover, it holds \((H^\circ )^\circ = H\). For \(R > 0\) and \({{\bar{x}}} \in {\mathbb {R}}^N\) we define:
and
For simplicity of exposition, when \({{\bar{x}}} = 0\), we set: \({\mathcal {B}}_R^H={\mathcal {B}}_R^H(0)\), \({\mathcal {B}}_R^{H^\circ }={\mathcal {B}}_R^{H^\circ }(0)\). In literature \({\mathcal {B}}_R^H\) and \({\mathcal {B}}_R^{H^\circ }\) are also called “Wulff shape” and “Frank diagram” respectively. We remark that there holds the following identities:
and
We refer the reader to [3, 8] for further details. Observe also that H is a norm equivalent to the euclidean one, i.e. there exist \(\alpha _1, \alpha _2 > 0\) such that:
Moreover, recalling that H is 1-homogeneous, by the Euler’s Theorem it follows
Since H is 1-homogeneous, we have that \(\nabla H\) is 0-homogeneous and it satisfies
Hence, by the previous equality, we infer that there exists \(M>0\) such that
For the same reasons there exists a constant \({{\overline{M}}}>0\) such that:
where \(|\cdot |\) denotes the usual Euclidean norm of a matrix, and
We start with some elliptic estimates that can be proved in the same spirit of the Euclidean framework.
Proposition 2.1
For any \(p>1\) and \(\eta , \eta ' \in {\mathbb {R}}^N\) such that \(|\eta |+|\eta '|>0\), it holds
Moreover, any \(p \ge 2\) it holds the following inequality
for any \(\eta , \eta ' \in {\mathbb {R}}^N\). Furthermore, if \(1<p<2\) we have that
for any \(\eta , \eta ' \in {\mathbb {R}}^N\) such that \(|\eta |+|\eta '|>0\).
Proof
We start the proof showing (2.9). First of all we note that (2.9) is symmetric in \(\eta \), \(\eta '\). Hence, without loss of generality, we can assume that \(|\eta '| \ge |\eta | > 0\). We note that for \(j=1, \dots ,N\):
By (2.12), using (2.4), (2.6) and (2.7) we have
where \(|\cdot |\) denotes the standard matrix Euclidean norm.
Now, we observe that
and, since \(|\eta '| \ge |\eta |\), we have either
or, putting \(\displaystyle t_0:= \frac{|\eta '|}{|\eta -\eta '|} \in (0,2)\),
If \(p>2\), using (2.14) in (2.13) we have
where \({{\tilde{C}}}_p=(p-1)\alpha _2^{p-2}M^2+\alpha _2^{p-1}{{\overline{M}}}\). Hence (2.9) holds.
If \(p \le 2\) and (2.15) holds, by (2.13) we obtain
where \({{\tilde{C}}}_p=((p-1)\alpha _2^{p-2}M^2+\alpha _2^{p-1}\overline{M})/4^{p-2}\). Hence (2.9) holds
If \(p \le 2\) and (2.16) holds, by (2.13) we obtain
where \({{\tilde{C}}}_p=(2(p-1)\alpha _2^{p-2}M^2+2\alpha _2^{p-1}{{\overline{M}}})/4^{p-2}\).
Collecting the estimates above, we deduce that inequality (2.9) holds for every \(p>1\) and for \(\tilde{C}_p=((p-1)\alpha _2^{p-2}M^2+\alpha _2^{p-1}{{\overline{M}}}) \cdot \max \{1, 4^{2-p}, 2 \cdot 4^{2-p}\}\).
Now we will show (2.10) and (2.11). For \(\eta , \eta '\in {\mathbb {R}}^N\), we define
and Taylor’s formula yields
provided \(|\eta '+t(\eta -\eta ')|\ne 0\), for \(0\le t\le 1\). But the case when \(\eta '+t(\eta -\eta ')=0\) can be easily verified. By [10, Theorem 1.5] we obtain that
where C(p) is a constant depending on p. We remark that
If \(1<p<2\), by (2.14) we have
and using (2.22) we arrive at (2.11).
If \(p\ge 2\), using a similar argument as in the proof of the inequality (2.9), we obtain
with \({{\overline{C}}}(p)\) constant depending on p. Since \(|\eta -\eta '|\le |\eta |+|\eta '|\) and using (2.4), we get (2.10). \(\square \)
We state now the Hardy inequality for the anisotropic operator \(\Delta _p^Hu\), defined in (1.1). We refer to [28, Proposition 7.5].
Theorem 2.2
(Hardy inequality) For any H satisfying the assumption \((h_H)\) and any \(u\in {\mathcal {D}}^{1,p}({\mathbb {R}}^N)\) and \(1<p<N\),
where \(C_H=(({N-p})/{p})^p\) is optimal.
Now we prove a technical lemma that will be very important in the proof of the asymptotic estimates.
Lemma 2.3
Let \(p>1\) and \(a,b \ge 0\). Then, for all \(\delta >0\) there exist \({\mathcal {C}}_\delta >0\) such that
Proof
Let us consider \(p>1\) as follows:
where \(\lfloor \cdot \rfloor \) is the floor function and \(\{\cdot \}\) is the mantissa function. Without loss of generality we assume that \(\{p\} \ne 0\) and, moreover, we set \(m:= \lfloor p \rfloor \). Hence, we have
Noticing that \(0 \le \{p\} < 1\) it follows that
Using this inequality in (2.26), we deduce
where we used the fact that \(p=m+\{p\}\). Now, we can apply the weighted Young’s inequality to each member of the first sum with conjugate exponents \(\left( p/(p-k),p/k\right) \) and to each member of the second sum with conjugate exponents \(\left( p/(m-k),p/(k+\{p\})\right) \) as follows
Hence, using this estimate we deduce
where we renamed \({\mathcal {C}}_\delta := \left( 1+2^{p+1} {\mathcal {C}}_\delta \right) \). Collecting (2.26), (2.27) and (2.29), we deduce that
with \({\mathcal {C}}_\delta :={\mathcal {C}}_\delta /(1+2^{p+1}\delta )\), and hence the thesis (2.25). \(\square \)
Finally, we recall a lemma (see Lemma 4.19 in [16]) that will be very useful in the proofs of our results.
Lemma 2.4
[16] Let \({\mathcal {L}}\) and g be two nondecreasing functions on the interval \((0, {{\bar{R}}}]\), for some \({{\bar{R}}}>0\). Suppose that it holds
for some \(0< \sigma , \tau < 1\). Then, for any \(\mu \in (0,1)\) and \(R \le {{\bar{R}}}\) we have
where \(\alpha = \alpha (\sigma , \tau , \mu ) =(1-\mu ) \log \sigma / \log \tau \).
3 Preliminary asymptotic estimates and comparison principles
The aim of this section is to prove some preliminary estimates that will be crucial in the proofs of the main results.
Lemma 3.1
There exists a positive constant \(\tau \) depending only on N, p and \(\gamma \) such that for any \({\bar{R}}> 0\) and for any solution u to problem (\({\mathcal {P}}_H\)) satisfying
there exists a positive constant \({\mathcal {C}}\) depending only on \(N, p, \gamma \) and \({{\overline{R}}}\) such that
and that
where \(\sigma _1, \sigma _2\) are two positive constants depending on N, p and \(\gamma \).
Proof
We start proving (3.2). To this aim let us consider \( R>0\) and a cut-off function \(\eta \in C^\infty _c({\mathbb {R}}^N)\) such that
By density argument it is possible to put \(\varphi =\eta ^p u\) as test function in (1.2), so that we obtain
First of all, using Euler’s Theorem (2.5), the 0-homogeneity of \(\nabla H\) (2.6) and Schwarz’s inequality, equation (3.5) becomes
Recalling that H is 1-homogeneous function, using the weighted Young’s inequality \(ab \le \varepsilon a^{\frac{p}{p-1}} + {\mathcal {C}}_\varepsilon b^p\) on the first term of the right hand side of (3.6), for any \(0<\varepsilon <1\) we have
where \( C(p,M,\varepsilon ):=(pM)^p {\mathcal {C}}_\varepsilon \). Now, noticing that \(\nabla (\eta u) = u \nabla \eta + \eta \nabla u\), by the triangular inequality, we deduce that for every \(p>1\) it holds
Thanks to (3.8) and applying Lemma 2.3 with \(a=H(\eta \nabla u)\) and \(b=H(u\nabla \eta )\), we deduce that
Using (3.9) in (3.7) we obtain
Now, applying the anisotropic Hardy inequality (see Theorem 2.2 or [28]) and (2.4) we have
Let us fix \(\varepsilon , \delta >0\) sufficiently small such that \({\mathcal {C}}_1:=({1-\varepsilon })/({1+2^{p+1}\delta }) - {\gamma }/{C_H} > 0\), so that
where \({\mathcal {C}}_2:= C(p,M,\varepsilon ) + \alpha _2^p {\mathcal {C}}_\delta \). By (2.4) we have
Now, using the Sobolev inequality in the left hand side of (3.13), the Hölder inequality and (3.4) in the right hand side, we obtain
hence we deduce
where C(p, N) is a positive constant depending on p and N.
Setting
and choosing \({\overline{R}}>0\) sufficiently small such that (3.1) holds, then \(\Vert u\Vert _{L^{p^*}({\mathcal {B}}^{H^\circ }_R)}^{p^*-p}/(\alpha _1^p{\mathcal {C}}_1 {\mathcal {C}}_S^p) \le 1/2\) for all \(0<R \le {{\overline{R}}}\). Hence we obtain that
where \({{\bar{C}}}:= \left( (2{\mathcal {C}}_2 C(p,N))/({\alpha _1^p{\mathcal {C}}_1 {\mathcal {C}}_S^p})\right) ^{\frac{p^*}{p}}\) and it depends only on N, p and \(\gamma \). Denoting with \(\displaystyle {\mathcal {L}}(R):=\int _{{\mathcal {B}}^{H^\circ }_R}u^{p^*} \, dx\) for \(0 < R \le {{\bar{R}}}\), we get that
where \(\vartheta = {{\bar{C}}}/({{\bar{C}}} + 1)\in (0,1)\), depends only on N, p and \(\gamma \). Now, by Lemma 2.4 it follows that
where \(\sigma _1'=\frac{1}{2}\log (1 / \vartheta )/ \log 2\) depends only on \(\vartheta \), Now (3.2) follows by setting \(\sigma _1 = \sigma _1' / p^*\) and \({\mathcal {C}}=(\vartheta ^{-1}{{\overline{R}}}^{-\sigma _1'}{\mathcal {L}}(\bar{R}))^{1/p^*}\). In a similar way, we can deduce (3.3). \(\square \)
Now, we denote by \({\mathcal {A}}_R={\mathcal {B}}^{H^\circ }_{8R} {\setminus } {\mathcal {B}}^{H^\circ }_{R/8}\) and \({\mathcal {D}}_R={\mathcal {B}}^{H^\circ }_{4R} {\setminus } {\mathcal {B}}^{H^\circ }_{R/4}\) for \(R>0\).
Lemma 3.2
Let \(t \in (p^*, N / \mu _1)\). There exists a positive constant \(\sigma = \sigma (N,p,\gamma ,t)\) such that for any solution u to problem (\({\mathcal {P}}_H\)) and for any \({{\bar{R}}}> 0\) satisfying the following inequality
then
where 
and \({\mathcal {C}}\) is a positive constant depending only on \(N, p, \gamma , {{\overline{R}}}\) and t.
Proof
It is easy to see that, setting \({{\hat{u}}}(x)=u(Rx)\), for \(R>0\),
Let \(m \ge 1\) and set
Hence we can consider the weak formulation of the last equation as follows
Let define \({{\hat{u}}}_m:=\min ({{\hat{u}}}, m)\) for \(m \ge 1\). By density argument, for any \(\eta \in C^\infty _c({\mathcal {A}}_1)\) it is possible to choose \(\varphi =\eta ^p {{\hat{u}}}_m^{p(s-1)}{{\hat{u}}}\), with \(s\ge 1\), as test function in (3.19), so that, using (3.18) and (2.5), we can compute \(I_1\) and \(I_2\)
In the same way, we obtain
Collecting both (3.20), (3.21), using the Scwharz’s inequality and recalling (2.6) we obtain
Now we can apply the weighted Young’s inequality to the first two terms in the right hand side of (3.22) with conjugate exponent \(\left( p/(p-1), p\right) \) in order to obtain
where \({\mathcal {C}}_{\varepsilon _1}(p,s,M)\) and \({\mathcal {C}}_{\varepsilon _2}(p,M)\) are two positive constants. Hence we obtain
Thanks to (3.8) and applying Lemma 2.3, with \(a=H(\eta {{\hat{u}}}^{s-1}\nabla {{\hat{u}}})\) and \(b=H({{\hat{u}}}^s\nabla \eta )\), we deduce that there hold the following inequalities in the sets \({\mathcal {A}}\) and \({\mathcal {B}}\) respectively:
By (3.25) and (3.26), we obtain
Using (2.4), we deduce that
where \(\hat{{\mathcal {C}}}\) depends on \(\delta _1, \delta _2, \varepsilon _1, \varepsilon _2, p, s, M\). Using Hardy’s and Hölder’s inequality in the right hand side of (3.28), and the definition of the sets \({\mathcal {A}}\) and \({\mathcal {B}}\), we obtain
where \(\chi =p^*/p\). Finally, we deduce
Noticing that \((p(s-1)+1)/s^p > \gamma /C_H\) for all \(s \in \left( (N-p)/(p \mu _2), (N-p)(p \mu _1)\right) \), we can fix \(\delta _1, \varepsilon _1>0\) sufficiently small such that
and \(\delta _2, \varepsilon _2>0\) sufficiently small such that
Hence by (3.30) we get
where \(\bar{{\mathcal {C}}}\) is a positive constant depending on \(\varepsilon _1,\varepsilon _2,\delta _1,\delta _2, p,s,\gamma ,C_H\). In conclusion, by (2.4) and Sobolev inequality we have
In order to apply the Moser’s iteration method we need to rewrite the last inequality as follows
where \({\mathcal {C}}_1:=\mathcal {{{\hat{C}}}}/(\alpha _1^p \bar{{\mathcal {C}}} {\mathcal {C}}_S^p)\), \({\mathcal {C}}_2:=1/(\alpha _1^p \bar{{\mathcal {C}}} {\mathcal {C}}_S^p)\) and \(\chi ={p^*}/{p}\).
Fix \(t\in (p^*,N/\mu _1)\) and \(k\in {\mathbb {N}}\) so that \(p\chi ^k\le t\le p\chi ^{k+1}\). Then there exist positive constants \({\mathcal {C}}_1\) and \({\mathcal {C}}_2\) such that (3.33) holds for all \(1\le s\le \min \left\{ (N-p)/(p\mu _1),\chi ^k\right\} \). Now, we set \(\sigma =(1 / (2{\mathcal {C}}_2))^\frac{1}{p^*-p}\) and choosing \({{\overline{R}}}\) sufficiently small such that (3.16) holds, we get
for all \(1\le s\le \min \left\{ (N-p)/(p\mu _1),\chi ^k\right\} \) and \({\mathcal {C}}={\mathcal {C}}_1/2.\) Applying Moser’s iteration method (see e.g. [16, 31] for further details), we conclude, after finitely many times of iteration,
for any \(t\in (p^*,N/\mu _1).\) This proves (3.17). \(\square \)
Let us now prove the following:
Theorem 3.3
Let u be a weak solution of (\({\mathcal {P}}_H\)). Then there exists a positive constant \(C=C(N,p,\gamma ,u)\) such that
and that
where \(\sigma _1, \sigma _2\) are givem in Lemma 3.1 and \( R_1, R_2 >0\) are constants depending on \(N, p, \gamma \) and u.
Proof
Let us fix \(t:= (p^*+ N/\mu _1)/{2} \in \left( p^*, N/\mu _1\right) \) as in Lemma 3.2 and \(\kappa := \min \{\tau , \sigma \}\), where \(\tau \) and \(\sigma \) are respectively as in Lemma 3.1 and Lemma 3.2. Let \({{\bar{R}}} > 0\) such that (3.1) holds for \(\kappa \) and let us consider \({{\hat{u}}}(x)=u(Rx)\), for \(R>0\) fixed. We note that \({{\hat{u}}}\) satisfies the equation
where
We note that \(\displaystyle H^\circ (x)^{-p}\) is bounded in \({\mathcal {D}}_1\) and \(V_R(x):=R^p{{\hat{u}}} ^{p^*-p}(x)\in L^q(\mathcal D_1)\) with \(q=t/(p^*-p)>N/p\) due to Lemma 3.2. Hence, as in the proof of [22, Theorem 1] a classical Moser iteration argument yields
for any ball \({\mathcal {B}}^{H^\circ }_{2r}(x) \subset {\mathcal {D}}_1\), where \({\textbf{C}}={\textbf{C}}(N,p,\gamma ,\Vert V_R\Vert _{L^q(\mathcal D_1)})\). We claim that \(\Vert V_R\Vert _{L^q({\mathcal {D}}_1)}\) is uniformly bounded with respect to R. Indeed from Lemma 3.2, since
we get
where \({\mathcal {C}}\) is a positive constant depending on \(N, p, \gamma , q\) and \({{\overline{R}}}\).
Using a covering argument we deduce that
Noticing that \({{\hat{u}}}(x)= u(Rx)\), by (3.38) we obtain that
for each \(0 < R \le {{\bar{R}}}/8\) or \(R \ge 8/{{\bar{R}}}\). By applying the Hölder’s inequality in (3.39), we get
for each \(0< R \le {{\bar{R}}}/8\) or \(R \ge 8/{{\bar{R}}}\) and \({\textbf{C}}\) depends only on \(N, p, \gamma , q, {{\bar{R}}}\) and \(\Vert u \Vert _{L^{p^*} ({\mathbb {R}}^N)}\).
Now we note that, since \({\mathcal {A}}_R \subset {\mathcal {B}}^{H^\circ }_{{{\bar{R}}}}\) for any \(0<R\le {{\bar{R}}}/8\) and \({\mathcal {A}}_R \subset ({\mathcal {B}}^{H^\circ }_{1/{{\bar{R}}}})^c\) for any \(R \ge 8/{{\bar{R}}}\), there exist, by Lemma 3.1, \(\sigma _1, \sigma _2>0\) depending only on \(N, p, \gamma \) such that
and that
Now, if we set \(R_1= {{\bar{R}}}/8\) and \(R_2=8/{{\bar{R}}}\), by (3.40) we get the thesis. \(\square \)
The next result is devoted to show the existence of some special supersolutions of our problem, in order to perform a comparison between them and the solutions of the doubly critical equation (\({\mathcal {P}}_H\)).
Proposition 3.4
Given two constants \(A>0\) and \(\alpha < p\), there exist constants \(0<\varepsilon ,\delta <1\), depending on \(N,p,\gamma ,A,\alpha \), such that
is a positive supersolution to equation
for some positive constant \(0<R_1<1\) depending only on \(N,p,\gamma , A\) and \(\alpha \), where g(x) is a positive function that belongs to \(L^{\frac{N}{p}}({\mathcal {B}}^{H^\circ }_{R_1})\) such that
In a similar way, given \(A>0\) and \(\alpha > p\), there exist \(0<\varepsilon , \delta <1\) such that
is a positive supersolution to equation
for some positive constant \(R_2>1\) depending only on \(N,p,\gamma \) and \(\alpha \), where g(x) is a positive function that belongs to \(L^{\frac{N}{p}}(({\mathcal {B}}^{H^\circ }_{R_2})^c)\) such that
Proof
Let us consider \(\mu , \delta , \varepsilon >0\) and let us define the function
It is easy to deduce that
where \(s(t):=t^{-\mu -1}[-\mu + \delta (\mu -\varepsilon )t^\varepsilon ]\). Using (2.2), we now compute
where in the last line we used the fact that \(\langle \nabla H^\circ (x), \nabla H(\nabla H^\circ (x)) \rangle =1\) and
Making standard computations on the right hand side of (3.48), one can deduce
where
and
We note that \(h(0)= - |\mu |^{p-2}[\mu ^2(p-1)-\mu (N-p)]-\gamma \). Hence, using the definition of \(\mu _1\) and \(\mu _2\), we deduce that \(h(0)=0\) when \(\mu =\mu _1\) and \(\mu =\mu _2\). Also we have \(h'(0)>0\) if \(\mu =\mu _1\), \(\varepsilon >0\) or \(\mu =\mu _2\), \(\varepsilon <0\). This implies there exist \(0<\delta _h<1\) such that
We set \(\delta =\min \{\delta _h, 1/2\}\), \(\varepsilon =(p-\alpha )/2\) and
It easy to check that \(v(H^\circ (x))=(1-\delta [H^\circ (x)]^\varepsilon )[H^\circ (x)]^{-\mu _1} \in {\mathcal {D}}^{1,p}({\mathcal {B}}^{H^\circ }_{R_1})\) is positive, which thanks to (3.51), \(g(x)\in L^{\frac{N}{p}}({\mathcal {B}}^{H^\circ }_{R_1}) \) and it satisfies (3.43). The other case is similar. \(\square \)
Now, we consider the following equation
where \(\Omega \) is an open subset of \({\mathbb {R}}^N\), \(w>0\) and \(w \in {\mathcal {D}}^{1,p}(\Omega )\). Let us start with a comparison principle in bounded domains.
The first result is given by the following pointwise estimate, in the same spirit of [20, 31].
Proposition 3.5
Let u, v two weakly differentiable strictly positive functions on a domain \(\Omega \).Footnote 1 We have that:
-
(i) if \(p \ge 2\), then
$$\begin{aligned} \begin{aligned}&H^{p-1}(\nabla u) \langle \nabla H (\nabla u), \nabla \left( u - \frac{v^p}{u^p} u \right) \rangle + H^{p-1}(\nabla v) \langle \nabla H (\nabla v), \nabla \left( v - \frac{u^p}{v^p} v \right) \rangle \\&\quad \ge C_{p}(u^p+v^p) H^p(\nabla (\ln u -\ln v)), \end{aligned}\nonumber \\ \end{aligned}$$(3.53)for some positive constant \(C_{p}\) depending only on p;
-
(ii) if \(1<p<2\), then
$$\begin{aligned} \begin{aligned}&H^{p-1}(\nabla u) \langle \nabla H (\nabla u), \nabla \left( u - \frac{v^p}{u^p} u \right) \rangle + H^{p-1}(\nabla v) \langle \nabla H (\nabla v), \nabla \left( v - \frac{u^p}{v^p} v \right) \rangle \\&\quad \ge C_p (u^p+v^p)\left[ H(\nabla \ln u) + H(\nabla \ln v)\right] ^{p-2}H^2(\nabla (\ln u -\ln v)), \end{aligned}\nonumber \\ \end{aligned}$$(3.54)for some positive constant \(C_{p}\) depending only on p.
Proof
Let u, v two weakly differentiable positive functions and consider the following
Then, thanks to the Euler’s theorem for 1-homogeneous functions, and since \(\nabla H\) is 0-homogeneus, we deduce that
and
(i) \(p \ge 2\). We recall that when \(p \ge 2\), it holds (2.10), i.e.
Hence we can apply this inequality, in order to give an estimate from below for (3.55) and (3.56):
Adding both these two inequalities, we obtain
(ii) \(1< p < 2\). We recall that when \(1< p < 2\), it holds (2.11), i.e.
where \(C_p\) is a positive constant depending only on p.
Now we proceed exactly as in the previous case to get an estimate from below for (3.55) and (3.56):
Adding both these two inequalities, we obtain
\(\square \)
Now, we are ready to prove the comparison principles in bounded and exteriors domains.
Proposition 3.6
Let \(\Omega \) be an open bounded smooth domain of \({\mathbb {R}}^N\) and \(f\in L^{\frac{N}{p}}(\Omega )\). Let \(u \in {\mathcal {D}} ^{1,p}(\Omega )\) be a weak positive subsolution to (3.52) and \(v\in {\mathcal {D}}^{1,p}(\Omega )\) be a weak positive supersolution of
with \(g\in L^{\frac{N}{p}}(\Omega )\). Assume that \(\inf _\Omega v >0\) and \(f\le g\) in \(\Omega .\) If \(u \le v\) on \(\partial \Omega \), then
Proof
We will give the proof of this result in the case \(p \ge 2\). The case \(1<p<2\) is similar.
Let us define
where \(m>1\). It is quite standard to show that \(\eta _1\) and \(\eta _2\) are good test function that we can use in the weak formulations of (3.52) and (3.65). Taking both these test function and subtracting the two equations, we obtain
since \(f\le g\) in \(\Omega \). Hence, setting \(\Omega _1:= \{x \in \Omega \ | \ 0 \le u^p-v^p \le m\}\) and \(\Omega _2:= \{x \in \Omega \ | \ u^p-v^p \ge m\}\), we deduce that
Applying (3.53) in (3.67) and making some computations we obtain
but this implies that
For the right hand side of (3.69), we have
where \(\Omega _2':=\{x \in \Omega \ | \ u^p \ge m\}\) and it holds that
Hence, passing to the limit for \(m \rightarrow + \infty \) in (3.69) we obtain that
which clearly implies that
for some positive constant K. By our assumptions \(\inf _{x \in \Omega } v > 0\) and \( u \le v\) on \(\partial \Omega \), hence it follows that \(K=1\). But this implies that
and this complete the proof of this result in the case \(p \ge 2\). The case \(1<p<2\) follows repeating verbatim the proof of the case \(p\ge 2\), but applying inequality (3.54) instead of (3.53). \(\square \)
Now we want to prove the corresponding result of Proposition 3.6 in exterior domains.
Proposition 3.7
Let \(\Omega \) be an exterior domain such that \({\mathbb {R}}^N \setminus \Omega \) is bounded and \(f\in L^{\frac{N}{p}}(\Omega )\). Let \(u \in \mathcal D^{1,p}(\Omega ) \) be a weak positive subsolutions to (3.52) and \(v\in {\mathcal {D}}^{1,p}(\Omega )\) be a positive supersolution of
with \(g\in L^{\frac{N}{p}}(\Omega )\). Assume that \(\inf _\Omega v >0\) and \(f\le g\) in \(\Omega \). If \(u \le v\) on \(\partial \Omega \) and
then
Proof
In the same spirit of Proposition 3.6 we prove our result in the case \(p \ge 2\). The other case is similar and it can be shown using the same arguments. To this aim, let \(\varphi _R \in C^\infty _c({B}_{2R})\) be a standard cut-off function such that
Let us define
where \(m>1\). As pointed out in the proof of previous proposition, it is possible to show, by standard arguments, that \(\eta _1\) and \(\eta _2\) are good test functions for the weak formulations (3.52) and (3.72). Hence, we obtain
since \(f\le g\) in \(\Omega .\) Now we explicitly compute the left hand side of (3.74).
where \(\Omega _1:= \{x \in \Omega \ | \ 0 \le u^p-v^p \le m\}\) and \(\Omega _2:= \{x \in \Omega \ | \ u^p-v^p \ge m\}\). By Proposition 3.5 and using the definition of \(\varphi _R\), it follows that there exits a positive constant depending only on p such that
Now we are going to give estimates for \(I_2, I_3\) and \(I_4\). We start with \(I_2\). Using (2.6) and the Cauchy–Schwarz inequality, setting \({\tilde{\Omega }}_1:=\{x\in \Omega : v^p\le u^p\},\) we have
where in the last line we applied the Hölder inequality with conjugate exponent \(\left( N,p/(p-1),p^*\right) \) and \({\mathcal {C}}:=2M\). Passing to the limit for R that goes to \(+\infty \) in the right hand side of (3.77), using also assumption (3.73) and (2.4), we deduce that \(I_2\) goes to zero.
Now we proceed with the estimate of the term \(I_3\). By (2.5), we have
Therefore
where \({\tilde{\Omega }}_2:= \{x \in \Omega \ | \ u^p \ge m\}\). Using this definition and also the properties of \(\varphi _R\) we deduce that
where \({\mathcal {C}}:=2M\). Passing to the limit for m, R that go to \(+\infty \), we deduce that
For the last term \(I_4\), by (2.5), recalling \({\tilde{\Omega }}_2:= \{x\in \Omega : u^p\ge m\}\), we have
Hence, passing to the limit the right hand side of (3.80), by (3.73) we have that \(I_4\) goes to zero when R tends to \(+\infty \).
Finally, if we combine all the estimates (3.76), (3.77), (3.79), (3.80) and we pass to the limit for \(m, R \rightarrow + \infty \) we deduce that
which implies \(u \le v\) in \(\Omega \) as we concluded in the proof of Proposition 3.6. \(\square \)
4 Proof of the main results
This section is dedicated to the proof of our main results: Theorems 1.1, 1.3 and 1.4.
Proof of Theorem 1.1
We start by proving (1.4). To this aim, let us consider u a solution of
In particular, we have that u is a subsolution of (4.1) in any bounded domain \(\Omega ={\mathcal {B}}_{R_1}^{H^\circ }\). We note that \(f(x):=u^{p^*-p}\in L^{\frac{N}{p}}({\mathcal {B}}_{R_1}^{H^\circ })\) and satisfies \(|f(x)|\le A[H^{\circ }(x)]^{-\alpha }\) with \(\alpha =\left( (N-p)/{p}-\sigma _1\right) (p^*-p)<p\) for \(x\in {\mathcal {B}}_{R_1}^{H^\circ }\) due to Theorem 3.3. By Proposition 3.4 we have that the function
is a positive supersolution of (3.65) in \({\mathcal {B}}^{H^\circ }_{R_1} \subset \Omega \), with \(g\in L^{\frac{N}{p}}({\mathcal {B}}^{H^\circ }_{R_1})\) satisfying \(g(x)\ge A [H^{\circ }(x)]^{-\alpha }\) and where \(0<\delta , \varepsilon , R_1<1\) are positive constants depending only on \(N, p, \gamma , A\) and \(\alpha .\)
Let us consider \(\Gamma >0\), \({\mathcal {M}}= \sup _{\partial {\mathcal {B}}_{R_1}^{H^\circ }} u\), \({\mathcal {N}}= \sup _{\partial {\mathcal {B}}_{R_1}^{H^\circ }} 1/v\) and define
It is easy to chek that w is a positive supersolution of (3.65) in \({\mathcal {B}}^{H^\circ }_{R_1} \subset \Omega \) and \(\inf _{{\mathcal {B}}^{H^\circ }_{R_1}} w = {\mathcal {M}}+\Gamma >0\) and \(u \le w\) on \(\partial {\mathcal {B}}^{H^\circ }_{R_1}\). Hence, by Proposition 3.6 we deduce that
Passing to the limit for \(\Gamma \rightarrow 0\) we obtain that
where \(C={\mathcal {M}} \cdot {\mathcal {N}}\).
Now we have to show the estimate from below. Let u be a weak solution of (\({\mathcal {P}}_H\)), then u is a supersolution of
We set \(c_1:=\inf _{ {\mathcal {B}}_{R_1}^{H^\circ }} u>0\).
Now, we define
where \(c= \inf _{\partial {\mathcal {B}}^{H^\circ }_{R_1}} [H^\circ (x)]^{\mu _1}=R_1^{\mu _1}\). The function \({{\tilde{w}}}\) is a subsolution to (4.2) in \({\mathcal {B}}^{H^\circ }_{R_1}\). Since, it holds that \(u \ge {{\tilde{w}}}\) in \(\partial {\mathcal {B}}^{H^\circ }_{R_1}\), we conclude by using Proposition 3.6 to obtain
and hence combining the estimates from above and below we deduce that (1.4) is proved.
Now, our aim is to prove (1.5). Let us consider u a subsolution of
We note that \(f(x):=u^{p^*-p}\in L^{\frac{N}{p}}(({\mathcal {B}}_{R_2}^{H^\circ })^c)\) and by Theorem 3.3 we have \(|f(x)|\le A[H^{\circ }(x)]^{-\alpha }\) with \(\alpha =\left( (N-p)/{p}+\sigma _2\right) (p^*-p)>p\) for \(x\in ({\mathcal {B}}_{R_2}^{H^\circ })^c\). By Proposition 3.4 the function
is a positive supersolution of (3.72) in \(({\mathcal {B}}^{H^\circ }_{R_2})^c \subset \Omega \), with \(g\in L^{\frac{N}{p}}(({\mathcal {B}}^{H^\circ }_{R_2})^c)\) satisfying \(g(x)\ge A [H^{\circ }(x)]^{-\alpha }\) and where \(0<\delta , \varepsilon <1\) and \(R_2>1\) are positive constants depending only on \(N, p, \gamma , A\) and \(\alpha .\)
Let us consider \(\Gamma >0\), \({\mathcal {M}}= \sup _{(\partial {\mathcal {B}}_{R_2}^{H^\circ })^c} u\), \({\mathcal {N}}= \sup _{(\partial {\mathcal {B}}_{R_2}^{H^\circ })^c} 1/v\) and define
We note that w is a positive supersolution of (3.72) in \(({\mathcal {B}}^{H^\circ }_{R_2})^c \subset \Omega \) and \(\inf _{({\mathcal {B}}^{H^\circ }_{R_2})^c} w = {\mathcal {M}}+\Gamma >0\) and \(u \le w\) on \((\partial {\mathcal {B}}^{H^\circ }_{R_2})^c\). We verify the condition (3.73). Since \(|\nabla \log v(x)|\le C|x|^{-1}\), by Hölder inequality we have
where C is a constant independent of R.
Hence, by Proposition 3.7 we deduce that
Passing to the limit for \(\Gamma \rightarrow 0\) we obtain that
where \(C={\mathcal {M}} \cdot {\mathcal {N}}\)
We conclude with the estimate from below. Let u be a weak solution of (\({\mathcal {P}}_H\)). Then u is a supersolution of
We claim that
for R sufficiently large and constant C independent of R. Indeed, by (4.5), we have
Therefore, considering the test function \(\eta =\zeta ^p u^{1-p}\), with \(\zeta \in C^1_c({\mathbb {R}}^N)\) nonnegative function, and taking in (4.7) we have
By Hölder inequality and by 0-homogeneity of \(\nabla H\) we get
Taking a standard cutoff function \(\zeta \) in (4.9) we get the claim (4.6).
Now we set \(c_2=\inf _{(\partial {\mathcal {B}}^{H^\circ }_{R_2})^c} u>0\) and \(c=\inf _{(\partial {\mathcal {B}}^{H^\circ }_{R_2})^c}[H^{\circ }(x)]^{\mu _2}\). We note that \(v={c_2}c{[H^{\circ }(x)]^{-\mu _2}}\) is a weak solution of (4.5). Moreover the condition (3.73) is verified. Indeed by Hölder inequality and (4.6) we have
since \(\mu _2> (N-p)/{p}.\) Applying the Proposition 3.7 we conclude that
and therefore the thesis. \(\square \)
Now we prove Theorem 1.3 that will be essential to prove the asymptotic behavior of the gradient of solutions to (\({\mathcal {P}}_H\)). For the reader convenience we state a more detailed statement contained in the following:
Theorem 4.1
Let \(v\in C^{1,\alpha }_{loc}({\mathbb {R}}^N\setminus \{0\})\) be a positive weak solution of the equation
where \(0 \le \gamma < C_H\). Assume that there exist two positive constants C and c such that
and suppose that there exists a positive constant \({{\hat{C}}}\) such that
then
with
On the other hand, suppose that there exist two positive constants \({{\tilde{C}}}\) and \({{\tilde{c}}}\) such that
and suppose that there exists a positive constant \({{\overline{C}}}\) such that
then
with
Proof
We prove this result only for \(p\ge 2.\) The other case is similar. Suppose that (4.12) holds. From definition of \({{\overline{c}}}_1\), there exist r (sufficiently small) such that
given \(a_n\rightarrow 0\).
On the other hand, there exist sequences of radii \(R_n\) and points \(x_n\) with \(R_n\) tending to zero and \(H^{\circ }(x_n)=R_n\), such that
Now we set
By (4.12), it follows that \(w_n\) is uniformly bounded in \(L^{\infty }({\mathcal {B}}_{2}^{H^\circ }{\setminus } {\mathcal {B}}_{1/2}^{H^\circ })\) and, since \(w_n\) satisfies the equation (4.11), by [2, 17, 18], it is also uniformly bounded in \(C^{1,\alpha }(K),\) for \(0<\alpha <1\) and for any compact set \(K\subset {\mathcal {B}}_{2}^{H^\circ }{\setminus } {\mathcal {B}}_{1/2}^{H^\circ }\). Moreover, from Ascoli-Arzela’s Theorem, we deduce that \(w_n\rightarrow w_{\infty }\) in the norm \(\Vert \cdot \Vert _{C^{1,\alpha }(K)}\), for any compact set \(K\subset {\mathcal {B}}_{2}^{H^\circ }{\setminus } {\mathcal {B}}_{1/2}^{H^\circ }.\) By (4.18), we have
and by (4.19), there exist a point \({{\overline{x}}}\in \partial {\mathcal {B}}_{1}^{H^\circ }\) such that \(w_{\infty }({{\overline{x}}})={{\overline{c}}}_1\). By the strong comparison principle [11], that holds under our assumption on H (see Sect. 2), we have
Now we set
and we note that it solves (4.11). Fix \(R>0\) sufficiently large and let \(\varphi _R \in C^\infty _c({B}_{2R})\) be a standard cut-off function such that
Let us define
where \(m>1\).
We remark that \(\varphi _1\) and \(\varphi _2\) are good test function in any domain \(0\not \in {\overline{\Omega }}.\) Testing (4.24) in (4.11) on the domain \({\mathbb {R}}^N{\setminus } {\mathcal {B}}_{R_n}^{H^\circ }\), and, since the stress field \(H(\nabla u)^{p-1}\nabla H(\nabla u)\in W^{1,2}_{loc}({\mathbb {R}}^N{\setminus } {\mathcal {B}}_{R_n}^{H^\circ })\) (see [19]), exploiting the divergence theorem, we have
where \(\eta _n\) is the inner unite normal vector at \(\partial {\mathcal {B}}_{R_n}^{H^\circ }.\) Now we set \({\mathcal {A}}:= \{ 0\le v^p-{{\hat{v}}}^p\le m\}\) and \({\mathcal {B}}:=\{ v^p-{{\hat{v}}}^p\ge m\}\). Then (4.25) becomes:
Now we set \(\Omega _1:= ({\mathbb {R}}^N{\setminus } {\mathcal {B}}_{R_n}^{H^\circ })\cap {\mathcal {A}}\) and \(\Omega _2:=({\mathbb {R}}^N{\setminus } {\mathcal {B}}_{R_n}^{H^\circ })\cap \mathcal B\). By Proposition 3.5, it follows that there exits a positive constant depending only on p such that
Now we estimate \(I_3+I_4\). Using (2.6) and the Hölder inequality with conjugate exponent \(\left( N,{p}/({p-1}),p^*\right) \) we have
By (4.22) we have \(H(\nabla \ln {{\hat{v}}}(x))\le C[H^{\circ }(x)]^{-1}\), where \(C=\mu _2\), we have that
where we used the Hölder inequality. Since v has the right summability at the infinity, for R that goes to infinity in the right hand side of (4.28), we obtain that \(I_3+I_4\) goes to zero.
Now we proceed with the estimate of the term \(I_5\). Recalling that \(v^p \ge m\) in \(\Omega _2\), we get
Using the properties of \(\varphi _R\) we obtain that
Passing to the limit for m, R that go to \(+\infty \), we deduce that \(I_5\) goes to zero since the set \( \Omega _2\) vanishes as \(m \rightarrow +\infty \).
For the last term \(I_6\) we obtain
Hence, passing to the limit in the right hand side of (4.31) and using (4.29) we have that the right hand of (4.31) tends to zero when R tends to the infinity.
Now we estimate the left hand of (4.26). By (4.24) and Proposition 2.1 (in particular see (2.11)), we get
where, in the last line, we used (2.4) and (2.6).
We set \(x=R_ny\), with \(y\in \partial {\mathcal {B}}_{1}^{H^\circ }\). Using this change of variables and recalling (4.12), (4.20) and (4.21), \(J_1\) can be estimated as
where we choose \(m=R_n^{-\epsilon }\), for \(\epsilon >0\) fixed sufficiently small, and \({{\tilde{C}}}_1\) is a positive constant.
In a similar way \(J_2\) is estimated as
where \({{\tilde{C}}}_2\) is a positive constant.
Finally, if we combine all the estimates (4.27), (4.28), (4.30), (4.31), (4.33), (4.34) and passing to the limit for \(R_n\rightarrow 0\), and then, exploiting the Fatou’s lemma, for \(R\rightarrow +\infty \), we deduce that
which implies \(v \le {{\hat{v}}}\) as we concluded in the proof of Proposition 3.6.
To prove that \({{\hat{v}}}\le v\), let us consider
where \(m>1\) and \(\varphi _R\) is the standard cutoff function defined in (4.23). Using (4.35) in the weak formulation of (4.11), proceeding in a similar way as above we obtain (4.14). The only thing to check is that
Assumption (4.13) ensures (4.36), and therefore we are done.
Now, assume that (4.15) holds and suppose that \(p\ge 2\), the other case is similar. The proof is similar to the previous one and, for this reason, we omit some details. From definition of \(\overline{c}_2\), there exist r (sufficiently large) such that
given \(a_n\rightarrow 0\).
On the other hand, there exist sequences of radii \(R_n\) and points \(x_n\) with \(R_n\) tending to infinity and \(H^{\circ }(x_n)=R_n\), such that
Now we set
As in the previous case we obtain
Now we set
and we remark that it solves (4.11). We show that \(v={{\hat{v}}}\) in \({\mathbb {R}}^N\). To this aim, fix \(\varepsilon >0\) sufficiently small and let \(\varphi _\varepsilon \in C^\infty ({\mathbb {R}}^N)\) be a function such that
Let us define
where \(m>1\).
We note that \(\varphi _1\) and \(\varphi _2\) are good test function in any bounded domain \(0\in \Omega .\) Testing (4.42) in (4.11) on the domain \({\mathcal {B}}_{R_n}^{H^\circ }\) we get
where \(\eta _n\) is the outward unite normal vector at \(\partial {\mathcal {B}}_{R_n}^{H^{\circ }}.\) By (4.42) and Proposition 2.1, the right hand of (4.43) becomes
where, in the last line, we used (2.4) and (2.6).
We consider the following change of variables \(x=y/R_n\), with \(y\in \partial {\mathcal {B}}_{1}^{H^\circ }\). Using this fact, by (4.39) and (4.40), we have
where \( {{\overline{C}}}_1\) is a positive constant.
In a similar way \({{\tilde{J}}}_2\) is estimated as
where \({{\overline{C}}}_2\) is a positive constant. We note that \(-N+\mu _1p+p<0\) since \(0\le \mu _1<(N-p)/p\). Proceeding as in the case (4.12), we prove that
which implies \(v \le {{\hat{v}}}\) in \(\Omega \) as we concluded in the proof of Proposition 3.6.
To prove that \({{\hat{v}}}\le v\), let us consider
where \(m>1\) and \(\varphi _\varepsilon \) is defined as (4.41). Testing (4.48) in (4.11), using the assumption (4.16) and proceeding in a similar way as above we get (4.17). \(\square \)
At this point we are ready to prove
Proof of Theorem 1.4
We prove (1.12), the other case is similar and it can be proved in a similar way. We start by showing the estimate from above. For \(R_n\) tending to infinity, let us consider
We remark that \(w_n\) is uniformly bounded in \(L^{\infty }({\mathcal {B}}_{2}^{H^\circ }{\setminus } {\mathcal {B}}_{1/2}^{H^\circ })\) and it weakly solves
Since \(\mu _2(p-1)+p-\mu _2(p^*-1)<0\), by [2, 17, 18], \(w_n\) is also uniformly bounded in \(C^{1,\alpha }(K),\) for \(0<\alpha <1\) and for any compact set \(K\subset {\mathcal {B}}_{2}^{H^\circ }{\setminus } {\mathcal {B}}_{1/2}^{H^\circ }\). For \(R_n\) sufficiently large we get the estimate from above in (1.12).
Now we prove the estimate from below. Suppose by contradiction that there exist sequences of points \(x_n\) such that
For \(0<a<A\) fixed, let us consider
For n sufficiently large, from Theorem 1.1, relabeling the constants, we have
Furthermore, recalling the estimate from above of the gradients of the weak solution u, proved previously, we get
For a, A fixed, by [2, 17, 18], \(w_n\) is also uniformly bounded in \(C^{1,\alpha }(K),\) for \(0<\alpha <1\) and for any compact set \(K\subset {\mathcal {B}}_{A}^{H^\circ }{\setminus } {\mathcal {B}}_{a}^{H^\circ }\). Moreover
in the norm \(C^{1,\alpha '}\), for \(0<\alpha ' <\alpha .\) Moreover, since \(w_n\) weakly solves
we deduce that
Now we take \(a_j=1/j\) and \(A_j=j\), for \(j\in {\mathbb {N}}\) and we construct \(w_{a_j,A_j}\) as above. For j goes to infinity, using a standard diagonal process, we construct a limiting profile \(w_{\infty }\) so that
with \(w_{\infty }\equiv w_{a_j,A_j}\) in \({\mathcal {B}}_{A_j}^{H^\circ }\setminus {\mathcal {B}}_{a_j}^{H^\circ }\).
Since \(w_{\infty }\) satisfies the assumptions (4.12), (4.13) of the Theorem 4.1, we get
where we set \(\overline{c_2}:=\limsup _{|x|\rightarrow 0} [H^{\circ }(x)]^{\mu _2}w_{\infty }(x)\).
Now we set \(y_n=x_n/R_n\), and by (4.51), we deduce that \(|\nabla w_n(y_n)|\) tends to zero as \(R_n\) tends to infinity. This fact and the uniform convergence of the gradients imply that there exist \({{\overline{y}}}\in \partial {\mathcal {B}}_{1}^{H^\circ }\) such that
This is an absurd since the solution \(w_{\infty }\) has no critical points. \(\square \)
Data Availability
All data generated or analyzed during this study are included in this published article.
Notes
We mean that \(u,v \ge C > 0\) a.e. in \(\Omega \).
References
Alexandrov, A.D.: A characteristic property of the spheres. Ann. Mat. Pura Appl. (4) 58, 303–315 (1962)
Antonini, C.A., Ciraolo, G., Farina, A.: Interior regularity results for inhomogeneous anisotropic quasilinear equations. Math. Ann. (2022). https://doi.org/10.1007/s00208-022-02500-x
Bellettini, G., Paolini, M.: Anisotropic motion by mean curvature in the context of Finsler geometry. Hokkaido Math. J. 25(3), 537–566 (1996)
Brezis, H., Lieb, E.: A relation between pointwise convergence of functions and convergence of functionals. Proc. Am. Math. Soc. 88(3), 486–490 (1983)
Caffarelli, L.A., Gidas, B., Spruck, J.: Asymptotic symmetry and local behaviour of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42(3), 271–297 (1989)
Castorina, D., Riey, G., Sciunzi, B.: Hopf Lemma and regularity results for quasilinear anisotropic elliptic equations. Calc. Var. Partial Differ. Equ. 58(3), 95, 18 (2019)
Catino, G., Monticelli, D.D., Roncoroni, A.: On the critical p-Laplace equation. Preprint arXiv:2204.06940
Cianchi, A., Salani, P.: Overdetermined anisotropic elliptic problems. Math. Ann. 345(4), 859–881 (2009)
Ciraolo, G., Figalli, A., Roncoroni, A.: Symmetry results for critical anisotropic p-Laplacian equations in convex cones. Geom. Funct. Anal. 30(3), 770–803 (2020)
Cozzi, M., Farina, A., Valdinoci, E.: Monotonicity formulae and classification results for singular, degenerate, anisotropic PDEs. Adv. Math. 293, 343–381 (2016)
Damascelli, L.: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. H. Poincaré Anal. Non Linéaire 15(4), 493–516 (1998)
Damascelli, L., Merchán, S., Montoro, L., Sciunzi, B.: Radial symmetry and applications for a problem involving the operator \(-\Delta _p ( \cdot )\) and critical nonlinearity in \({\mathbb{R} }^N\). Adv. Math. 265, 313–335 (2014)
Di Benedetto, E.: \(C^{1+\alpha }\) local regularity of weak solutions of degenerate elliptic equations. Nonlinear Anal. 7(8), 827–850 (1983)
Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry of positive solutions of nonlinear elliptic equations in \({{\mathbb{R} }}^N\). Mathematical analysis and applications, Part A, pp. 369–402, Adv. in Math. Suppl. Stud., 7a. Academic Press, New York (1981)
Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34(4), 525–598 (1981)
Han, Q., Lin, F.: Elliptic partial differential equations. Courant Lect. Notes Math., 1 New York University, Courant Institute of Mathematical Sciences, New York American Mathematical Society, Providence, RI, x+144 pp (1997)
Lieberman, G.M.: Boundary regularity for solutions of degenerate elliptic equations. Nonlinear Anal. 12(11), 1203–1219 (1988)
Lieberman, G.M.: The natural generalization of the natural conditions of Ladyzhenskaya and Ural’tseva for elliptic equations. Commun. Partial Differ. Equ. 16(2–3), 311–361 (1991)
Montoro, L., Sciunzi, B.: Pohozaev identity for Finsler anisotropic problems. NoDEA Nonlinear Differ. Equ. Appl. 30(3), 33, 15 (2023)
Oliva, F., Sciunzi, B., Vaira, G.: Radial symmetry for a quasilinear elliptic equation with a critical Sobolev growth and Hardy potential. J. Math. Pures Appl. (9) 140, 89–109 (2020)
Ou, Q.: On the classification of entire solutions to the critical \(p\)-Laplace equation. Preprint arXiv:2210.05141
Serrin, J.: Local behaviour of solutions of quasilinear equations. Acta Math. 111, 247–302 (1964)
Serrin, J.: A symmetry problem in potential theory. Arch. Ration. Mech. Anal. 43, 304–318 (1971)
Sciunzi, B.: Classification of positive \(D^{1, p}({{\mathbb{R} }}^N)\)-solutions to the critical \(p\)-Laplace equation in \({{\mathbb{R} }}^N\). Adv. Math. 291, 12–23 (2016)
Struwe, M.: Variational methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Fourth edition, Ergeb. Math. Grenzgeb. (3), 34 [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics] Springer, Berlin, xx+302 pp (2008)
Terracini, S.: On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv. Differ. Equ. 1(2), 241–264 (1996)
Tolksdorf, P.: Regularity for a more general class of quasilinear elliptic equations. J. Differ. Equ. 51(1), 126–150 (1984)
Van Schaftingen, J.: Anisotropic symmetrization. Ann. Inst. H. Poincaré Anal. Non Linéaire 23(4), 539–565 (2006)
Vétois, J.: A priori estimates and application to the symmetry of solutions for critical \(p\)-Laplace equations. J. Differ. Equ. 260(1), 149–161 (2016)
Vétois, J.: A note on the classification of positive solutions to the critical \(p\)-Laplace equation in \({{\mathbb{R} }}^n\). Preprint arXiv:2304.02600
Xiang, C.L.: Asymptotic behaviors of solutions to quasilinear elliptic equations with critical Sobolev growth and Hardy potential. J. Differ. Equ. 259(8), 3929–3954 (2015)
Xiang, C.L.: Gradient estimates for solutions to quasilinear elliptic equations with critical Sobolev growth and Hardy potential. Acta Math. Sci. Ser. B 37(1), 58–68 (2017)
Funding
Open access funding provided by Università della Calabria within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by M. del Pino.
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The authors are members of INdAM. The authors are partially supported by PRIN project 2017JPCAPN (Italy): Qualitative and quantitative aspects of nonlinear PDEs. L. Montoro is partially supported by Agencia Estatal de Investigación (Spain): project PDI2019-110712GB-100.
Appendix A
Appendix A
In this section we prove the existence of a weak solution to problem (\({\mathcal {P}}_H\)) by means of a minimization problem. For this reason we define the following minimization problem: Let \(S(\gamma )\) defined as
where
Thanks to the Hardy inequality (2.24) we deduce that \(S(\gamma )\ge 0\).
Theorem 5.1
Let \(0\le \gamma < C_H\). The problem (\({\mathcal {P}}_H\)) has a positive weak solution that it minimizes the quotient (A.1).
Proof
Let \(\{u_n\}\) be a minimizing sequence to (A.1). Without loss of generality, because of the homogeneity of the quotient in (A.1), we assume that
By Hardy inequality (observe that H is a norm equivalent to the euclidean one), \(\mathcal L(\cdot )^{1/p}\) is an equivalent norm to the standard one \(\Vert \cdot \Vert _{{\mathcal {D}}^{1,p}({\mathbb {R}}^N)}\), we have that
with C that does not depend on n. Hence up to a subsequence \(u_n \rightharpoonup u_0\) in \({\mathcal {D}}^{1,p}({\mathbb {R}}^N)\). Moreover let us also assume that (we will prove it later)
Recalling (A.1), using the weak lower semicontinuity of the norm and (A.3), we deduce
Using Sobolev inequality we have that \(u_n\rightarrow u_0\) a.e. in \({\mathbb {R}}^N\). Therefore by the Breis-Lieb result [4], it follows that
Then from (A.5) we obtain
From (A.1), we deduce
and
Using these last inequalities in (A.6) we get
where in the last line we used (A.3). Taking the limit superior of (A.7) we get a contradicition, i.e. \(S(\gamma )<S(\gamma )\) unless the sequence \(\mathcal L(u_0-u_n)\rightarrow 0\). Hence, since \({\mathcal {L}}(\cdot )^{1/p}\) is an equivalent norm to \(\Vert \cdot \Vert _{{\mathcal {D}}^{1,p}({\mathbb {R}}^N)}\), we finally get that \(u_n\rightarrow u_0\) in \({\mathcal {D}}^{1,p}({\mathbb {R}}^N)\). Therefore passing to the limit in (A.1), we obtain
namely \(u_0\) (eventually redefining it as \(Cu_0\), for some positive constant) is a weak solution to (\({\mathcal {P}}_H\)).
Now we prove that actually (A.4) holds, concluding indeed the proof. Let \(\{u_n\}\) the minimizing sequence such that (A.3) holds. For every n let us take a sequence of radii \(R_n\) such that
and let us define the rescaled sequence
Using assumptions \((h_H)\), (A.8) and (A.9) we deduce that
Now we prove the following
Lemma 5.2
Let \(\{u_n\}\) be a minimizing sequence, weakly converging to zero. Then, for every ball \(B_r\) and for every \(\varepsilon \in (-r,r)\) there exists \(\rho \in (0,\varepsilon )\cup (\varepsilon , 0)\) such that for a subsequence
Proof
By the homogeneity of the quotient in (A.1) we can assume that the minimizing sequence \(\{u_n\}\) is such that \(\Vert u_n\Vert _{L^{p^*}({\mathbb {R}}^N)}=1\), so that \(\mathcal L(u_n)\rightarrow S(\gamma )\). By Ekeland’s \(\varepsilon -\)principle, we can suppose that the minimizing sequence has the Palais-Smale property, that is
for all \(\varphi \in {\mathcal {D}}^{1,p}({\mathbb {R}}^N)\) We have that
is bounded. Then we can find \(\rho \in (0,\varepsilon )\) such that for infinitely many n’s it holds
for some positive constant C and hence up to redefining the constant
Therefore (see [25, Theorem A.8]) since
with both embedding compact, we can assume that a subsequence converges strongly to some limit, say u in the trace space \(W^{1-\frac{1}{p},p}((r+\rho ) S^{N-1})\). Using the fact that the trace operator has a continuous embedding from \(W^{1,p}(B_{r+\rho })\) into \(W^{1-\frac{1}{p},p}((r+\rho ) S^{N-1})\), by the weak convergence to zero of \(\{u_n\}\), we deduce that indeed \(u\equiv 0\).
Now we show the following
Claim: Let \(\Omega \subset {\mathbb {R}}^N\) a generic smooth bounded domain. The inverse operator
is continuous. Indeed we consider a succession \(g_n\rightarrow g\) in \(W^{1-\frac{1}{p},p}(\Omega )\), and let \(u_n,u\in W^{1,p}(\Omega )\) be the solutions to
The solution to (A.14) can be obtained minimizing the functional
on the set \(\left\{ \{g\}+ W^{1,p}_0(\Omega )\right\} \), \(\left\{ \{g_n\}+ W^{1,p}_0(\Omega )\right\} \) respectively. Since \((u-g),(u_n-g_n)\in W^{1,p}_0(\Omega )\), integrating by parts (A.14) and subtracting the equations, we obtain
We recall (see [6, Lemma 4.1]) that for \(x\in {\mathbb {R}}^N{\setminus } \{0\}\), \(y\in {\mathbb {R}}^N\) there exist a constant \(C>0\) such that
Therefore, by (A.15) and (2.9) we get
where in the last inequality we have used the Hölder inequality.
We recall that by \(W^{1-\frac{1}{p},p}(\partial \Omega )\) we denote the space of traces \(u_{|\partial \Omega }\), namely the set (of equivalence classes) \(\left\{ \{u\}+ W^{1,p}_0(\Omega ), u\in W^{1,p}(\Omega )\right\} \), endowed with the trace norm
Hence using (A.16) letting the boundary data \(g_n\rightarrow g\) in the sense of (A.17) we obtain the claim.
Let us define the two auxiliary sequences of \({\mathcal {D}}^{1,p}({\mathbb {R}}^N)\) as follows:
and
where \(w_{1,n}\) respectively \(w_{2,n}\) denote the solutions to
respectively
Since \(u_n \rightarrow 0\) on \(\partial B_{r+\rho }\) in the \(W^{1-\frac{1}{p},p}\) norm, see (A.13), by the above claim we immediately get that both
Using \(u_{1,n}\) as test function in (A.12) we obtain
and recalling the definition of \(u_{1,n}\) and by (A.21), we obtain
In the same way, using \(u_{2,n}\) as test function in (A.12) we obtain
Moreover, by definition (A.2), using the two sequences \(\{u_{1,n}\}\) and \(\{u_{2,n}\}\) we infer that actually
and
so that
and \(\Vert u_n\Vert _{p^*}^{p^*}=\Vert u_{1,n}\Vert _{p^*}^{p^*}+\Vert u_{2,n}\Vert _{p^*}^{p^*}+ o(1).\) Let us assume for example, that \(\{u_{1,n}\}\) does not converges to zero. Since \(\{u_n\}\) is a minimizing sequence we have that (see (A.1))
and also that \({\mathcal {L}}(u_{2,n})\ge S(\gamma )\Vert u_{2,n}\Vert ^p_{p^*}\) and
By (A.22) we deduce that
by some computations we deduce that actually
a contradiction with (A.1) unless \({\mathcal {L}}(u_{2,n})\) tends to zero. Using Hardy inequality in (A.2), recalling (A.18), we obtain
The other case of (A.11) can be proved arguing in the same as we have done above, assuming that \(\{u_{2,n}\}\) does not converge to zero. This concludes the proof of the lemma. \(\square \)
Using the invariance of the problem under the scaling \(R_n^{{(p-N)}/{p}}u({x}/{R_n})\), the sequence \(\{w_n\}\) (see (A.9) and (A.10)) is still a minimizing sequence bounded in \({\mathcal {D}}^{1,p}({\mathbb {R}}^N)\): hence it admits (up to a subsequence) a weakly convergence sequence. We want to show that the weak limit cannot be zero. We argue by contradiction and we apply Lemma 5.2 twice choosing \(r=1\) and \(\varepsilon =\pm 1/4\) respectively. We find the existence of \(\rho ^+\in (0,1/4)\) and \(\rho ^-\in (-1/4, 0)\) such that (A.11) holds. Using the alternative (A.11) together with (A.10), we obtain that
Since \(w_n\rightharpoonup 0\) using the strong convergence on compacts K of \(w_n \rightarrow 0\) in \(L^p(K)\) we deduce that
Let us take a smooth cut-off function \(\eta \), with \(0\le \eta \le 1\), such that \(\eta \equiv 1\) in \(B(0)_{ 5/4}{\setminus } B(0)_{3/4}\) and \(\eta \equiv 0\) in \({\mathbb {R}}^N{\setminus } (B(0)_{ 3/2}{\setminus } B(0)_{1/2})\). Take in to account Hardy inequality, (A.23) and (A.24) we deduce that
for \(n\rightarrow +\infty \) and therefore we infer that
Then, recalling also (A.1) we get
Passing to the limit we obtain
We claim that (A.25) is not possible. Indeed let \(u_0\) be a minimizer of (A.1) for \(\gamma =0\) ( [9]). Therefore we clearly deduce the following
Inequality (A.26) gives the desired contradiction, concluding the proof. \(\square \)
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Esposito, F., Montoro, L., Sciunzi, B. et al. Asymptotic behaviour of solutions to the anisotropic doubly critical equation. Calc. Var. 63, 77 (2024). https://doi.org/10.1007/s00526-024-02682-z
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00526-024-02682-z