-
Decay estimates for defocusing energy-critical Hartree equation Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-05-02 Miao Chen, Hua Wang, Xiaohua Yao
In this paper, we are devoted to establishing the point-wise decay estimates for solution to the 5D defocusing energy-critical Hartree equation with an initial data in H 2 ( R 5 ) ∩ L 1 ( R 5 ) ${H}^{2}\left({\mathbb{R}}^{5}\right)\cap {L}^{1}\left({\mathbb{R}}^{5}\right)$ . We show that the nonlinear solution has the same time decay rate as the linear one. The main new ingredient is that we used the
-
Existence of ground states to quasi-linear Schrödinger equations with critical exponential growth involving different potentials Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-04-30 Caifeng Zhang, Maochun Zhu
The purpose of this paper is three-fold. First, we establish singular Trudinger–Moser inequalities with less restrictive constraint: (0.1) sup u ∈ H 1 ( R 2 ) , ∫ R 2 ( | ∇ u | 2 + V ( x ) u 2 ) d x ≤ 1 ∫ R 2 e 4 π 1 − β 2 u 2 − 1 | x | β d x < + ∞ , $$\underset{u\in {H}^{1}({\mathbb{R}}^{2}),\underset{{\mathbb{R}}^{2}}{\int }(\vert \nabla u{\vert }^{2}+V(x){u}^{2})\mathrm{d}x\le 1}{\mathrm{sup}}\
-
A degenerate migration-consumption model in domains of arbitrary dimension Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-04-29 Michael Winkler
In a smoothly bounded convex domain Ω ⊂ R n ${\Omega}\subset {\mathbb{R}}^{n}$ with n ≥ 1, a no-flux initial-boundary value problem for u t = Δ u ϕ ( v ) , v t = Δ v − u v , $$\begin{cases}_{t}={\Delta}\left(u\phi \left(v\right)\right),\quad \hfill \\ {v}_{t}={\Delta}v-uv,\quad \hfill \end{cases}$$ is considered under the assumption that near the origin, the function ϕ suitably generalizes the prototype
-
New multiplicity results in prescribing Q-curvature on standard spheres Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-04-20 Mohamed Ben Ayed, Khalil El Mehdi
In this paper, we study the problem of prescribing Q-Curvature on higher dimensional standard spheres. The problem consists in finding the right assumptions on a function K so that it is the Q-Curvature of a metric conformal to the standard one on the sphere. Using some pinching condition, we track the change in topology that occurs when crossing a critical level (or a virtually critical level if it
-
Increase of power leads to a bilateral solution to a strongly nonlinear elliptic coupled system Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-04-20 Francisco Ortegón Gallego, Mohamed Rhoudaf, Hajar Talbi
In this paper, we analyze the following nonlinear elliptic problem A ( u ) = ρ ( u ) | ∇ φ | 2 in Ω , div ( ρ ( u ) ∇ φ ) = 0 in Ω , u = 0 on ∂ Ω , φ = φ 0 on ∂ Ω . $\begin{cases}A\left(u\right)=\rho \left(u\right)\vert \nabla \varphi {\vert }^{2}\,\text{in}\,{\Omega},\quad \hfill \\ \text{div}\left(\rho \left(u\right)\nabla \varphi \right)=0\,\text{in}\,{\Omega},\quad \hfill \\ u=0\,\text{on}\,\partial
-
Curvature conditions, Liouville-type theorems and Harnack inequalities for a nonlinear parabolic equation on smooth metric measure spaces Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-04-12 Ali Taheri, Vahideh Vahidifar
In this paper we prove gradient estimates of both elliptic and parabolic types, specifically, of Souplet-Zhang, Hamilton and Li-Yau types for positive smooth solutions to a class of nonlinear parabolic equations involving the Witten or drifting Laplacian on smooth metric measure spaces. These estimates are established under various curvature conditions and lower bounds on the generalised Bakry-Émery
-
Multiple concentrating solutions for a fractional (p, q)-Choquard equation Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-04-04 Vincenzo Ambrosio
We focus on the following fractional (p, q)-Choquard problem: ( − Δ ) p s u + ( − Δ ) q s u + V ( ε x ) ( | u | p − 2 u + | u | q − 2 u ) = 1 | x | μ * F ( u ) f ( u ) in R N , u ∈ W s , p ( R N ) ∩ W s , q ( R N ) , u > 0 in R N , $\begin{cases}{\left(-{\Delta}\right)}_{p}^{s}u+{\left(-{\Delta}\right)}_{q}^{s}u+V\left(\varepsilon x\right)\left(\vert u{\vert }^{p-2}u+\vert u{\vert }^{q-2}u\right)=
-
On the large solutions to a class of k-Hessian problems Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-04-02 Haitao Wan
In this paper, we consider the k-Hessian problem S k (D 2 u) = b(x)f(u) in Ω, u = +∞ on ∂Ω, where Ω is a C ∞-smooth bounded strictly (k − 1)-convex domain in R N ${\mathbb{R}}^{N}$ with N ≥ 2, b ∈ C∞(Ω) is positive in Ω and may be singular or vanish on ∂Ω, f ∈ C[0, ∞) ∩ C 1(0, ∞) (or f ∈ C 1 ( R ) $f\in {C}^{1}\left(\mathbb{R}\right)$ ) is a positive and increasing function. We establish the first
-
Michael-Simon type inequalities in hyperbolic space H n + 1 ${\mathbb{H}}^{n+1}$ via Brendle-Guan-Li’s flows Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-04-01 Jingshi Cui, Peibiao Zhao
In the present paper, we first establish and verify a new sharp hyperbolic version of the Michael-Simon inequality for mean curvatures in hyperbolic space H n + 1 ${\mathbb{H}}^{n+1}$ based on the locally constrained inverse curvature flow introduced by Brendle, Guan and Li (“An inverse curvature type hypersurface flow in H n + 1 ${\mathbb{H}}^{n+1}$ ,” (Preprint)) as follows (0.1) ∫ M λ ′ f 2 E 1
-
A note on the classification of positive solutions to the critical p-Laplace equation in R n ${\mathbb{R}}^{n}$ Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-29 Jérôme Vétois
In this note, we obtain a classification result for positive solutions to the critical p-Laplace equation in R n ${\mathbb{R}}^{n}$ with n ≥ 4 and p > p n for some number p n ∈ n 3 , n + 1 3 ${p}_{n}\in \left(\frac{n}{3},\frac{n+1}{3}\right)$ such that p n ∼ n 3 + 1 n ${p}_{n}\sim \frac{n}{3}+\frac{1}{n}$ , which improves upon a similar result obtained by Ou (“On the classification of entire solutions
-
Global structure of positive and sign-changing periodic solutions for the equations with Minkowski-curvature operator Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-29 Ruyun Ma, Zhongzi Zhao, Xiaoxiao Su
We show the existence of unbounded connected components of 2π-periodic positive solutions for the equations with one-dimensional Minkowski-curvature operator − u ′ 1 − u ′ 2 ′ = λ a ( x ) f ( u , u ′ ) , x ∈ R , $-{\left(\frac{{u}^{\prime }}{\sqrt{1-{u}^{\prime 2}}}\right)}^{\prime }=\lambda a\left(x\right)f\left(u,{u}^{\prime }\right), x\in \mathbb{R},$ where λ > 0 is a parameter, a ∈ C ( R , R )
-
Moving planes and sliding methods for fractional elliptic and parabolic equations Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-19 Wenxiong Chen, Yeyao Hu, Lingwei Ma
In this paper, we summarize some of the recent developments in the area of fractional elliptic and parabolic equations with focus on how to apply the sliding method and the method of moving planes to obtain qualitative properties of solutions. We will compare the two methods and point out the pros and cons of each. We will demonstrate how to modify the ideas and techniques in studying fractional elliptic
-
On constant higher order mean curvature hypersurfaces in H n × R ${\mathbb{H}}^{n}{\times}\mathbb{R}$ Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-15 Barbara Nelli, Giuseppe Pipoli, Giovanni Russo
We classify hypersurfaces with rotational symmetry and positive constant r-th mean curvature in H n × R ${\mathbb{H}}^{n}{\times}\mathbb{R}$ . Specific constant higher order mean curvature hypersurfaces invariant under hyperbolic translation are also treated. Some of these invariant hypersurfaces are employed as barriers to prove a Ros–Rosenberg type theorem in H n × R ${\mathbb{H}}^{n}{\times}\mathbb{R}$
-
Infinitely many free or prescribed mass solutions for fractional Hartree equations and Pohozaev identities Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-14 Silvia Cingolani, Marco Gallo, Kazunaga Tanaka
In this paper we study the following nonlinear fractional Hartree (or Choquard-Pekar) equation ( − Δ ) s u + μ u = ( I α * F ( u ) ) F ′ ( u ) in R N , ${\left(-{\Delta}\right)}^{s}u+\mu u=\left({I}_{\alpha }{\ast}F\left(u\right)\right){F}^{\prime }\left(u\right)\quad \text{in} {\mathbb{R}}^{N},$ (*) where μ > 0, s ∈ (0, 1), N ≥ 2, α ∈ (0, N), I α ∼ 1 | x | N − α ${I}_{\alpha }\sim \frac{1}{\vert x{\vert
-
An upper bound for the least energy of a sign-changing solution to a zero mass problem Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-14 Mónica Clapp, Liliane Maia, Benedetta Pellacci
We give an upper bound for the least possible energy of a sign-changing solution to the nonlinear scalar field equation − Δ u = f ( u ) , u ∈ D 1,2 ( R N ) , $-{\Delta}u=f\left(u\right), u\in {D}^{1,2}\left({\mathrm{R}}^{N}\right),$ where N ≥ 5 and the nonlinearity f is subcritical at infinity and supercritical near the origin. More precisely, we establish the existence of a nonradial sign-changing
-
Sharp affine weighted L 2 Sobolev inequalities on the upper half space Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-14 Jingbo Dou, Yunyun Hu, Caihui Yue
We establish some sharp affine weighted L 2 Sobolev inequalities on the upper half space, which involves a divergent operator with degeneracy on the boundary. Moreover, for some certain exponents cases, we also characterize the extremal functions and best constants. Our approach only relies on the L 2 structure of gradient norm, affine invariance and a class of weighted L 2 Sobolev inequality on the
-
The existence and multiplicity of L 2-normalized solutions to nonlinear Schrödinger equations with variable coefficients Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-14 Norihisa Ikoma, Mizuki Yamanobe
The existence of L 2–normalized solutions is studied for the equation − Δ u + μ u = f ( x , u ) in R N , ∫ R N u 2 d x = m . $-{\Delta}u+\mu u=f\left(x,u\right)\quad \quad \text{in} {\mathbf{R}}^{N},\quad {\int }_{{\mathbf{R}}^{N}}{u}^{2} \mathrm{d}x=m.$ Here m > 0 and f(x, s) are given, f(x, s) has the L 2-subcritical growth and (μ, u) ∈ R × H 1(R N ) are unknown. In this paper, we employ the argument
-
A C 2,α,β estimate for complex Monge–Ampère type equations with conic sigularities Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-13 Liding Huang, Gang Tian, Jiaxiang Wang
In this paper, we give an alternative approach to the C 2,α,β estimate for complex Monge-Ampère equations with cone singularities along simple normal crossing divisors.
-
Geometry of branched minimal surfaces of finite index Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-12 William H. Meeks, Joaquín Pérez
Given I , B ∈ N ∪ { 0 } $I,B\in \mathbb{N}\cup \left\{0\right\}$ , we investigate the existence and geometry of complete finitely branched minimal surfaces M in R 3 ${\mathbb{R}}^{3}$ with Morse index at most I and total branching order at most B. Previous works of Fischer-Colbrie (“On complete minimal surfaces with finite Morse index in 3-manifolds,” Invent. Math., vol. 82, pp. 121–132, 1985) and
-
Liouville theorems of solutions to mixed order Hénon-Hardy type system with exponential nonlinearity Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-12 Wei Dai, Shaolong Peng
In this paper, we are concerned with the Hénon-Hardy type systems with exponential nonlinearity on a half space R + 2 ${\mathbb{R}}_{+}^{2}$ : ( − Δ ) α 2 u ( x ) = | x | a u p 1 ( x ) e q 1 v ( x ) , x ∈ R + 2 , ( − Δ ) v ( x ) = | x | b u p 2 ( x ) e q 2 v ( x ) , x ∈ R + 2 , $\begin{cases}{\left(-{\Delta}\right)}^{\frac{\alpha }{2}}u\left(x\right)=\vert x{\vert }^{a}{u}^{{p}_{1}}\left(x\right){
-
Segregated solutions for nonlinear Schrödinger systems with a large number of components Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-12 Haixia Chen, Angela Pistoia
In this paper we are concerned with the existence of segregated non-radial solutions for nonlinear Schrödinger systems with a large number of components in a weak fully attractive or repulsive regime in presence of a suitable external radial potential.
-
A semilinear Dirichlet problem involving the fractional Laplacian in R+ n Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-12 Yan Li
We investigate the Dirichelt problem involving the fractional Laplacian in the upper half-space R + n = x ∈ R n ∣ x 1 > 0 ${\mathbb{R}}_{+}^{n}=\left\{x\in {\mathbb{R}}^{n}\mid {x}_{1}{ >}0\right\}$ : ( − Δ ) s u ( x ) = f ( u ( x ) ) , x ∈ R + n , u ( x ) > 0 , x ∈ R + n , u ( x ) = 0 , x ∉ R + n . \begin{cases}\quad \hfill & {\left(-{\Delta}\right)}^{s}u\left(x\right)=f\left(u\left(x\right)\right)
-
Sliding methods for dual fractional nonlinear divergence type parabolic equations and the Gibbons’ conjecture Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-11 Yahong Guo, Lingwei Ma, Zhenqiu Zhang
In this paper, we consider the general dual fractional parabolic problem ∂ t α u ( x , t ) + L u ( x , t ) = f ( t , u ( x , t ) ) in R n × R . ${\partial }_{t}^{\alpha }u\left(x,t\right)+\mathcal{L}u\left(x,t\right)=f\left(t,u\left(x,t\right)\right) \text{in} {\mathbb{R}}^{n}{\times}\mathbb{R}.$ We show that the bounded entire solution u satisfying certain one-direction asymptotic assumptions must
-
Infinite energy harmonic maps from quasi-compact Kähler surfaces Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-11 Georgios Daskalopoulos, Chikako Mese
We construct infinite energy harmonic maps from a quasi-compact Kähler surface with a Poincaré-type metric into an NPC space. This is the first step in the construction of pluriharmonic maps from quasiprojective varieties into symmetric spaces of non-compact type, Euclidean and hyperbolic buildings and Teichmüller space.
-
Nonlinear problems inspired by the Born–Infeld theory of electrodynamics Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-08 Yisong Yang
It is shown that nonlinear electrodynamics of the Born–Infeld theory type may be exploited to shed insight into a few fundamental problems in theoretical physics, including rendering electromagnetic asymmetry to energetically exclude magnetic monopoles, achieving finite electromagnetic energy to relegate curvature singularities of charged black holes, and providing theoretical interpretation of equations
-
Annuloids and Δ-wings Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-08 David Hoffman, Francisco Martín, Brian White
We describe new annular examples of complete translating solitons for the mean curvature flow and how they are related to a family of translating graphs, the Δ-wings. In addition, we will prove several related results that answer questions that arise naturally in this investigation. These results apply to translators in general, not just to graphs or annuli.
-
Liouville type theorems involving fractional order systems Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-08 Qiuping Liao, Zhao Liu, Xinyue Wang
In this paper, let α be any real number between 0 and 2, we study the following semi-linear elliptic system involving the fractional Laplacian: ( − Δ ) α / 2 u ( x ) = f ( u ( x ) , v ( x ) ) , x ∈ R n , ( − Δ ) α / 2 v ( x ) = g ( u ( x ) , v ( x ) ) , x ∈ R n . $\begin{cases}{\left(-{\Delta}\right)}^{\alpha /2}u\left(x\right)=f\left(u\left(x\right),v\left(x\right)\right), x\in {\mathbb{R}}^{n},\quad
-
Eigenvalue lower bounds and splitting for modified Ricci flow Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-07 Tobias Holck Colding, William P. Minicozzi II
We prove sharp lower bounds for eigenvalues of the drift Laplacian for a modified Ricci flow. The modified Ricci flow is a system of coupled equations for a metric and weighted volume that plays an important role in Ricci flow. We will also show that there is a splitting theorem in the case of equality.
-
Comparison formulas for total mean curvatures of Riemannian hypersurfaces Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-01 Mohammad Ghomi
We devise some differential forms after Chern to compute a family of formulas for comparing total mean curvatures of nested hypersurfaces in Riemannian manifolds. This yields a quicker proof of a recent result of the author with Joel Spruck, which had been obtained via Reilly’s identities.
-
A Liouville theorem for superlinear parabolic equations on the Heisenberg group Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-01 Juncheng Wei, Ke Wu
We consider a parabolic nonlinear equation on the Heisenberg group. Applying the Gidas–Spruck type estimates, we prove that under suitable conditions, the equation does not have positive solutions. As an application of the nonexistence result, we provide optimal universal estimates for positive solutions.
-
Stability and critical dimension for Kirchhoff systems in closed manifolds Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-01 Emmanuel Hebey
The Kirchhoff equation was proposed in 1883 by Kirchhoff [Vorlesungen über Mechanik, Leipzig, Teubner, 1883] as an extension of the classical D’Alembert’s wave equation for the vibration of elastic strings. Almost one century later, Jacques Louis Lions [“On some questions in boundary value problems of mathematical physics,” in Contemporary Developments in Continuum Mechanics and PDE’s, G. M. de la
-
On subsolutions and concavity for fully nonlinear elliptic equations Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-01 Bo Guan
Subsolutions and concavity play critical roles in classical solvability, especially a priori estimates, of fully nonlinear elliptic equations. Our first primary goal in this paper is to explore the possibility to weaken the concavity condition. The second is to clarify relations between weak notions of subsolution introduced by Székelyhidi and the author, respectively, in attempt to treat equations
-
On the functional ∫Ωf + ∫Ω*g Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-01 Qiang Guang, Qi-Rui Li, Xu-Jia Wang
In this paper, we consider a class of functionals subject to a duality restriction. The functional is of the form J ( Ω , Ω * ) = ∫ Ω f + ∫ Ω * g $\mathcal{J}\left({\Omega},{{\Omega}}^{{\ast}}\right)={\int }_{{\Omega}}f+{\int }_{{{\Omega}}^{{\ast}}}g$ , where f, g are given nonnegative functions in a manifold. The duality is a relation α(x, y) ≤ 0 ∀ x ∈ Ω, y ∈ Ω*, for a suitable function α. This model
-
Rigidity properties of Colding–Minicozzi entropies Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-01 Jacob Bernstein
We show certain rigidity for minimizers of generalized Colding–Minicozzi entropies. The proofs are elementary and work even in situations where the generalized entropies are not monotone along mean curvature flow.
-
Non-homogeneous fully nonlinear contracting flows of convex hypersurfaces Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-01 Pengfei Guan, Jiuzhou Huang, Jiawei Liu
We consider a general class of non-homogeneous contracting flows of convex hypersurfaces in R n + 1 ${\mathbb{R}}^{n+1}$ , and prove the existence and regularity of the flow before extincting to a point in finite time.
-
Perturbation compactness and uniqueness for a class of conformally compact Einstein manifolds Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2024-03-01 Sun-Yung Alice Chang, Yuxin Ge, Xiaoshang Jin, Jie Qing
In this paper, we establish compactness results for some classes of conformally compact Einstein metrics defined on manifolds of dimension d ≥ 4. In the special case when the manifold is the Euclidean ball with the unit sphere as the conformal infinity, the existence of such class of metrics has been established in the earlier work of C. R. Graham and J. Lee (“Einstein metrics with prescribed conformal
-
Chemotaxis-Stokes interaction with very weak diffusion enhancement: Blow-up exclusion via detection of absorption-induced entropy structures involving multiplicative couplings Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2022-01-01 Michael Winkler
Abstract The chemotaxis–Stokes system n t + u ⋅ ∇ n = ∇ ⋅ ( D ( n ) ∇ n ) − ∇ ⋅ ( n S ( x , n , c ) ⋅ ∇ c ) , c t + u ⋅ ∇ c = Δ c − n c , u t = Δ u + ∇ P + n ∇ Φ , ∇ ⋅ u = 0 , \left\{\begin{array}{l}{n}_{t}+u\cdot \nabla n=\nabla \cdot (D\left(n)\nabla n)-\nabla \cdot (nS\left(x,n,c)\cdot \nabla c),\\ {c}_{t}+u\cdot \nabla c=\Delta c-nc,\\ {u}_{t}=\Delta u+\nabla P+n\nabla \Phi ,\hspace{1.0em}\nabla
-
On the L q -reflector problem in ℝ n with non-Euclidean norm Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2022-01-01 Jing Wang
Abstract In this article, we introduce a class of geometric optics measure-the L q Σ {L}_{q}\hspace{0.33em}\Sigma -reflector measure which arises from an L q {L}_{q} extension of the Σ \Sigma -reflector measure. And we ask a Minkowski-type problem for this class of measure, called the L q Σ {L}_{q}\hspace{0.33em}\Sigma -reflector problem. It is shown that the foundations of such measure have been laid
-
Non-degeneracy of bubble solutions for higher order prescribed curvature problem Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2022-01-01 Yuxia Guo,Yichen Hu
Abstract In this article, we are concerned with the following prescribed curvature problem involving polyharmonic operator on S N {{\mathbb{S}}}^{N} : D m u = K ( ∣ y ∣ ) u m ∗ − 1 , u > 0 in S N , u ∈ H m ( S N ) , {D}^{m}u=K\left(| y| ){u}^{{m}^{\ast }-1},\hspace{1.0em}u\gt 0\hspace{0.33em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{S}}}^{N},\hspace{1.0em}u\in {H}^{m}\left({{\mathbb{S}}}^{N})
-
Weighted critical exponents of Sobolev-type embeddings for radial functions Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2022-01-01 Jiabao Su,Cong Wang
Abstract In this article, we prove the upper weighted critical exponents for some embeddings from weighted Sobolev spaces of radial functions into weighted Lebesgue spaces. We also consider the lower critical exponent for certain embedding.
-
Existence and asymptotic behavior of solitary waves for a weakly coupled Schrödinger system Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2022-01-01 Xiaoming An,Jing Yang
Abstract This paper deals with the following weakly coupled nonlinear Schrödinger system − Δ u 1 + a 1 ( x ) u 1 = ∣ u 1 ∣ 2 p − 2 u 1 + b ∣ u 1 ∣ p − 2 ∣ u 2 ∣ p u 1 , x ∈ R N , − Δ u 2 + a 2 ( x ) u 2 = ∣ u 2 ∣ 2 p − 2 u 2 + b ∣ u 2 ∣ p − 2 ∣ u 1 ∣ p u 2 , x ∈ R N , \left\{\begin{array}{ll}-\Delta {u}_{1}+{a}_{1}\left(x){u}_{1}=| {u}_{1}{| }^{2p-2}{u}_{1}+b| {u}_{1}{| }^{p-2}| {u}_{2}{| }^{p}{u}_{1}
-
Asymptotic mean-value formulas for solutions of general second-order elliptic equations Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2022-01-01 Pablo Blanc,Fernando Charro,Juan J. Manfredi,Julio D. Rossi
Abstract We obtain asymptotic mean-value formulas for solutions of second-order elliptic equations. Our approach is very flexible and allows us to consider several families of operators obtained as an infimum, a supremum, or a combination of both infimum and supremum, of linear operators. The families of equations that we consider include well-known operators such as Pucci, Issacs, and k-Hessian operators
-
Large solutions of a class of degenerate equations associated with infinity Laplacian Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2022-01-01 Cuicui Li,Fang Liu
Abstract In this article, we investigate the boundary blow-up problem Δ ∞ h u = f ( x , u ) , in Ω , u = ∞ , on ∂ Ω , \left\{\begin{array}{ll}{\Delta }_{\infty }^{h}u=f\left(x,u),& {\rm{in}}\hspace{0.33em}\Omega ,\\ u=\infty ,& {\rm{on}}\hspace{0.33em}\partial \Omega ,\end{array}\right. where Δ ∞ h u = ∣ D u ∣ h − 3 ⟨ D 2 u D u , D u ⟩ {\Delta }_{\infty }^{h}u=| Du\hspace{-0.25em}{| }^{h-3}\langle
-
On fractional logarithmic Schrödinger equations Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2022-01-01 Qi Li,Shuangjie Peng,Wei Shuai
Abstract We study the following fractional logarithmic Schrödinger equation: ( − Δ ) s u + V ( x ) u = u log u 2 , x ∈ R N , {\left(-\Delta )}^{s}u+V\left(x)u=u\log {u}^{2},\hspace{1em}x\in {{\mathbb{R}}}^{N}, where N ≥ 1 N\ge 1 , ( − Δ ) s {\left(-\Delta )}^{s} denotes the fractional Laplace operator, 0 < s < 1 0\lt s\lt 1 and V ( x ) ∈ C ( R N ) V\left(x)\in {\mathcal{C}}\left({{\mathbb{R}}}^{N})
-
A sharp global estimate and an overdetermined problem for Monge-Ampère type equations Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2022-01-01 Ahmed Mohammed, Giovanni Porru
We consider Monge-Ampère type equations involving the gradient that are elliptic in the framework of convex functions. Through suitable symmetrization we find sharp estimates to solutions of such equations. An overdetermined problem related to our Monge-Ampère type operators is also considered and we show that such a problem may admit a solution only when the underlying domain is a ball.
-
Frontmatter Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-11-01
Article Frontmatter was published on November 1, 2021 in the journal Advanced Nonlinear Studies (volume 21, issue 4).
-
Sharp Trudinger–Moser Inequality and Ground State Solutions to Quasi-Linear Schrödinger Equations with Degenerate Potentials in ℝ n Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-11-01 Lu Chen, Guozhen Lu, Maochun Zhu
The main purpose of this paper is to establish the existence of ground-state solutions to a class of Schrödinger equations with critical exponential growth involving the nonnegative, possibly degenerate, potential V: - div ( | ∇ u | n - 2 ∇ u ) + V ( x ) | u | n - 2 u = f ( u ) . -\operatorname{div}(\lvert\nabla u\rvert^{n-2}\nabla u)+V(x)\lvert u\rvert^{n-% 2}u=f(u). To this end, we
-
Sharp Blow-Up Profiles of Positive Solutions for a Class of Semilinear Elliptic Problems Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-11-01 Wan-Tong Li, Julián López-Gómez, Jian-Wen Sun
This paper analyzes the behavior of the positive solution θ ε {\theta_{\varepsilon}} of the perturbed problem { - Δ u = λ m ( x ) u - [ a ε ( x ) + b ε ( x ) ] u p = 0 in Ω , B u = 0 on ∂ Ω , \left\{\begin{aligned} \displaystyle{}{-\Delta u}&\displaystyle=\lambda m(x)u-% [a_{\varepsilon}(x)+b_{\varepsilon}(x)]u^{p}=0&&\displaystyle\text{in}\ \Omega% ,\\ \displaystyle Bu&\disp
-
Nonlocal Differential Equations with Convolution Coefficients and Applications to Fractional Calculus Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-11-01 Christopher S. Goodrich
The existence of at least one positive solution to a large class of both integer- and fractional-order nonlocal differential equations, of which one model case is - A ( ( b * u q ) ( 1 ) ) u ′′ ( t ) = λ f ( t , u ( t ) ) , t ∈ ( 0 , 1 ) , q ≥ 1 , -A((b*u^{q})(1))u^{\prime\prime}(t)=\lambda f(t,u(t)),\quad t\in(0,1),\,q\geq 1, is considered. Due to the coefficient A ( ( b * u q )
-
Up-to-Boundary Pointwise Gradient Estimates for Very Singular Quasilinear Elliptic Equations with Mixed Data Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-11-01 Tan Duc Do, Le Xuan Truong, Nguyen Ngoc Trong
This paper establishes pointwise estimates up to boundary for the gradient of weak solutions to a class of very singular quasilinear elliptic equations with mixed data { - div ( 𝐀 ( x , D u ) ) = g - div f in Ω , u = 0 on ∂ Ω , \left\{\begin{aligned} \displaystyle-\operatorname{div}(\mathbf{A}(x,Du))&% \displaystyle=g-\operatorname{div}f&&\displaystyle\text{in }\Omega,\\ \displaystyle
-
Singular Finsler Double Phase Problems with Nonlinear Boundary Condition Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-11-01 Csaba Farkas, Alessio Fiscella, Patrick Winkert
In this paper, we study a singular Finsler double phase problem with a nonlinear boundary condition and perturbations that have a type of critical growth, even on the boundary. Based on variational methods in combination with truncation techniques, we prove the existence of at least one weak solution for this problem under very general assumptions. Even in the case when the Finsler manifold reduces
-
Existence and Convergence of Solutions to Fractional Pure Critical Exponent Problems Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-11-01 Víctor Hernández-Santamaría, Alberto Saldaña
We study existence and convergence properties of least-energy symmetric solutions (l.e.s.s.) to the pure critical exponent problem ( - Δ ) s u s = | u s | 2 s ⋆ - 2 u s , u s ∈ D 0 s ( Ω ) , 2 s ⋆ := 2 N N - 2 s , (-\Delta)^{s}u_{s}=\lvert u_{s}\rvert^{2_{s}^{\star}-2}u_{s},\quad u_{s}\in D^% {s}_{0}(\Omega),\,2^{\star}_{s}:=\frac{2N}{N-2s}, where s is any positive number, Ω is either ℝ N
-
Blow-Up Phenomena and Asymptotic Profiles Passing from H 1-Critical to Super-Critical Quasilinear Schrödinger Equations Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-11-01 Daniele Cassani, Youjun Wang
We study the asymptotic profile, as ℏ → 0 {\hbar\rightarrow 0} , of positive solutions to - ℏ 2 Δ u + V ( x ) u - ℏ 2 + γ u Δ u 2 = K ( x ) | u | p - 2 u , x ∈ ℝ N , -\hbar^{2}\Delta u+V(x)u-\hbar^{2+\gamma}u\Delta u^{2}=K(x)\lvert u\rvert^{p-2% }u,\quad x\in\mathbb{R}^{N}, where γ ⩾ 0 {\gamma\geqslant 0} is a parameter with relevant physical interpretations, V and K are given potentials
-
Well-Posedness and Uniform Decay Rates for a Nonlinear Damped Schrödinger-Type Equation Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-11-01 Marcelo M. Cavalcanti, Valéria N. Domingos Cavalcanti
In this paper we study the existence as well as uniform decay rates of the energy associated with the nonlinear damped Schrödinger equation, i u t + Δ u + | u | α u - g ( u t ) = 0 in Ω × ( 0 , ∞ ) , iu_{t}+\Delta u+|u|^{\alpha}u-g(u_{t})=0\quad\text{in }\Omega\times(0,\infty), subject to Dirichlet boundary conditions, where Ω ⊂ ℝ n {\Omega\subset\mathbb{R}^{n}} , n ≤ 3 {n\leq 3} , is a bounded
-
The Moving Plane Method for Doubly Singular Elliptic Equations Involving a First-Order Term Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-11-01 Francesco Esposito, Berardino Sciunzi
In this paper we deal with positive singular solutions to semilinear elliptic problems involving a first-order term and a singular nonlinearity. Exploiting a fine adaptation of the well-known moving plane method of Alexandrov–Serrin and a careful choice of the cutoff functions, we deduce symmetry and monotonicity properties of the solutions.
-
Concentration-Compactness Principle for Trudinger–Moser’s Inequalities on Riemannian Manifolds and Heisenberg Groups: A Completely Symmetrization-Free Argument Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-11-01 Jungang Li, Guozhen Lu, Maochun Zhu
The concentration-compactness principle for the Trudinger–Moser-type inequality in the Euclidean space was established crucially relying on the Pólya–Szegő inequality which allows to adapt the symmetrization argument. As far as we know, the first concentration-compactness principle of Trudinger–Moser type in non-Euclidean settings, such as the Heisenberg (and more general stratified) groups where the
-
Liouville Theorems for Fractional Parabolic Equations Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-11-01 Wenxiong Chen, Leyun Wu
In this paper, we establish several Liouville type theorems for entire solutions to fractional parabolic equations. We first obtain the key ingredients needed in the proof of Liouville theorems, such as narrow region principles and maximum principles for antisymmetric functions in unbounded domains, in which we remarkably weaken the usual decay condition u → 0 u\to 0 at infinity to a polynomial growth
-
Existence and Uniqueness of Multi-Bump Solutions for Nonlinear Schrödinger–Poisson Systems Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-07-31 Mingzhu Yu, Haibo Chen
In this paper, we study the following Schrödinger–Poisson equations: { - ε 2 Δ u + V ( x ) u + K ( x ) ϕ u = | u | p - 2 u , x ∈ ℝ 3 , - ε 2 Δ ϕ = K ( x ) u 2 , x ∈ ℝ 3 , \left\{\begin{aligned} &\displaystyle{-}\varepsilon^{2}\Delta u+V(x)u+K(x)\phi u% =\lvert u\rvert^{p-2}u,&\hskip 10.0ptx&\displaystyle\in\mathbb{R}^{3},\\ &\displaystyle{-}\varepsilon^{2}\Delta\phi=K(x)u^{2}
-
Maximum Principles and ABP Estimates to Nonlocal Lane–Emden Systems and Some Consequences Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-07-28 Edir Junior Ferreira Leite
This paper deals with maximum principles depending on the domain and ABP estimates associated to the following Lane–Emden system involving fractional Laplace operators: { ( - Δ ) s u = λ ρ ( x ) | v | α - 1 v in Ω , ( - Δ ) t v = μ τ ( x ) | u | β - 1 u in Ω , u = v = 0 in ℝ n ∖ Ω , \left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=\lambda\rho(x% )\lvert v\
-
Existence of Ground States of Fractional Schrödinger Equations Adv. Nonlinear Stud. (IF 1.8) Pub Date : 2021-07-28 Li Ma, Zhenxiong Li
We consider ground states of the nonlinear fractional Schrödinger equation with potentials ( - Δ ) s u + V ( x ) u = f ( x , u ) , s ∈ ( 0 , 1 ) , (-\Delta)^{s}u+V(x)u=f(x,u),\quad s\in(0,1), on the whole space ℝ N {\mathbb{R}^{N}} , where V is a periodic non-negative nontrivial function on ℝ N {\mathbb{R}^{N}} and the nonlinear term f has some proper growth on u. Under uniform bounded assumptions