Elliptic problems with mixed nonlinearities and potentials singular at the origin and at the boundary of the domain

Abstract

We are interested in the following Dirichlet problem:

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u}{\textrm{dist}(x,\mathbb {R}^N \setminus \Omega )^2} = f(x,u) &{} \quad \text{ in } \Omega \\ u = 0 &{} \quad \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$

on a bounded domain $\Omega \subset \mathbb {R}^N$ with $0 \in \Omega$. We assume that the nonlinear part is superlinear on some closed subset $K \subset \Omega$ and asymptotically linear on $\Omega \setminus K$. We find a solution with the energy bounded by a certain min–max level, and infinitely, many solutions provided that f is odd in u. Moreover, we study also the multiplicity of solutions to the associated normalized problem.

1 Introduction

We are interested in the problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u}{\textrm{dist}(x,\mathbb {R}^N \setminus \Omega )^2} = f(x,u) &{} \quad \text{ in } \Omega \\ u = 0 &{} \quad \text{ on } \partial \Omega , \end{array} \right. \end{aligned}$$

(1.1)

where $\lambda , \mu \in \mathbb {R}$ are real parameters, $f : \Omega \times \mathbb {R}\rightarrow \mathbb {R}$, $\Omega \subset \mathbb {R}^N$ is a bounded domain in $\mathbb {R}^N$ with $0 \in \Omega$.

Semilinear problems of general form

$$\begin{aligned} -\Delta u = h(x,u) \end{aligned}$$

appear when one looks for stationary states of time-dependent problems, including the heat equation $\frac{\partial u}{\partial t} - \Delta u = h(x,u)$ or the wave equation $\frac{\partial ^2 u}{ \partial t^2} - \Delta u = h(x,u)$. In nonlinear optics, the nonlinear Schrödinger equation is studied

$$\begin{aligned} {\textbf{i}} \frac{\partial \Psi }{\partial t} + \Delta \Psi = h(x, |\Psi |) \Psi , \quad (t,x) \in \mathbb {R}\times \Omega \end{aligned}$$

(1.2)

and looking for standing waves $\Psi (t,x) = e^{i\lambda t} u(x)$ leads then to a semilinear problem.

The time-dependent equation [Eq. (1.2)] appears in physical models in the case of bounded domains $\Omega$ ([11, 12, 24]), as well as in the case $\Omega = \mathbb {R}^N$ ([8, 18]). Two points of view of solutions to (1.1) are possible; either $\lambda$ may be prescribed or may be considered as a part of the unknown. In the latter case, a natural additional condition is the prescribed mass $\int _\Omega u^2 \, dx$. In the paper, we will consider both cases; namely, we will look for solutions for the unconstrained problem (1.1) as well as the constrained one; see (1.3).

Equation (1.1) (and systems of such equations) on bounded domains has been studied in the presence of bounded potentials [4] and singular at the origin [13]; see also [10, 14, 15] for the case of unbounded $\Omega$. Its constrained counterpart without the potential has been studied, e.g., in [19, 20], where (1.3) was studied with $f(x,u)=|u|^{p-2}u$, $\nu =\mu =0$ in the mass-subcritical, mass-critical, and mass-supercritical cases. In this paper, we are interested in the presence of a potential

$$\begin{aligned} V(x) = -\frac{\mu }{|x|^2} - \frac{\nu }{\textrm{dist}\,(x, \mathbb {R}^N \setminus \Omega )^2}, \end{aligned}$$

which is singular in $\Omega$ as well as on the whole boundary $\partial \Omega$. We mention here that Schrödinger operators were studied with potentials being singular at the point on the boundary [7], as well as with potentials being singular on the whole boundary [16, 17]. We assume that $\Omega$ is a domain satisfying the following condition:

• (C) ${-}\Delta d \ge 0$ in $\Omega$, in the sense of distributions, where $d(x) := \textrm{dist}(x, \mathbb {R}^N \setminus \Omega )$.

This condition allows us to study the singular potential by means of Hardy-type inequalities (Sect. 2). As we will see in Sect. 2 (see Proposition 2.1), any convex domain $\Omega$ satisfies (C).

We impose the following condition on parameters appearing in the problem:

• (N) $\mu ,\nu \ge 0$, $\frac{\mu }{(N-2)^2} + \nu < \frac{1}{4}$, $N \ge 3$.

In what follows, $\lesssim$ denotes the inequality up to a multiplicative constant. Moreover, C denotes a generic constant which may vary from one line to another.

On the nonlinear part of Eq. (1.1), we propose the following assumptions.

• (F1) $f : \Omega \times \mathbb {R}\rightarrow \mathbb {R}$ is a Carathéodory function and there is $2< p < 2^*$, such that

  $$\begin{aligned} |f(x,u)| \lesssim 1+|u|^{p-1}, \quad u \in \mathbb {R}, \ x \in \Omega ; \end{aligned}$$

• (F2) $f(x,u) = o(u)$ uniformly in $x \in \Omega$ as $u \rightarrow 0$;

• (F3) $F(x,u)/|u|^2 \rightarrow +\infty$ as $|u| \rightarrow +\infty$ for $x \in K$, where $F(x,u) := \int _0^u f(x,s) \, ds$ and $K \subset \Omega$ is a closed set with $| \textrm{int}\, K | > 0$;

• (F4) f(x, u)/|u| is nondecreasing on $(-\infty , 0)$ and on $(0,\infty )$;

• (F5) $\lim _{|u|\rightarrow +\infty } \frac{f(x,u)}{u} = \Theta (x)$ uniformly in $x \in \Omega \setminus K$, where $\Theta \in L^\infty (\Omega \setminus K)$.

A simple example satisfying all foregoing conditions is the following:

$$\begin{aligned} f(x,u) = \left\{ \begin{array}{ll} \Gamma (x) |u|^{p-2}u, &{} \quad x \in K, \\ \frac{|u|^2}{1+|u|^2} u &{} \quad x \in \Omega \setminus K, \end{array} \right. \end{aligned}$$

where $\Gamma \in L^\infty (K)$ is a nonnegative function. Note that the foregoing example is a combination of nonlinearities appearing in nonlinear optics ([8, 11, 18]). Namely, for materials with the Kerr effect, one has $f(x,u) = |u|^{p-2}u$ with $p=4$, and in the saturation effect, the nonlinearity is asymptotically linear and $f(x,u) = \frac{|u|^2}{1+|u|^2} u$. Therefore, we can deal with media being composites of materials with different polarizations.

To show the boundedness of minimizing sequences to the problem (1.1), we impose the following abstract condition:

• (A) $-\lambda$ is not an eigenvalue of $-\Delta - \frac{\mu }{|x|^2} - \frac{\nu }{d(x)^2} - \Theta (x)$ with Dirichlet boundary conditions on $L^2(\Omega \setminus K)$.

As we will see in Sect. 2, (A) is satisfied if, e.g., $\lambda \ge |\Theta |_\infty$ (cf. Theorem 2.2).

Theorem 1.1

Suppose that (C), (N), (F1)–(F5), (A) are satisfied and $\lambda \ge 0$. Then, there is a nontrivial weak solution to (1.1) with the energy level c satisfying (3.1).

Theorem 1.2

Suppose that (C), (N), (F1)–(F5), (A) hold, $\lambda \ge 0$, and f is odd in $u \in \mathbb {R}$. Then, there is infinitely many weak solutions to (1.1).

In the last section, we also study the normalized problem

$$\begin{aligned} \left\{ \begin{array}{ll} -\Delta u + \lambda u - \mu \frac{u}{|x|^2} - \nu \frac{u}{\textrm{dist}(x, \mathbb {R}^N\setminus \Omega )^2} = f(x,u) &{} \quad \text{ in } \Omega \\ u = 0 &{} \quad \text{ on } \partial \Omega , \\ \int _\Omega u^2 \, dx = \rho > 0, \end{array} \right. \end{aligned}$$

(1.3)

where $\rho$ is fixed and $(\lambda , u) \in \mathbb {R}\times H^1_0(\Omega )$ is an unknown. Then, we obtain the following multiplicity result in the so-called mass-subcritical case.

Theorem 1.3

Suppose that (C), (N), (F1) hold with $p < 2_* := 2+\frac{4}{N}$, and f is odd in $u \in \mathbb {R}$. Then, there is infinitely many weak solutions to (1.3).

2 The domain $\Omega$nd the singular Schrödinger operator

We recall that if $A \subset \mathbb {R}^N$ is a closed, nonempty set, we can define the distance function $\textrm{dist}(\cdot , A) : \mathbb {R}^N \rightarrow [0,\infty )$ by

$$\begin{aligned} \textrm{dist}(x, A) := \inf _{y \in A} |x-y|, \quad x \in \mathbb {R}^N. \end{aligned}$$

We collect the following properties of the distance function:

• (i) $|\textrm{dist}(x, A) - \textrm{dist}(x',A) \le |x-x'|$ for all $x,x' \in \mathbb {R}^N$,

• (ii) if $x \in \mathbb {R}^N \setminus A$, then $\textrm{dist}(\cdot , A)$ is differentiable at x if and only if there is unique $x \in A$, such that $\textrm{dist}(x,A) = |x-y|$ and then $\nabla \textrm{dist}(x, A) = \frac{x-y}{|x-y|}$.

Now, we consider $A := \mathbb {R}^N \setminus \Omega$. It is clear that A is a closed subset of $\mathbb {R}^N$. We recall that we denote $d(x) = \textrm{dist}(x, \mathbb {R}^N \setminus \Omega )$. Observe that, due to Rademacher’s theorem ([26, Theorem 2.2.1]) and (i), d is differentiable almost everywhere and, from (ii) $|\nabla d| = 1$, almost everywhere. We remind that the assumption (C) says that

$$\begin{aligned} -\Delta d \ge 0 \text{ in } \Omega \end{aligned}$$

holds in the sense of distributions. We note the following fact.

Proposition 2.1

If $\Omega \subset \mathbb {R}^N$ is a convex domain in $\mathbb {R}^N$, then $d \big |_\Omega$ is concave and satisfies (C).

Proof

First note that $d \big |_\Omega : \Omega \rightarrow [0,\infty )$ is a concave function. Indeed, fix any $x,y \in \Omega$ and $\alpha \in [0,1]$. Let $z = \alpha x + (1-\alpha )y$. Choose $z_0 \in \partial \Omega$, such that $d(z) = |z-z_0|$. Let $T_{z_0} := z_0 + \textrm{span} \{ z-z_0 \}^\perp$ be an affine subspace of $\mathbb {R}^N$ orthogonal to $z-z_0$ containing $z_0$. Define $x_0, y_0$ as orthogonal projections of x, y, respectively, onto $T_{z_0}$. Then

$$\begin{aligned} d(z) = |z-z_0| = \alpha |x-x_0| + (1-\alpha )|y-y_0| \ge \alpha d(x) + (1-\alpha ) d(y), \end{aligned}$$

which completes the proof of concavity. Moreover, since d is concave on $\Omega$, from [9, Theorem 6.8], there is a nonnegative Radon measure $\mu$ on $\Omega$ satisfying

$$\begin{aligned} -\Delta d = \mu \quad \text{ in } \text{ the } \text{ sense } \text{ of } \text{ distributions, } \end{aligned}$$

namely

$$\begin{aligned} \int _{\Omega } \nabla d \cdot \nabla \varphi \, dx = \int _\Omega \varphi \, d\mu \quad \text{ for } \varphi \in {{\mathcal {C}}}_0^\infty (\Omega ). \end{aligned}$$

Clearly, for $\varphi \ge 0$, we get

$$\begin{aligned} \int _{\Omega } \nabla d \cdot \nabla \varphi \, dx \ge 0, \end{aligned}$$

and condition (C) holds. $\square$

To study singular terms in (1.1), we recall the following Hardy-type inequalities. If $u \in H^1_0 (\Omega )$, where $\Omega$ is a domain in $\mathbb {R}^N$ with finite Lebesgue measure and $0 \in \Omega$, then (see [6])

$$\begin{aligned} \frac{(N-2)^2}{4} \int _\Omega \frac{u^2}{|x|^2} \, dx \le \int _\Omega |\nabla u|^2 \, dx. \end{aligned}$$

(2.1)

Now, let $\Omega \subset \mathbb {R}^N$ be a bounded domain satisfying (C). Then, for $u \in H^1_0(\Omega )$, the following Hardy inequality involving the distance function holds (see [2]):

$$\begin{aligned} \frac{1}{4} \int _\Omega \frac{u^2}{d(x)^2} \, dx \le \int _{\Omega } |\nabla u|^2 \, dx. \end{aligned}$$

(2.2)

We consider the operator ${\mathcal {A}} := -\Delta - \frac{\mu }{|x|^2} - \frac{\nu }{d(x)^2} - \Theta (x)$ on $L^2 (\Omega \setminus K)$ with Dirichlet boundary conditions. Then, the domain is ${{\mathcal {D}}}({{\mathcal {A}}}) := H^2 (\Omega \setminus K) \cap H^1_0 (\Omega \setminus K)$.

Theorem 2.2

The operator ${{\mathcal {A}}}: {{\mathcal {D}}}(A) \subset L^2 (\Omega \setminus K) \rightarrow L^2 (\Omega \setminus K)$ is elliptic, self-adjoint on $L^2 (\Omega \setminus K)$ and has compact resolvents. Moreover, the spectrum $\sigma ({{\mathcal {A}}}) \subset (-|\Theta |_\infty , +\infty )$ and consists of eigenvalues $-|\Theta |_\infty < \lambda _1 \le \lambda _2 \le \ldots$ with $\lambda _n \rightarrow +\infty$ as $n \rightarrow +\infty$.

Proof

It is well known that ${{\mathcal {D}}}({{\mathcal {A}}})$ is closed in $L^2(\Omega \setminus K)$. It is easy to check that ${{\mathcal {A}}}$ is self-adjoint. The compactness of its resolvents easily follows from the Rellich–Kondrachov theorem. Hence, its spectrum consists of eigenvalues $\lambda _n$ with $\lambda _n \rightarrow +\infty$. To see that $\sigma ({{\mathcal {A}}}) \subset (-|\Theta |_\infty , +\infty )$, suppose that $\lambda$ is an eigenvalue of ${{\mathcal {A}}}$ with an associated eigenfunction $u \in H^1_0(\Omega \setminus K)$. We can treat u as a function in $H^1_0(\Omega )$ being zero on K. Then, using (2.1) and (2.2)

$$\begin{aligned} \lambda \int _{\Omega \setminus K} u^2 \, dx&\ge - \int _\Omega |\nabla u|^2 + \mu \frac{u^2}{|x|^2} + \nu \frac{u^2}{d(x)^2} \, dx + \lambda \int _{\Omega \setminus K} u^2 \, dx \\ {}&= -\int _{\Omega \setminus K} \Theta (x) u^2 \, dx \ge - |\Theta |_\infty \int _{\Omega \setminus K} u^2 \, dx. \end{aligned}$$

Hence, $\lambda \ge -|\Theta |_\infty$. To show that $\lambda \ne -|\Theta |_\infty$, suppose by contradiction that there is $u \in H^1_0 (\Omega \setminus K)$ with

$$\begin{aligned} \int _{\Omega \setminus K} |\nabla u|^2 - \mu \frac{u^2}{|x|^2} - \nu \frac{u^2}{d(x)^2} - \Theta (x) u^2 \, dx = -|\Theta |_\infty \int _{\Omega \setminus K} u^2 \, dx. \end{aligned}$$

Thus

$$\begin{aligned} \int _{\Omega \setminus K} |\nabla u|^2 - \mu \frac{u^2}{|x|^2} - \nu \frac{u^2}{d(x)^2} \, dx = \int _{\Omega \setminus K} (\Theta (x) - |\Theta |_\infty ) u^2 \, dx \le 0. \end{aligned}$$

However, using (2.1) and (2.2), we get

$$\begin{aligned} \int _{\Omega \setminus K} |\nabla u|^2 - \mu \frac{u^2}{|x|^2} - \nu \frac{u^2}{d(x)^2} \, dx \ge \left( 1 - \frac{4\mu }{(N-2)^2} - 4 \nu \right) \int _{\Omega \setminus K} |\nabla u|^2 \, dx. \end{aligned}$$

Hence, $\int _{\Omega \setminus K} |\nabla u|^2 \, dx = 0$ and $u = 0$, which is a contradiction. Hence, $\sigma ({{\mathcal {A}}}) \subset (-|\Theta |_\infty , +\infty )$. $\square$

3 Variational setting and critical point theory

Suppose that $(E, \Vert \cdot \Vert )$ is a Hilbert space and ${{\mathcal {J}}}: E \rightarrow \mathbb {R}$ is a nonlinear functional of the general form

$$\begin{aligned} {{\mathcal {J}}}(u) = \frac{1}{2} \Vert u\Vert ^2 - {{\mathcal {I}}}(u), \end{aligned}$$

where ${{\mathcal {I}}}$ is of ${{\mathcal {C}}}^1$ class and ${{\mathcal {I}}}(0)=0$. We introduce the so-called Nehari manifold

$$\begin{aligned} {{\mathcal {N}}}:= \{ u \in E \setminus \{ 0 \} \ : \ {{\mathcal {J}}}'(u)(u) = 0 \}. \end{aligned}$$

Observe that ${{\mathcal {I}}}'(u)(u) > 0$ on ${{\mathcal {N}}}$. Indeed

$$\begin{aligned} 0 = {{\mathcal {J}}}'(u)(u) = \Vert u\Vert ^2 - {{\mathcal {I}}}'(u)(u), \quad u \in {{\mathcal {N}}}. \end{aligned}$$

To utilize the mountain pass approach, we consider the following space of paths:

$$\begin{aligned} \Gamma := \{ \gamma \in {{\mathcal {C}}}([0,1], E) \ : \ \gamma (0) = 0, \ \Vert \gamma (1)\Vert > r, \ {{\mathcal {J}}}(\gamma (1)) < 0 \} \end{aligned}$$

for the radius $r > 0$ given in (J1) below, and the following mountain pass level:

$$\begin{aligned} c := \inf _{\gamma \in \Gamma } \sup _{t \in [0,1]} {{\mathcal {J}}}(\gamma (t)). \end{aligned}$$

Moreover, we set

$$\begin{aligned} \Gamma _Q := \Gamma \cap {{\mathcal {C}}}([0,1],Q), \end{aligned}$$

where Q is a vector subspace of E given in (J2) below.

We propose an abstract theorem which is a combination of [3, Theorem 5.1] and [5, Theorem 2.1]. The proof is a straightforward modification of proofs of mentioned theorems; however, we include it here for the reader’s convenience.

Theorem 3.1

Suppose that

• (J1) there is $r > 0$, such that

  $$\begin{aligned} a := \inf _{\Vert u\Vert =r} {{\mathcal {J}}}(u) > 0; \end{aligned}$$

• (J2) there is a closed vector subspace $Q \subset E$, such that $\frac{{{\mathcal {I}}}(t_n u_n)}{t_n^2} \rightarrow +\infty$ for $t_n \rightarrow +\infty$, $u_n \in Q$ and $u_n \rightarrow u \ne 0$;

• (J3) for all $t > 0$ and $u \in {{\mathcal {N}}}$, there holds

  $$\begin{aligned} \frac{t^2-1}{2} {{\mathcal {I}}}'(u)(u) - {{\mathcal {I}}}(tu) + {{\mathcal {I}}}(u) \le 0. \end{aligned}$$

Then, $\Gamma _Q \ne \emptyset$, ${{\mathcal {N}}}\cap Q \ne \emptyset$ and

$$\begin{aligned} 0 < \inf _{\Vert u\Vert =r} {{\mathcal {J}}}(u) \le c \le \inf _{\gamma \in \Gamma _Q} \sup _{t \in [0,1]} {{\mathcal {J}}}(\gamma (t)) = \inf _{{{\mathcal {N}}}\cap Q} {{\mathcal {J}}}= \inf _{u \in Q \setminus \{0\}} \sup _{t \ge 0} {{\mathcal {J}}}(tu). \nonumber \\ \end{aligned}$$

(3.1)

Moreover, there is a Cerami sequence for ${{\mathcal {J}}}$ on the level c, i.e., a sequence $\{ u_n \}_n \subset E$, such that

$$\begin{aligned} {{\mathcal {J}}}(u_n) \rightarrow c, \quad (1+\Vert u_n\Vert ) {{\mathcal {J}}}'(u_n) \rightarrow 0. \end{aligned}$$

Proof

Observe that there exists $v \in Q \setminus \{0\}$ with $\Vert v\Vert > r$, such that ${{\mathcal {J}}}(v) < 0$. Indeed, fix $u \in Q \setminus \{0\}$ and from (J2), there follows that

$$\begin{aligned} \frac{{{\mathcal {J}}}(tu)}{t^2} = \frac{1}{2} \Vert u\Vert ^2 - \frac{{{\mathcal {I}}}(tu)}{t^2} \rightarrow - \infty \quad \text{ as } t \rightarrow +\infty , \end{aligned}$$

(3.2)

and we may take $v := t u$ for sufficiently large $t > 0$. In particular, the family of paths $\Gamma _Q$ is nonempty. Moreover, ${{\mathcal {J}}}(tu) \rightarrow 0$ as $t \rightarrow 0^+$, and for $t = \frac{r}{\Vert u\Vert } > 0$, we get ${{\mathcal {J}}}(tu) > 0$. Hence, taking (3.2) into account, $(0,+\infty ) \ni t \mapsto {{\mathcal {J}}}(tu) \in \mathbb {R}$ has a local maximum, which is a critical point of ${{\mathcal {J}}}(tu)$ and $tu \in {{\mathcal {N}}}$. Hence, ${{\mathcal {N}}}\cap Q \ne \emptyset$. Suppose that $u \in {{\mathcal {N}}}\cap Q$. Then, from (J3)

$$\begin{aligned} {{\mathcal {J}}}(tu) = {{\mathcal {J}}}(tu) - \frac{t^2-1}{2} {{\mathcal {J}}}'(u)(u) \le {{\mathcal {J}}}(u), \end{aligned}$$

and therefore, u is a maximizer (not necessarily unique) of ${{\mathcal {J}}}$ on $\mathbb {R}_+ u := \{ su \ : \ s > 0 \}$. Hence, for any $u \in {{\mathcal {N}}}\cap Q$, there are $0 < t_{\min } (u) \le 1 \le t_{\max }(u)$, such that $t u \in {{\mathcal {N}}}\cap Q$ for any $t \in [t_{\min }(u), t_{\max } (u)]$ and

$$\begin{aligned}{}[t_{\min }(u), t_{\max } (u)] \ni t \mapsto {{\mathcal {J}}}(tu) \in \mathbb {R}\end{aligned}$$

is constant. Moreover, ${{\mathcal {J}}}'(tu)(u) > 0$ for $t \in (0, t_{\min }(u))$ and ${{\mathcal {J}}}'(tu)(u) < 0$ for $t \in (t_{\max } (u), +\infty )$, $Q \setminus {{\mathcal {N}}}$ consists of two connected components and any path $\gamma \in \Gamma _Q$ intersects ${{\mathcal {N}}}\cap Q$. Thus

$$\begin{aligned} \inf _{\gamma \in \Gamma _Q} \sup _{t \in [0,1]} {{\mathcal {J}}}(\gamma (t)) \ge \inf _{{{\mathcal {N}}}\cap Q} {{\mathcal {J}}}. \end{aligned}$$

Since

$$\begin{aligned} \inf _{{{\mathcal {N}}}\cap Q} {{\mathcal {J}}}= \inf _{u \in Q \setminus \{0\}} \sup _{t > 0} {{\mathcal {J}}}(tu), \end{aligned}$$

there follows, under (J1), that:

$$\begin{aligned} c = \inf _{\gamma \in \Gamma } \sup _{t \in [0,1]} {{\mathcal {J}}}(\gamma (t)) \le \inf _{\gamma \in \Gamma _Q} \sup _{t \in [0,1]} {{\mathcal {J}}}(\gamma (t)) = \inf _{{{\mathcal {N}}}\cap Q} {{\mathcal {J}}}= \inf _{u \in Q \setminus \{0\}} \sup _{t > 0} {{\mathcal {J}}}(tu). \end{aligned}$$

The existence of a Cerami sequence follows from the mountain pass theorem. $\square$

To study the multiplicity of solutions, we will recall the symmetric mountain pass theorem. We consider the following condition:

• (J4) there exists a sequence of subspaces $\widetilde{E}_1 \subset \widetilde{E_2} \subset \ldots \subset E$, such that $\dim \widetilde{E}_k=k$ for every $k \ge 1$ and there is a radius $R_k$, such that $\sup _{u \in \widetilde{E}_k,\ \Vert u\Vert \ge R_k} {{\mathcal {J}}}\le 0$.

Then, the following theorem holds.

Theorem 3.2

([1, Corolarry 2.9], [21, Theorem 9.12]) Suppose that ${{\mathcal {J}}}$, as above, is even and satisfies (J1), (J4) and a Palais–Smale condition (namely, any Palais–Smale sequence for ${{\mathcal {J}}}$ contains a convergent subsequence). Then, ${{\mathcal {J}}}$ has an unbounded sequence of critical values.

We work in the usual Sobolev space $H^1_0(\Omega )$ being the completion of ${{\mathcal {C}}}_0^\infty (\Omega )$ with respect to the norm

$$\begin{aligned} \Vert u \Vert _{H^1} := \left( \int _{\Omega } |\nabla u|^2 + u^2 \, dx \right) ^{1/2}. \end{aligned}$$

Define the bilinear form $B : H^1_0(\Omega ) \times H^1_0 (\Omega ) \rightarrow \mathbb {R}$ by

$$\begin{aligned} B(u,v) := \int _{\Omega } \nabla u \cdot \nabla v + \lambda uv \, dx - \mu \int _{\Omega } \frac{uv}{|x|^2} \, dx - \nu \int _\Omega \frac{uv}{d(x)^2} \, dx, \quad u,v \in H^1_0(\Omega ). \end{aligned}$$

Lemma 3.3

B defines an inner product on $H^1_0 (\Omega )$. Moreover, the associated norm is equivalent with the usual one.

Proof

To check that B is positive-definite, we utilize (2.1), (2.2), and (N) to get

$$\begin{aligned} B(u,u)&= \int _{\Omega } |\nabla u |^2 + \lambda u^2 \, dx - \mu \int _{\Omega } \frac{u^2}{|x|^2} \, dx - \nu \int _\Omega \frac{u^2}{d(x)^2} \, dx\\&\ge \left( 1 - \frac{4 \mu }{(N-2)^2} - 4\nu \right) \int _\Omega |\nabla u|^2 \, dx + \lambda u^2 \, dx\\ {}&\ge \left( 1 - \frac{4 \mu }{(N-2)^2} - 4\nu \right) \int _\Omega |\nabla u|^2 \, dx, \end{aligned}$$

and the statement follows from the Poincaré inequality. Moreover, from

$$\begin{aligned} \int _\Omega |\nabla u|^2 + \lambda u^2 \, dx \ge B(u,u) \ge \left( 1 - \frac{4 \mu }{(N-2)^2} - 4\nu \right) \int _\Omega |\nabla u|^2 \, dx, \end{aligned}$$

there follows that B generates a norm on $H^1_0(\Omega )$ equivalent to the standard one. $\square$

Let $\Vert \cdot \Vert$ denote the norm generated by B, namely

$$\begin{aligned} \Vert u\Vert := \sqrt{B(u,u)}, \quad u \in H^1_0(\Omega ). \end{aligned}$$

Then, we can define the energy functional ${{\mathcal {J}}}: H^1_0 (\Omega ) \rightarrow \mathbb {R}$ by

$$\begin{aligned} {{\mathcal {J}}}(u) := \frac{1}{2} \Vert u\Vert ^2 - \int _\Omega F(x,u) \, dx, \end{aligned}$$

(3.3)

where $G(x,u) := \int _0^u g(x,s) \, ds$ and F is given in (F3). It is well known that under (F1), (F2) the functional is of ${{\mathcal {C}}}^1$ class and

$$\begin{aligned} {{\mathcal {J}}}'(u)(v) = B(u,v) - \int _\Omega f(x,u)v \, dx, \quad u,v \in H^1_0(\Omega ). \end{aligned}$$

Hence, its critical points are weak solutions to (1.1).

4 Verification of (J1)–(J4)

Observe that (F1), (F2) imply that for every $\varepsilon > 0$, one can find $C_\varepsilon > 0$, such that

$$\begin{aligned} |f(x,u)| \le \varepsilon |u| + C_{\varepsilon }|u|^{p-1}. \end{aligned}$$

There follows also a similar inequality for F, namely:

$$\begin{aligned} F(x,u) \le \varepsilon u^2 + C_\varepsilon |u|^p. \end{aligned}$$

(4.1)

We note also that if, in addition, (F4) holds, then $F(x,u) \ge 0$. Moreover, we recall that the functional ${{\mathcal {J}}}$ is defined by (3.3).

Lemma 4.1

Suppose that (C), (N), (F1)–(F5) hold. Then, ${{\mathcal {J}}}$ satisfies (J1)–(J3) in Theorem 3.1 and (J4) in Theorem 3.2.

• (J1) Using (4.1) and Sobolev embeddings, we obtain

  $$\begin{aligned} \int _\Omega F(x,u)\,dx\le \varepsilon |u|_2^2+C_\varepsilon |u|_p^p\lesssim \varepsilon \Vert u\Vert ^2+C_\epsilon \Vert u\Vert ^p. \end{aligned}$$

  Hence, can choose $\varepsilon >0$ and $r>0$, such that

  $$\begin{aligned} \int _\Omega F(x,u)\,dx\le \frac{1}{4}\Vert u\Vert ^2 \end{aligned}$$

  for all $\Vert u\Vert \le r$. Then, we get

  $$\begin{aligned} {\mathcal {J}}(u)&= \frac{1}{2}\Vert u\Vert ^2-\int _\Omega F(x,u)\,dx \ge \frac{1}{4}\Vert u\Vert ^2=\frac{r^2}{4}>0 \end{aligned}$$

  for all $\Vert u\Vert =r.$

• (J2) Let $Q := H^1_0 (\textrm{int}\, K)$. Let $t_n\rightarrow +\infty$, $u_n \in Q$ and $u_n\rightarrow u\ne 0$. Then, from Fatou’s lemma and (F3)

  $$\begin{aligned} \frac{{\mathcal {I}}(t_nu_n)}{t_n^2}=\frac{\int _K F(x,t_nu_n)\,dx}{t_n^2} \rightarrow +\infty \quad \text{ as } n \rightarrow +\infty . \end{aligned}$$

• (J3) Fix $u \in {{\mathcal {N}}}$. Define

  $$\begin{aligned} (0,\infty ) \ni t \mapsto \varphi (t):=\frac{t^2-1}{2}{\mathcal {I}}'(u)(u)-{\mathcal {I}}(tu)+{\mathcal {I}}(u)\in \mathbb {R}. \end{aligned}$$

  Note that $\varphi (1)=0$. Moreover

  $$\begin{aligned} \varphi '(t)&=t {\mathcal {I}}'(u)(u)-{\mathcal {I}}'(tu)(u)=\int _\Omega f(x,u)tu\,dx-\int _\Omega f(x,tu)u\,dx. \end{aligned}$$

  Suppose that $t\in (0,1)$. Then, (F4) implies that for a.e. $x \in \Omega$, $f(x,tu)u \le tf(x,u)u$ and, therefore, $\varphi '(t) \ge 0$. Similarly, $\varphi '(t)\le 0$ for $t>1$, which implies that $\varphi (t)\le \varphi (0)=0$ for all $t>0$.

• (J4) Let $\widetilde{E} \subset H^1_0(\textrm{int}\,K) \subset H^1_0(\Omega )$ be a finite-dimensional subspace. Note that on $\widetilde{E}$, all norms are equivalent. Suppose, by contradiction, that there is a sequence $(u_n) \subset \widetilde{E}$, such that $\Vert u_n\Vert \rightarrow +\infty$ and ${{\mathcal {J}}}(u_n) > 0$. Let $w_n(x) := u_n(x) / \Vert u_n\Vert$. It is clear that $\Vert w_n\Vert =1$ and, since $\widetilde{E}$ is finite-dimensional, there is $w \in \widetilde{E} \setminus \{0\}$, such that $\Vert w_n-w\Vert \rightarrow 0$. In particular $|\textrm{supp}\,(w) \cap K| > 0$. Then, for a.e. $x \in \textrm{supp}\,(w) \cap K$, we have that

  $$\begin{aligned} u_n(x)^2 = \Vert u_n\Vert ^2 w_n(x)^2 \rightarrow +\infty . \end{aligned}$$

  Hence, by the Fatou’s lemma and (F3)

  $$\begin{aligned} 0 < \frac{{{\mathcal {J}}}(u_n)}{\Vert u_n\Vert ^2}&= \frac{1}{2} - \int _{\Omega } \frac{F(x,u_n)}{\Vert u_n\Vert ^2} \, dx \\ {}&\le \frac{1}{2} - \int _{\textrm{supp}\,(w) \cap K} \frac{F(x,u_n)}{u_n^2} w_n^2 \, dx \rightarrow -\infty , \end{aligned}$$

  which is a contradiction.

5 Cerami sequences and proofs of main theorems

Lemma 5.1

Any Cerami sequence for ${{\mathcal {J}}}$ is bounded.

Proof

Suppose that $\Vert u_n\Vert \rightarrow +\infty$ up to a subsequence. We define $v_n = \frac{u_n}{\Vert u_n\Vert }$. Then, $\Vert v_n\Vert =1$ and $v_n \rightharpoonup v_0$ in $H^1_0(\Omega )$. From compact Sobolev embeddings, $v_n \rightarrow v_0$ in $L^2(\Omega )$, in $L^p (\Omega )$ and almost everywhere.

We consider three cases.

• Suppose that $v_0 = 0$. Define $F_\varepsilon (x,u) := F(x,u) + \frac{\varepsilon }{p} |u|_p^p$. Then, for every $\varepsilon > 0$, the following condition holds:

  $$\begin{aligned} \frac{F_\varepsilon (x,u)}{u^2} \rightarrow +\infty \quad \text{ as } \ |u|\rightarrow +\infty \ \text{ uniformly } \text{ in } \Omega . \end{aligned}$$

  Then, usual arguments (cf. [23, Lemma 2.2]) show that

  $$\begin{aligned} {{\mathcal {J}}}_\varepsilon (u) \ge {{\mathcal {J}}}_\varepsilon (tu) - \frac{t^2-1}{2} {{\mathcal {J}}}_\varepsilon '(u)(u), \quad u \in H^1_0 (\Omega ), \end{aligned}$$

  where ${{\mathcal {J}}}_\varepsilon (u) := {{\mathcal {J}}}(u) - \frac{\varepsilon }{p} |u|_p^p$. Taking $\varepsilon \rightarrow 0^+$, we get the following inequality:

  $$\begin{aligned} {{\mathcal {J}}}(u) \ge {{\mathcal {J}}}(tu) - \frac{t^2-1}{2} {{\mathcal {J}}}'(u)(u). \end{aligned}$$

  Taking $t := \frac{t}{\Vert u_n\Vert }$ and $u := u_n$ above, we obtain that

  $$\begin{aligned} {{\mathcal {J}}}(u_n) \ge {{\mathcal {J}}}\left( \frac{t}{\Vert u_n\Vert }u_n\right) - \frac{\frac{t^2}{\Vert u_n\Vert ^2}-1}{2} {{\mathcal {J}}}'(u_n)(u_n) = {{\mathcal {J}}}(t v_n) + o(1). \end{aligned}$$

  Hence

  $$\begin{aligned} {{\mathcal {J}}}(u_n) \ge \frac{t^2}{2} - \int _\Omega F(x, tv_n) \, dx + o(1). \end{aligned}$$

  Moreover

  $$\begin{aligned} \left| \int _\Omega F(x,tv_n) \, dx \right| \le \varepsilon t^2 \int _\Omega |v_n|^2 \, dx + C_\varepsilon t^p \int _\Omega |v_n|^p \, dx \rightarrow 0 \quad \text{ as } n \rightarrow \infty . \end{aligned}$$

  Thus

  $$\begin{aligned} c + o(1) = {{\mathcal {J}}}(u_n) \ge \frac{t^2}{2} + o(1) \end{aligned}$$

  for any $t > 0$—a contradiction.

• Now, we suppose that $v_0 \ne 0$ and $|\textrm{supp}\,v_0 \cap K | > 0$. Then

  $$\begin{aligned} o(1)=\frac{{{\mathcal {J}}}(u_n)}{\Vert u_n\Vert ^2}&= \frac{1}{2} - \int _\Omega \frac{F(x,u_n)}{\Vert u_n\Vert ^2} \, dx = \frac{1}{2} - \int _\Omega \frac{F(x,u_n)}{u_n^2} v_n^2 \, dx \\ {}&\le \frac{1}{2} - \int _{\textrm{supp}\,v_0 \cap K} \frac{F(x,u_n)}{u_n^2} v_n^2 \, dx \rightarrow -\infty , \end{aligned}$$

  a contradiction.

• Suppose that $v_0 \ne 0$ and $|\textrm{supp}\,v_0 \cap K| = 0$. Then, $\textrm{supp}\,v_0 \subset \Omega \setminus K$. Fix $\varphi \in {{\mathcal {C}}}_0^\infty (\Omega \setminus K)$ and note that

  $$\begin{aligned} o(1) = {{\mathcal {J}}}'(u_n)(\varphi ) = \langle u_n, \varphi \rangle - \int _{\Omega } f(x,u_n) \varphi \, dx. \end{aligned}$$

  Observe that

  $$\begin{aligned} \int _{\Omega } f(x,u_n)\varphi \, dx&= \Vert u_n\Vert \int _{\Omega } \frac{f(x,u_n)}{u_n} v_n \varphi \, dx \\ {}&= \Vert u_n\Vert \left( \int _{\textrm{supp}\,\varphi \cap \textrm{supp}\,v} \frac{f(x,u_n)}{u_n} v_n \varphi \, dx + o(1) \right) . \end{aligned}$$

  Observe that for almost every $x \in \textrm{supp}\,\varphi \cap \textrm{supp}\,v$, we obtain that ${|u_n(x)| = |v_n(x)| \Vert u_n\Vert \rightarrow +\infty }$. From (F5)

  $$\begin{aligned} \frac{f(x,u_n(x))}{u_n(x)} v_n(x) \varphi (x) \rightarrow \Theta (x) v_0(x) \varphi (x) \end{aligned}$$

  pointwise, a.e. on $\textrm{supp}\,\varphi \cap \textrm{supp}\,v$. Combining (F4) and (F5), we also get that

  $$\begin{aligned} \left| \frac{f(x,u_n(x))}{u_n(x)} \right| ^2 \le |\Theta |_\infty ^2 \end{aligned}$$

  and $\frac{f(\cdot ,u_n)}{u_n} \rightarrow \Theta$ in $L^2(\textrm{supp}\,\varphi \cap \textrm{supp}\,v)$. Thus, from Lebesgue dominated convergence theorem and the Hölder inequality, we get

  $$\begin{aligned} \int _{\textrm{supp}\,\varphi \cap \textrm{supp}\,v} \frac{f(x,u_n)}{u_n} v_n \varphi \, dx \rightarrow \int _{\Omega } \Theta (x) v \varphi \, dx. \end{aligned}$$

  Hence

  $$\begin{aligned} \langle v, \varphi \rangle = \int _\Omega \Theta (x) v \varphi \, dx. \end{aligned}$$

  In particular, 0 is an eigenvalue of the operator $-\Delta + \lambda - \frac{\mu }{|x|^2} - \frac{\nu }{d(x)^2} - \Theta (x)$ with Dirichlet boundary conditions on $\Omega \setminus K$, which contradicts (A).

$\square$

Proof of Theorem 1.1

Since Cerami sequence $u_n$ is bounded, we have following convergences (up to a subsequence):

$$\begin{aligned} u_n\rightharpoonup u_0 \quad&\text{ in } H^1_0(\Omega ),\\ u_n\rightarrow u_0 \quad&\text{ in } L^2(\Omega ), \text{ and } \text{ in } L^p(\Omega ),\\ u_n\rightarrow u_0\quad&\text{ a.e. } \text{ on } \Omega . \end{aligned}$$

Hence, for any $\varphi \in {{\mathcal {C}}}^\infty _0(\Omega )$

$$\begin{aligned} {{\mathcal {J}}}'(u_n)(\varphi )-{{\mathcal {J}}}'(u_0)(\varphi )=\langle u_n-u_0,\varphi \rangle -\int _\Omega \left( f(x,u_n)-f(x,u_0)\right) \varphi \,dx\rightarrow 0, \end{aligned}$$

because obviously weak convergence of $u_n$ implies that

$$\begin{aligned} \langle u_n-u_0,\varphi \rangle \rightarrow 0, \end{aligned}$$

and we will use the Vitali convergence theorem to prove that

$$\begin{aligned} \int _\Omega \left( f(x,u_n)-f(x,u_0)\right) \varphi \,dx\rightarrow 0. \end{aligned}$$

Hence, we need to check the uniform integrability of the following family $\left\{ \left( f(x,u_n)-f(x,u_0)\right) \varphi \right\} _n$. Using (F1) and Lemma 5.1 we obtain that for any measurable set $E \subset \Omega$

$$\begin{aligned}&\int _E|f(x,u_n)-f(x,u_0)|\varphi \,dx \le \int _E|f(x,u_n)\varphi |\,dx+\int _E|f(x,u_0)\varphi |\,dx\\&\quad \lesssim \int _E|\varphi |\,dx+\int _E|u_n|^{p-1}|\varphi |\,dx+\int _E|\varphi |\,dx+\int _E|u_0|^{p-1}|\varphi |\,dx\\&\quad \lesssim |\varphi \chi _E|_1+ |u_n|_p^{p-1} |\varphi \chi _E|_p +|u_0|_p^{p-1} |\varphi \chi _E|_p \\&\quad \lesssim |\varphi \chi _E|_1+|\varphi \chi _E|_p. \end{aligned}$$

Then, for any $\varepsilon > 0$, we can choose $\delta > 0$ small enough that

$$\begin{aligned} \int _E|f(x,u_n)-f(x,u_0)|\varphi \,dx < \varepsilon \end{aligned}$$

for $|E|<\delta$. Hence ${{\mathcal {J}}}'(u_n)(\varphi )\rightarrow {{\mathcal {J}}}'(u_0)(\varphi )$, and ${{\mathcal {J}}}'(u_0)=0$. $\square$

Proof of Theorem 1.2

The statement follows directly from Theorem 3.2 and Lemma 4.1. $\square$

6 Multiple solutions to the mass-subcritical normalized problem

In what follows we are interested in the normalized problem (1.3), where $\lambda$ is not prescribed anymore and is the part of the unknown $(\lambda , u) \in \mathbb {R}\times H^1_0(\Omega )$. Then, solutions are critical point of the energy functional

$$\begin{aligned} {{\mathcal {J}}}_0 (u) := \frac{1}{2} |\nabla u|_2^2 - \frac{\mu }{2} \int _\Omega \frac{u^2}{|x|^2} \, dx - \frac{\nu }{2} \int _\Omega \frac{u^2}{d(x)^2} \, dx - \int _\Omega F(x,u) \, dx \end{aligned}$$

restricted to the $L^2$-sphere in $H^1_0(\Omega )$

$$\begin{aligned} {{\mathcal {S}}}:= \left\{ u \in H^1_0(\Omega ) \ : \ \int _\Omega u^2 \, dx = \rho \right\} \end{aligned}$$

and $\lambda$ arises as a Lagrange multiplier.

We recall the well-known Gagliardo-Nirenberg inequality

$$\begin{aligned} |u|_p \le C_{p,N} |\nabla u|_2^{\delta _p} |u|_2^{1-\delta _p}, \quad u \in H^1_0(\Omega ), \end{aligned}$$

(6.1)

where $\delta _p := N \left( \frac{1}{2} - \frac{1}{p} \right)$ and $C_{p,N} > 0$ is the optimal constant.

Lemma 6.1

${{\mathcal {J}}}_0$ is coercive and bounded from below on ${{\mathcal {S}}}$.

Proof

Using (F1), (2.1), (2.2), and (6.1), we obtain

$$\begin{aligned} {{\mathcal {J}}}_0(u)&= \frac{1}{2} |\nabla u|_2^2 - \frac{\mu }{2} \int _\Omega \frac{u^2}{|x|^2} \, dx - \frac{\nu }{2} \int _\Omega \frac{u^2}{d(x)^2} \, dx - \int _\Omega F(x,u) \, dx \\&\ge \frac{1}{2} \left( 1 - \frac{4\mu }{(N-2)^2} - 4\nu \right) |\nabla u|_2^2 -C_1 |\Omega | - C_1|u|_p^p \\&\ge \frac{1}{2} \left( 1 - \frac{4\mu }{(N-2)^2} - 4\nu \right) |\nabla u|_2^2 - C_1|\Omega |- C\left( |\nabla u|_2^{\delta _p} |u|_2^{1-\delta _p} \right) ^p\\&\ge \frac{1}{2} \left( 1 - \frac{4\mu }{(N-2)^2} - 4\nu \right) |\nabla u|_2^2 -C_1|\Omega |+ C|\nabla u|_2^{\delta _pp}, \end{aligned}$$

where

$$\begin{aligned} \delta _p p = N \left( \frac{1}{2} - \frac{1}{p} \right) p = N \left( \frac{p}{2}-1 \right) < N \cdot \frac{2}{N} = 2. \end{aligned}$$

Thus, ${{\mathcal {J}}}_0$ is coercive and bounded from below on ${{\mathcal {S}}}$. $\square$

Lemma 6.2

${{\mathcal {J}}}_0$ satisfies the Palais-Smale condition on ${{\mathcal {S}}}$, i.e., any Palais–Smale sequence for ${{\mathcal {J}}}_0 |_{{\mathcal {S}}}$ has a convergent subsequence.

Proof

Let $(u_n) \subset {{\mathcal {S}}}$ be a Palais–Smale sequence for ${{\mathcal {J}}}_0 |_{{\mathcal {S}}}$. Then, Lemma 6.1 implies that $(u_n)$ is bounded in $H_0^1 (\Omega )$. Hence, we may assume that (up to a subsequence)

$$\begin{aligned} u_n \rightharpoonup u \quad&\text{ in } H^1_0(\Omega ), \\ u_n \rightarrow u \quad&\text{ in } L^p (\Omega ), \\ u_n \rightarrow u \quad&\text{ a.e. } \text{ on } \Omega . \end{aligned}$$

Moreover

$$\begin{aligned} {{\mathcal {J}}}_0'(u_n) + \lambda _n u_n \rightarrow 0 \quad \text{ in } H^{-1}(\Omega ) := (H^1_0 (\Omega ))^* \end{aligned}$$

for some $\lambda _n \in \mathbb {R}$. In particular

$$\begin{aligned} {{\mathcal {J}}}_0'(u_n)(u_n) + \lambda _n |u_n|_2^2 \rightarrow 0. \end{aligned}$$

Note that

$$\begin{aligned} \lambda _n = - \frac{{{\mathcal {J}}}_0'(u_n)(u_n)}{|u_n|_2^2} + o(1)= - \frac{\Vert u_n\Vert ^2 - \int _\Omega f(x,u_n)u_n \, dx}{|u_n|_2^2} + o(1). \end{aligned}$$

Observe that from (F1)

$$\begin{aligned} \left| \int _\Omega f(x,u_n)u_n \, dx \right| \lesssim 1 + |u_n|_p^p \lesssim 1. \end{aligned}$$

Therefore, $(\lambda _n) \subset \mathbb {R}$ is bounded, and (up to a subsequence) $\lambda _n \rightarrow \lambda _0$. Therefore, up to a subsequence

$$\begin{aligned} o(1)&= {{\mathcal {J}}}_0'(u_n)(u_n) + \lambda _n |u_n|_2^2 - {{\mathcal {J}}}_0'(u_n)(u) - \lambda _n \int _{\Omega } u_n u \, dx \\&= {{\mathcal {J}}}_0'(u_n)(u_n) - {{\mathcal {J}}}_0'(u_n)(u) + \lambda _n \int _{\Omega } u_n (u_n-u) \, dx \\&= \Vert u_n\Vert ^2 - \langle u_n, u \rangle - \int _{\Omega } f(x,u_n) (u_n-u) \, dx + \lambda _n \int _\Omega u_n (u_n-u) \, dx. \end{aligned}$$

It is clear that $\langle u_n, u\rangle \rightarrow \Vert u\Vert ^2$ and that

$$\begin{aligned} \left| \lambda _n \int _\Omega u_n (u_n-u) \, dx \right| \lesssim |u_n|_2^2 |u_n-u|_2^2 \rightarrow 0. \end{aligned}$$

Moreover, from (F1)

$$\begin{aligned} \left| \int _\Omega f(x,u_n)(u_n-u) \, dx \right| \lesssim |u_n-u|_2 + |u_n|_p^{p-1} |u_n-u|_p \rightarrow 0. \end{aligned}$$

Hence, $\Vert u_n\Vert \rightarrow \Vert u\Vert$ and, therefore, $u_n \rightarrow u$ in $H^1_0(\Omega )$. $\square$

Proof of Theorem 1.3

From [22, Theorem II.5.7], we obtain that ${{\mathcal {J}}}_0$ has at least ${\hat{\gamma }}({{\mathcal {S}}})$ critical points, where

$$\begin{aligned} {\hat{\gamma }} ({{\mathcal {S}}}) := \sup \{ \gamma (K) \ : \ K \subset {{\mathcal {S}}} \text{- } \text{ symmetric } \text{ and } \text{ compact } \}, \end{aligned}$$

and $\gamma$ denotes the Krasnoselskii’s genus for symmetric and compact sets. We will show that ${\hat{\gamma }}({{\mathcal {S}}}) = +\infty$. Indeed, fix $k \in \mathbb {N}$. It is sufficient to construct a symmetric and compact set $K \subset {{\mathcal {S}}}$ with $\gamma (K) = k$. Choose functions $w_1, w_2, \ldots , w_k \in {{\mathcal {C}}}_0^\infty (\Omega ) \cap {{\mathcal {S}}}$ with pairwise disjoint supports, namely, $w_i w_j = 0$ for $i \ne j$. Now, we set

$$\begin{aligned} K := \left\{ \sum _{i=1}^k \alpha _i w_i \in {{\mathcal {S}}}\ : \ \sum _{i=1}^k \alpha _i^2 = 1 \right\} . \end{aligned}$$

It is clear that $K \subset {{\mathcal {S}}}$ is symmetric and compact. We will show that $\gamma (K) = k$. In what follows $S^{m-1}$ denotes the $(m-1)$-dimensional sphere in $\mathbb {R}^m$ of radius 1 centered at the origin. Note that $h : K \rightarrow S^{k-1}$ given by

$$\begin{aligned} K \ni \sum _{i=1}^k \alpha _i w_i \mapsto h \left( \sum _{i=1}^k \alpha _i w_i \right) := (\alpha _1, \ldots , \alpha _k) \in S^{k-1} \end{aligned}$$

is a homeomorphism, which is odd. Hence, $\gamma (K) \le k$. Suppose by contradiction that $\gamma (K) < k$. Then, there is a continuous and odd function $\widetilde{h} : K \rightarrow S^{\gamma (K) -1}$. However, $\widetilde{h} \circ h^{-1} : S^{k-1} \rightarrow S^{\gamma (K)-1}$ is an odd, continuous map, which contradicts the Borsuk–Ulam theorem [22, Proposition II.5.2], [25, Theorem  D.17]. Hence, $\gamma (K) = k$. $\square$