On $p$-Laplacian Kirchhoff-Schrödinger-Poisson type systems with critical growth on the Heisenberg group

1.   Introduction

In this article, we investigate the following Kirchhoff-Schrödinger-Poisson type systems on the Heisenberg group:

where $a, b$ are positive real numbers, $\Omega\subset \mathbb{H}^N$ is a bounded region with smooth boundary, $1 < p < Q$, $Q = 2N + 2$ is the homogeneous dimension of the Heisenberg group $\mathbb{H}^N$, $Q^{\ast} = \frac{pQ}{Q-p}$, $q\in(2p, Q^{\ast})$, $\Delta_{H, p}u = \mbox{div}(|\nabla_{H} u|^{p-2}\nabla_{H} u)$ is known as the $p$-horizontal Laplacian, and $\mu$ and $\lambda$ are some positive real parameters.

In recent years, geometrical analysis of the Heisenberg group has found significant applications in quantum mechanics, partial differential equations and other fields, which has attracted the attention of many scholars who tried to establish the existence and multiplicity of solutions of partial differential equations on the Heisenberg group. For instance, in the subcase of problem (1.1), when $p = 2$ and $b = \mu = 0$, the existence of solutions for some nonlinear elliptic problems in bounded domains has been established. Tyagi ^[1] studied a class of singular boundary value problem on the Heisenberg group:

and under appropriate conditions, obtained some existence results using Bonanno's three critical point theorem. Goel and Sreenadh ^[2] dealt with a class of Choquard type equation on the Heisenberg group, and they established regularity of solutions and nonexistence of solutions invoking the mountain pass theorem, the linking theorem and iteration techniques and boot-strap method.

In the case $b\neq 0$ and $\mu = 0$, problem (1.1) becomes the Kirchhoff problem, which has also been widely studied. For example, Sun et al. ^[3] dealt with the following Choquard-Kirchhoff problem with critical growth:

where $f$ is a Carathéodory function, $M$ is the Kirchhoff function, $\Delta_{H}$ is the Kohn Laplacian on the Heisenberg group $\mathbb{H}^N$, $\mu > 0$ is a parameter and $Q_\lambda^\ast = \frac{2Q-\lambda}{Q-2}$ is the critical exponent. In their paper, a new version of the concentration-compactness principle on the Heisenberg group was established for the first time. Moreover, the existence of nontrivial solutions was obtained even under nondegenerate and degenerate conditions. Zhou et al. ^[4] proved the existence of solutions of Kirchhoff type nonlocal integral-differential operators with homogeneous Dirichlet boundary conditions on the Heisenberg group using the variational method and the mountain pass theorem. Deng and Xian ^[5] obtained the existence of solutions for Kirchhoff type systems involving the $Q$-Laplacian operator on the Heisenberg group with the help of the Trudinger-Moser inequality and the mountain pass theorem. For more related results, see ^{[6,7,8,9,10,11,12,13]}.

When $b = 0$, $p = 2$ and $\mu \neq 0$, problem (1.1) becomes the Schrödinger-Poisson system. This is a very interesting subject and has recently witnessed very profound results. For example, An and Liu ^[14] dealt with the following forms of Schrödinger-Poisson type system on the Heisenberg group:

where $\mu\in \mathbb{R}$ and $\lambda > 0$ are some real parameters and $1 < q < 2$. By applying the concentration compactness and the critical point theory, they found at least two positive solutions and a positive ground state solution.

Liang and Pucci ^[15] studied the following critical Kirchhoff-Poisson system on the Heisenberg group:

where $\Omega\subset \mathbb{H}^1$ is a smooth bounded domain, $\Delta_H$ is the Kohn-Laplacian on the first Heisenberg group $\mathbb{H}^1$ and $1 < q < 2$. By applying the symmetric mountain pass lemma, they obtained the multiplicity of solutions with $\lambda$ sufficiently small.

The Kirchhoff-Poisson system on the Heisenberg group with logarithmic and critical nonlinearity was considered by Pucci and Ye ^[13]:

Under suitable assumptions on the Kirchhoff function $M$ covering the degenerate case, they showed that for a sufficiently large $\lambda > 0,$ there exists a nontrivial solution to the above problem.

When $p\neq2$ and $\mu \neq 0$, as far as we know, for Kirchhoff-Schrodinger-Poisson systems (1.1) with critical nonlinearities on the Heisenberg group, existence and multiplicity results are not yet available. In the Euclidean case, Du et al. ^[16] first studied the existence results for the Kirchhoff-Poisson systems with $p$-Laplacian under the subcritical case using the mountain pass theorem. Later, Du et al. ^[17] studied quasilinear Schrödinger-Poisson systems. For the critical case, Du et al. ^[18] also obtained the existence of ground state solutions with the variational approach.

Inspired by the above achievements, we aim to establish some results on the existence and multiplicity of nontrivial solutions of the Kirchhoff-Schrödinger-Poisson systems (1.1). The major difficulties in dealing with problem (1.1) are the presence of a nonlocal term and critical nonlinearities making the study of this problem very challenging.

Before presenting the main results of this article, we first present some concepts of the Heisenberg group. The Heisenberg group is represented by $\mathbb{H}^N$. If $\xi = (x, y, t)\in\mathbb{H}^N$, then the definition of this group operation is

$\xi^{-1} = -\xi$ is the inverse, and therefore $(\xi')^{-1}\circ {\xi}^{-1} = (\xi\circ\xi')^{-1}$.

The definition of a natural group of dilations on $\mathbb{H}^N$ is $\delta_s(\xi) = (sx, sy, s^2t)$, for every $s > 0$. Hence, $\delta_s(\xi_0\circ\xi) = \delta_s(\xi_0)\circ \delta_s(\xi)$. It can be easily proved that the Jacobian determinant of dilatations $\delta_s: \mathbb{H}^N\to \mathbb{H}^N$ is constant and equal to $s^Q,$ for every $\xi = (x, y, t)\in \mathbb{H}^N$. The critical exponent is $Q^{*}: = \frac{pQ}{Q-p}$, where the natural number $Q = 2N+2$ is called the homogeneous dimension of $\mathbb{H}^N$. We define the Korányi norm as follows

and we derive this norm from the Heisenberg group's anisotropic dilation. Hence, the homogeneous degree of the Korányi norm is equal to 1, in terms of dilations

The set

denotes the Korányi open ball of radius $\mbox{r}$ centered at $\xi_{0}$. For the sake of simplicity, we shall denote $B_{r} = B_{r}(O)$, where $O = (0, 0)$ is the natural origin of $\mathbb{H}^{N}$.

The following vector fields

generate the real Lie algebra of left invariant vector fields for $j = 1, \cdots, n$, which forms a basis satisfying the Heisenberg regular commutation relation on $\mathbb{H}^{N}$. This means that

The so-called horizontal vector field is just a vector field with the span of $[X_{j}, Y_{j}]^{n}_{j = 1}$.

The Heisenberg gradient on $\mathbb{H}^{N}$ is

and the Kohn Laplacian on $\mathbb{H}^{N}$ is given by

The Haar measure is invariant under the left translations of the Heisenberg group and is $Q$-homogeneous in terms of dilations. More precisely, it is consistent with the $(2n + 1)$-dimensional Lebesgue measure. Hence, as shown Leonardi and Masnou ^[19], the topological dimension $2N + 1$ of $\mathbb{H}^{N}$ is strictly less than its Hausdorff dimension $Q = 2N + 2$. Next, $|\Omega|$ denotes the (2N + 1)-dimensional Lebesgue measure of any measurable set $\Omega\subseteq \mathbb{H}^{N}$. Hence,

Now, we can state the main result of the paper.

Theorem 1.1. Let $q\in(2p, Q^{\ast})$. Then there exist positive constants $\mu_1$ and $\lambda_1$ such that for every $\mu\in(0, \mu_1)$ and $\lambda \in (\lambda_1, +\infty)$, the following assertions hold:

$(I)$ Problem (1.1) has a nontrivial weak solution;

$(II)$ Problem (1.1) has infinitely many nontrivial weak solutions if parameter $a$ is large enough.

We can give the following example for problem (1.1) with $p = 3$ and $\Omega \subset \mathbb{H}^1$:

In this case, $N = 1$, $p = 3$ and $q = 8$, then $Q = 2N+2 = 4$, $Q^\ast = 12$. If positive parameters $\mu$ small enough and $\lambda$ large enough, by Theorem 1.1, we know that problem (1.1) has a nontrivial weak solution. Moreover, if in addition the parameter $a$ is large enough, problem (1.1) has infinitely many nontrivial weak solutions. It should be noted that the methods in An and Liu ^[14] and Liang and Pucci ^[15] do not seem to apply to problem (1.1).

Remark 1.1. Compared with previous results, this paper has the following key new features:

$1)$ The presence of the nonlocal term $\phi|u|^{p-2}u$;

$2)$ The lack of compactness caused by critical index;

$3)$ The presence of the $p$-Laplacian makes this problem more complex and interesting.

It is worth stressing that the nonlocal term and the critical exponent lead to the lack of compactness condition, and we use the concentration-compactness principle to overcome this difficulty. Moreover, we shall use some more refined estimates to overcome the presence of the $p$-Laplacian.

We need to emphasize here that despite the similarity of some properties between the classical Laplacian $\Delta$ and Kohn Laplacian $\Delta_{H}$, similarities can be misleading (see Garofalo and Lanconelli ^[20]), so there are still many properties that deserve further study. Moreover, for the case $p\neq 2$, it is difficult to prove the boundedness of Palais-Smale sequences. In order to overcome these difficulties, we use some more accurate estimates of relevant expression. Additionally, we use the concentration-compactness principle on the Heisenberg group to prove the compactness condition.

The paper is organized as follows. In Section 2, we introduce some notations and known facts. Moreover, we introduce some key estimates. In addition, we define the corresponding energy functional $I_{\lambda}$ and its derivative at $u$, that is, $I_{\lambda}'(u)$. In Section 3, we prove Theorem 1.1.

2.   Preliminaries and $(PS)_c$ condition

2.1.   Preliminaries

First of all, we collected some known facts, useful in the sequel. For additional background material, readers are advised to refer to Papageorgiou et al. ^[21].

Let

represent the usual $L^s$-norm.

Following Folland and Stein ^[22], we define the space $\mathring{S}_1^2(\Omega)$ as the closure of $C_{0}^{\infty}(\Omega)$ in $S_1^2(\mathbb{H}^{N})$. Then $\mathring{S}_1^2(\Omega)$ is a Hilbert space with the norm

We define the Folland-Stein space $S^{1, p}(\Omega)$ as the closure of $C_0^{\infty}(\Omega)$ with the norm

Then the embedding

is compact. However, if $s = Q^{*}$, the embedding is only continuous (see Vassiliev ^[23]).

Additionally, we say that $(u, \phi)\in S^{1, p}(\Omega)\times S^{1, p}(\Omega)$ is a solution of problem (1.1) if and only if

and

for every $v, \omega \in S^{1, p}(\Omega)\times S^{1, p}(\Omega)$. Moreover, $(u, \phi)\in S^{1, p}(\Omega)\times S^{1, p}(\Omega)$ is a positive solution of problem (1.1) if $u$ and $\phi$ are both positive. Therefore, in order to apply the critical point theory, we need to define the functional $J(u, \phi): S^{1, p}(\Omega)\times S^{1, p}(\Omega)\rightarrow \mathbb{R}$ as follows

for every $(u, \phi)\in S^{1, p}(\Omega)\times S^{1, p}(\Omega)$. Then $J$ is $C^{1}$ on $S^{1, p}(\Omega)\times S^{1, p}(\Omega)$ and its critical points are the solutions of problem (1.1). Indeed, the partial derivatives of $J$ at $(u, \phi)$ are denoted by $J_u'(u, \phi)$, $J_\phi'(u, \phi)$, namely for every $v, \omega \in S^{1, p}(\Omega)\times S^{1, p}(\Omega)$,

and

Standard computations show that $J_u'$ (respectively $J_\phi'$) continuously maps $S^{1, p}(\Omega)\times S^{1, p}(\Omega)$ into the dual of $S^{1, p}(\Omega)$. Moreover, the functional $J$ is $C^{1}$ on $S^{1, p}(\Omega)\times S^{1, p}(\Omega)$ and

if and only if $(u, \phi)$ is a solution of problem (1.1).

Lemma 2.1. Let $u\in S^{1, p}(\Omega)$. Then there is a unique nonnegative function $\phi_u\in \mathring{S}_1^2(\Omega)$ such that

Furthermore, $\phi_u\geq0$ and $\phi_u > 0$ if $u\neq0$. Also,

(i) $\phi_{tu} = t^{p}\phi_u,$ for every $t > 0$;

(ii) $\|\phi_u\|_{\mathring{S}_1^2(\Omega)}\leq \hat{C}\|u\|^{p}$, where $\hat{C} > 0$;

(iii) Let $u_n\rightharpoonup u$ in $S^{1, p}(\Omega)$. Then, $\phi_{u_{n}}\rightharpoonup\phi_u$ in $\mathring{S}_1^2(\Omega)$, and

Proof. For any $u\in\mathring{S}_1^2(\Omega)$, we define $W: \mathring{S}_1^2(\Omega)\rightarrow \mathbb{R}$,

Let $v_n\rightarrow v\in\mathring{S}_1^2(\Omega),$ as $n\rightarrow \infty$. It follows by the Hölder inequality that

where

is the best Sobolev constant. This implies that $W$ is a continuous linear functional. Using the Lax-Milgram theorem, we see that there is a unique $\phi_u\in\mathring{S}_1^2(\Omega)$ satisfying

Thus, $\phi_u\in\mathring{S}_1^2(\Omega)$ is the unique solution of problem (2.1). Moreover, applying the maximum principle, one has $\phi_u\geq0$ and $\phi_u > 0$ if $u\neq0$. Indeed, for every $t > 0$, one has

Hence $\phi_{tu} = t^{p}\phi_{u}$ due to the uniqueness of $\phi_{u}$.

Furthermore, since $\phi_{u}\in S^{1, p}(\Omega)$, we can view it as a text function in problem (2.1). Then by (2.4), the Sobolev inequality and the Hölder inequality, we have (henceforth $C_0$, $C_1$, $C_2$ will denote positive constants)

Therefore, we get $\|\phi_u\|_{\mathring{S}_1^2(\Omega)}\leq C_1\|u\|^{p}$.

Since $u_n\rightharpoonup u$ in $S^{1, p}(\Omega)$, we can conclude that $u_n\rightarrow u$ a.e. in $\Omega$ and $\{|u_n|^{p}\}$ is bounded in $L^{2}(\Omega)$. Moreover, we have $|u_n|^{p}\rightharpoonup|u|^{p}$ in $L^{2}(\Omega)$. Then for every $v\in\mathring{S}_1^2(\Omega)$, it follows that

Therefore, $\phi_{u_{n}}\rightharpoonup\phi_u$ in $\mathring{S}_1^2(\Omega)$. By the Hölder inequality, the Sobolev inequality and (ⅱ), one has

Hence, $\{\phi_{u_n} |u_{n}|^{p-2}u_n\}$ is bounded in $L^{\frac{2p}{2p-1}}(\Omega)$. Since

we get

The proof of Lemma 2.1 is complete.

By similar arguments as in An and Liu ^[14], we can get the following result.

Lemma 2.2. Let $\Psi(u) = \phi_u$ for every $u\in S^{1, p}(\Omega)$, where $\phi_u$ is as in Lemma 2.1, and let

Then $\Psi$ is $C^{1}$ and $\Upsilon$ is the graph of $\Psi$.

We define the corresponding energy functional $I_{\lambda}(u) = J(u, \phi_u)$ of problem (1.1) by

Based on the definition of $J$ and Lemma 2.2, we can conclude that $I_\lambda$ is of $C^{1}$.

Lemma 2.3. (see An and Liu ^[14]) Let $(u, \phi)\in S^{1, p}(\Omega)\times S^{1, p}(\Omega)$. Then $(u, \phi)$ is a critical point of $J$ if and only if $u$ is a critical point of $I_{\lambda}$ and $\phi = \Psi(u)$, where $\Psi$ was defined in Lemma 2.2.

According to Lemma 2.3, we know that a solution $(u, \phi_{u})$ of problem (1.1) corresponds to a critical point $u$ of the functional $I_\lambda$ with $\phi = \Psi(u)$ and

Therefore, based on the above arguments, we shall strive to use critical point theory and some analytical techniques to prove the existence of critical points of functional $I_\lambda$.

2.2.   $(PS)_c$ condition

In this subsection, our main focus will be on proving that the functional $I_{\lambda}$ satisfies the Palais-Smale condition.

Lemma 2.4. Let $q\in(2p, Q^{\ast})$. Then there exists $\mu_1 > 0$ such that for any $\mu < \mu_1$, the energy functional $I_\lambda$ satisfies $(PS)_c$ condition, where

and $S$ is the best Sobolev constant given by (2.3).

Proof. Let us assume that $\{u_{n}\}_{n}\subset S^{1, p}(\Omega)$ is a $(PS)_{c}$ sequence related to the functional $I_{\lambda}$, that is,

It follows that

where $\hat{C}$ is a positive constant given by Lemma 2.1(ⅱ). Let $\mu_1 = \frac{b}{\hat{C}}$. By (2.9), we know that $(PS)_{c}$ sequence $\{u_{n}\}_{n}\subset S^{1, p}(\Omega)$ is bounded for every $\mu < \mu_1$. Thus, we may assume that $u_n\rightharpoonup u$ weakly in $S^{1, p}(\Omega)$, and $u_n\rightarrow u$ in $L^{s}(\Omega)$ with $1 < s < Q^{*}$. Furthermore, since $I_\lambda(u_n) = I_\lambda(|u_n|)$, we may also assume that $u_n\geq0$ and $u\geq0$. Therefore, invoking the concentration compactness principle on the Heisenberg group (see Vassiliev [23, Lemma 3.5]), we obtain

where $\{x_j\}_{j\in\Lambda}\subset\Omega$ is the most a countable set of distinct points, $\omega$ and $\nu$ in $\mathbb{H}^{N}$ are two positive Radon measures, and $\{\omega_j\}_{j\in\Lambda}$, $\{\nu_j\}_{j\in\Lambda}$ are nonnegative numbers. Moreover, we have

Next, we shall show that $\Lambda = \emptyset$. Indeed, assume that the hypothesis $\omega_j\neq0$ holds for some $j\in\Lambda$. Then when $\varepsilon > 0$ is sufficiently small, we can find $0\leq\psi_{\varepsilon, j}\leq1$ satisfying the following

where $\psi_{\varepsilon, j}\in C_{0}^{\infty}(B_{H}(\xi_{j}, \varepsilon))$ is a cut-off function. Clearly, $(u_n\psi_{\varepsilon, j})_n$ is bounded in $S^{1, p}(\Omega)$. It follows from (2.8) and the boundedness of $(u_n\psi_{\varepsilon, j})_n$ that

that is,

It follows from the dominated convergence theorem that

Hence, letting $\varepsilon\rightarrow0$, we get

By Lemma 2.1,

and since $u_n\rightarrow u$ in $L^{s}(\Omega)$ with $1 < s < Q^{*}$, one has

Combining (2.15) with (2.16), we obtain that

thus

Since

applying the Hölder inequality, we obtain

By (2.10), we have

and

Therefore, by (2.13)–(2.19), one gets $\nu_j\geq a\omega_j$. It follows from (2.11) that

In fact, if $\nu_j\geq (aS)^{\frac{Q}{p}}$ holds, therefore by (2.8) and (2.10), for every $\mu < \mu_1$, we have

which contradicts (2.7). Thus, $\Lambda = \emptyset$. By (2.10) and $\Lambda = \emptyset$, we have

Let

If $A = 0$, then $u_n\rightarrow0$ in $S^{1, p}(\Omega)$. So assume now that $A > 0$. By (2.8), we get

Let $v = u$ in (2.22). Then

By (2.8), (2.10), (2.21) and Lemma 2.1, one also has

Thus, combining (2.23) and (2.24), we get $\lim_{n\rightarrow \infty}\|u_n\|^{p} = \|u\|^{p}$. Thus, we see that $u_n\rightarrow u$ in $S^{1, p}(\Omega)$ by the uniform convexity of $S^{1, p}(\Omega)$. This completes the proof of Lemma 2.4.

3.   Proof of Theorem 1.1

3.1.   Existence of a nontrivial weak solutions

We need the following auxiliary lemmas to prove our main result.

Lemma 3.1. Let $q\in(2p, Q^{\ast})$ and $\mu\in(0, \mu_1)$. Then functional $I_{\lambda}$ satisfies the mountain pass geometry, that is,

(i) There exist constants $\rho, \alpha > 0$ satisfying $I_{\lambda}(u)|_{\partial B_{\rho}}\geq \alpha,$ for every $u\in S^{1, p}(\Omega)$;

(ii) There exists $e\in S^{1, p}(\Omega)\backslash\overline{B_{\rho}}$ satisfying $I_\lambda(e) < 0$.

Proof. First, applying the Hölder inequality, we get

Let

We now show that there exists a constant $\rho > 0$ satisfying $f(\rho)\geq \frac{a}{p}$. We see that $f$ is a continuous function on $[0, +\infty)$ and $\lim_{t\rightarrow0^{+}}f(t) = \frac{a}{p}.$ Hence there exists $\rho$ such that $f(t)\geq \frac{a}{p}-\varepsilon_1,$ for every $0\leq t\leq \rho$, where $\rho$ is small enough such that $\|u\| = \rho$. If we choose $\varepsilon_1 = \frac{a}{2p}$, we have $f(t)\geq \frac{a}{2p},$ for every $0\leq t\leq \rho$. In particular, $f(\rho)\geq \frac{a}{2p}$ and we obtain $I_{\lambda}(u)\geq \frac{a}{2p}\rho^{p} = \alpha$ for $\|u\| = \rho$. Hence assertion (i) of Lemma 3.1 holds.

Next, we shall show that assertion (ⅱ) of Lemma 3.1 also holds:

Thus, we can deduce that $I_{\lambda}(s_{0}u) < 0$ and $s_{0}\|u\| > \rho,$ for every $s_{0}$ large enough. Let $e = s_{0}u$. Then $e$ is the desired function and the proof of (ⅱ) of Lemma 3.1 is complete.

Proof of Theorem 1.1(I). We claim that

where

Indeed, we can choose $v_{1}\in S^{1, p}(\Omega)\backslash\{0\}$ with $\|v_{1}\| = 1$. From (3.2), we have $\lim_{s\rightarrow +\infty}I_{\lambda}(sv_{1}) = -\infty.$ Then

So $s_{\lambda}$ satisfies

Next, we shall prove that $\{s_{\lambda}\}_{\mu > 0}$ is bounded. In fact, suppose that the following hypothesis $s_{\lambda}\geq1$ is satisfied for every $\lambda > 0$. Then it follows from (3.4) that

Since $2p < q < Q^{\ast}$, we can deduce that $\{s_{\lambda}\}_{\lambda > 0}$ is bounded.

Next, we shall demonstrate that $s_{\lambda}\rightarrow0,$ as $\lambda\rightarrow \infty$. Suppose to the contrary, that there exist $s_{\lambda} > 0$ and a sequence $(\lambda_{n})_{n}$ with $\lambda_{n}\rightarrow \infty,$ as $n\rightarrow \infty,$ satisfying $s_{\lambda_{n}}\rightarrow s_{\lambda},$ as $n\rightarrow \infty$. Invoking the Lebesgue dominated convergence theorem, we see that

It now follows that

Thus, invoking (3.4), we can show that this cannot happen. Therefore, $s_{\lambda}\rightarrow0,$ as $\lambda\rightarrow \infty$.

Furthermore, (3.4) implies that

and

Hence based on the definition of $I_{\lambda}$ and $s_{\lambda}\rightarrow0,$ as $\lambda\rightarrow \infty$, we get that

So, there is $\lambda_{1} > 0,$ satisfying for every $\lambda > \lambda_{1}$,

Letting $e = t_{1}v_{1}$ with $t_{1}$ large enough for $I_{\lambda}(e) < 0$, we get

Therefore

for $\lambda$ large enough. This completes the proof of Theorem 1.1(Ⅰ).

3.2.   Existence of infinitely many nontrivial weak solutions

In this subsection, we shall use the Krasnoselskii genus theory to prove Theorem 1.1(Ⅱ). To this end, let $E$ be a Banach space and denote by $\Lambda$ the class of all closed subsets $A \subset E\backslash \{0\}$ that are symmetric with respect to the origin, that is, $u\in E$ implies $-u\in E$. Moreover, suppose that $X$ is $k$-dimensional and $X = \hbox{span}\{z_{1}, \cdots, z_{k}\}$. For every $n\geq k$, inductively select $z_{n+1}\notin X_{n} = \hbox{span}\{z_{1}, \cdots, z_{n}\}$. Let $R_n = R(Z_n)$ and $\Upsilon_n = B_{R_{n}}\bigcap Z_n$. Define

and

where $\gamma(V)$ is the Krasnoselskii genus of $V$.

Theorem 3.1. (see Rabinowitz [24, Theorem 9.12]) Let $I\in C^{1}(E, \mathbb{R})$ be even with $I(0) = 0$ and let E be an infinite-dimensional Banach space. Assume that $X$ is a finite-dimensional space, $E = X\oplus Y$ and that I satisfies the following properties:

(i) There exists $\theta > 0$ such that $I$ satisfies $(PS)_{c}$ condition, for every $c\in(0, \theta)$;

(ii) There exist $\rho, \alpha > 0$ satifying $I(u)\geq \alpha,$ for every $u\in\partial B_{\rho}\bigcap Y$;

(iii) For every finite-dimensional subspace $\tilde{E}\subset E$, there exists $R = R(\tilde{E}) > \rho$ such that $I(u)\leq0$ on $\tilde{E}\setminus B_{R}$.

For every $i\in N$, let $c_i = \inf_{X\in\Gamma_{i}}\max_{u\in Z}I(u),$ hence, $0\leq c_{i}\leq c_{i+1}$ and $c_{i} < \theta,$ for every $i > k$. Then every $c_{i}$ is a critical value of $I$. Moreover, if $c_{i} = c_{i+1} = \cdots = c_{i+p} = c < \theta$ for $i > k$, then $\gamma(K_{c})\geq p+1$, where

Lemma 3.2. There is a nondecreasing sequence $\{s_n\}$ of positive real numbers, independent of $\lambda,$ such that for every $\lambda > 0$, we have

where $\Gamma_{n}$ was defined in Theorem 3.1.

Proof. By the definition of $\Gamma_{n}$, one has

therefore $s_n < \infty$ and $s_n\leq s_{n+1}$.

Proof of Theorem 1.1(II). We note that $I_{\lambda}$ satisfies $I_{\lambda}(0) = 0$ and $I_{\lambda}(-u) = I_{\lambda}(u)$. In the sequel, we shall divide the proof into the following three steps:

Step 1. We shall prove that $I_\lambda$ satisfies hypothesis (ⅱ) of Theorem 3.1. Indeed, similar to the proof of (ⅰ) in Lemma 3.1, we can easily prove that the energy functional $I_{\lambda}$ satisfies the hypothesis $(ii)$ of Theorem 3.1.

Step 2. We shall prove that $I_\lambda$ satisfies hypothesis (ⅲ) of Theorem 3.1. Indeed, let $Y$ be a finite-dimensional subspace of $S^{1, p}(\Omega)$. Since all norms in finite-dimensional space are equivalent, it follows that for every $u \in Y$, we have

for some positive constant $C > 0$. Also, because of $2p < Q^{*}$, we can choose a large $R > 0$ such that $I_{\lambda}(u) \leq 0$ on $S^{1, p}(\Omega)\setminus B_R$. This fact implies that the energy functional $I_{\lambda}$ satisfies the hypothesis $(iii)$ of Theorem 3.1.

Step 3. We shall prove that problem (1.1) has infinitely many nontrivial weak solutions. Indeed, applying the argument in Wei and Wu ^[25], we can choose $a_1$ large enough so that for every $a > a_1$,

that is,

Thus, one has

From Lemma 2.4, we know that $I_{\lambda}$ satisfies $(PS)_{c_i^\lambda} (i = 1, 2, \cdots, n)$ condition. This fact implies that the levels $c_{1}^{\lambda}\leq c_{2}^{\lambda}\leq\cdots\leq c_{n}^{\lambda}$ are critical values of $I_{\lambda}$, which be guaranteed by an application of the Rabinowitz result [24, Proposition 9.30].

If $c_{i}^{\lambda} = c_{i+1}^{\lambda}$ where $i = 1, 2, \cdots, k-1$, then applying the Ambrosetti and Rabinowitz result [26, Remark 2.12 and Theorem 4.2], we see that the set $K_{c_{i}^{\lambda}}$ consists of infinite number of different points, so problem (1.1) has infinite number of weak solutions. Hence, problem (1.1) has at least $k$ pairs of solutions. Since $k$ is arbitrary, we can conclude that problem (1.1) has infinitely many solutions. This completes the proof of Theorem 1.1(Ⅱ).

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

Song was supported by the National Natural Science Foundation of China (No. 12001061), the Science and Technology Development Plan Project of Jilin Province, China (No. 20230101287JC) and Innovation and Entrepreneurship Talent Funding Project of Jilin Province (No. 2023QN21). Repovš was supported by the Slovenian Research Agency program No. P1-0292 and grants Nos. J1-4031, J1-4001, N1-0278, N1-0114 and N1-0083.

Conflict of interest

The authors declare there is no conflict of interest.