$p$-biharmonic Kirchhoff equations with critical Choquard nonlinearity

TL;DR

This paper studies a complex nonlinear PDE involving $p$-biharmonic operators and critical Choquard nonlinearity, establishing existence and multiplicity of solutions using variational methods and concentration-compactness principles.

Contribution

It introduces new existence and multiplicity results for $p$-biharmonic Kirchhoff equations with critical Choquard nonlinearity, including a concentration-compactness framework.

Findings

Proved concentration-compactness principle for the $p$-biharmonic Choquard equation.

Established existence of solutions for various parameter ranges.

Demonstrated multiplicity of solutions depending on parameters $\lambda$ and $\alpha$.

Abstract

In this article, we deal with the following involving $p$-biharmonic critical Choquard-Kirchhoff equation $$ \left(a+b\left(\int_{\mathbb R^N}|\Delta u|^p dx\right)^{\theta-1}\right) \Delta_{p}^{2}u = \alpha \left(|x|^{-\mu}*u^{p^*_\mu}\right)|u|^{p^*_\mu-2}u+ \lambda f(x) |u|^{r-2} u \; \text{in}\; \mathbb R^N, $$ where $a\geq 0$, $b> 0$, $0<\mu<N$, $N>2p$, $p\geq 2$, $\theta\geq1$, $\alpha$ and $\lambda$ are positive real parameters, $p_{\mu}^{*}= \frac{p(2N-\mu)}{2(N-2p)}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. The function $f \in L^{t}(\mathbb R^N)$ with $t= \frac{p^{*}}{(p^* -r)}$ if $p<r<p^*:=\frac{Np}{N-2p}$ and $t=\infty$ if $r\geq p^{*}$. We first prove the concentration compactness principle for the $p$-biharmonic Choquard-type equation. Then using the variational method together with the concentration-compactness, we established the existence and multiplicity of solutions to the above problem with respect to parameters $\lambda$ and \(\alpha\) for different values of $r$. The results obtained here are new even for $p-$Laplacian.

Equations

$$\begin{array}[]{lr}\left(a+b\left(\int_{\mathbb{R}^{N}}|\Delta u|^{p}dx\right)^{\theta-1}\right)\Delta_{p}^{2}u=\alpha\left(|x|^{-\mu}*u^{p^{*}_{\mu}}\right)|u|^{p^{*}_{\mu}-2}u+\lambda f(x)|u|^{r-2}u\;\text{in}\;\mathbb{R}^{N},\end{array}$$

$$(\mathcal{P}_{\alpha,\lambda})\quad\left\{\begin{array}[]{lr}(a+b\|u\|^{p(\theta-1)})\Delta_{p}^{2}u=\alpha\left(|x|^{-\mu}*u^{p^{*}_{\mu}}\right)|u|^{p^{*}_{\mu}-2}u+\lambda f(x)|u|^{r-2}u\;\text{in}\;\mathbb{R}^{N},\end{array}\right.$$

$$-\Delta u+V(x)u=(I_{\alpha }\ast F(u))f(u)+g(x,u)\text{in }{\mathbb{R}}^N,$$

$$I_{\alpha }(x)=\frac{B_{\alpha }}{|x{|}^{N-\alpha }}\text{where}{B}_{\alpha }=\frac{\Gamma \left(\frac{N-\alpha }{2}\right)}{\Gamma \left(\frac{\alpha }{2}\right){\pi }^{N/2}{2}^{\alpha }},$$

$$\rho u_{tt}-\left(\frac{{\rho }_0}{h}+\frac{E}{2L}{\int }_0^L{\left|u_x\right|}^{2}dx\right)u_{xx}=0,$$

$$-(a+b\Vert u{\Vert }^2)\Delta u=\gamma \left(|x{|}^{-\mu }\ast u^{2_{\mu }^{\ast }}\right)|u{|}^{2_{\mu }^{\ast }-2}u+\lambda f(x)|u{|}^{r-2}u\text{in}{\mathbb{R}}^N,$$

$${\Delta }_p^{2}u=\lambda |u{|}^{p-2}u+|u{|}^{r-2}u\text{in}\Omega ;u=\nabla u=0\text{in}\partial \Omega .$$

$${\Delta }_p^{2}u-{\Delta }_{p}u+V(x)|u{|}^{p-2}u=\lambda f_1(x)|u{|}^{r-2}u+f_2(x)|u{|}^{q-2}u;x\in {\mathbb{R}}^N,$$

$${\Delta }_p^{2}u+M\left({\int }_{{\mathbb{R}}^N}{\varphi }_0(x,\nabla u)dx\right)\text{div}(\varphi (x,\nabla u))+V(x)|u{|}^{p-2}u=\lambda g(x,u);x\in {\mathbb{R}}^N,$$

$$b>\frac{2p_{\mu }^{\ast }-p}{p(\theta -1)}\left[{\left(\frac{ap(\theta -1)}{(p\theta -2p_{\mu }^{\ast })}\right)}^{\frac{2p_{\mu }^{\ast }-p\theta }{2p_{\mu }^{\ast }-p}}{\left(2^p{S}_{H,L}^{-\frac{2p_{\mu }^{\ast }}{p}}\right)}^{\frac{p(\theta -1)}{2p_{\mu }^{\ast }-p}}\right]:=b^1,$$

$$\Vert u\Vert \le {\left[\frac{\lambda \Vert f{\Vert }_{\frac{p^{\ast }}{p^{\ast }-r}}}{S^{\frac{r}{p}}\left(b-S_{H,L}^{-\frac{2p_{\mu }^{\ast }}{p}}\right)}\right]}^{\frac{1}{p\theta -r}}.$$

$$\begin{array}[]{lr}\left(a+b\left(\int_{\mathbb{R}^{N}}|\Delta u|^{p}dx\right)^{\theta-1}\right)\Delta_{p}^{2}u=\alpha\displaystyle\left(\int_{\mathbb{R}^{N}}\frac{|u(y)|^{p_{\mu}^{*}}}{|y|^{\beta}|x-y|^{\mu}}dy\right)\frac{|u(x)|^{p_{\mu}^{*}-2}}{|x|^{\beta}}u+\lambda f(x)|u|^{r-2}u\;\text{in}\;\mathbb{R}^{N},\end{array}$$

$$\Vert u\Vert :=\Vert u{\Vert }_{D^{2,p}({\mathbb{R}}^N)}={\left({\int }_{{\mathbb{R}}^N}|\Delta u{|}^{p}dx\right)}^{\frac{1}{p}}.$$

$$S:=\underset{u\in D^{2,p}({\mathbb{R}}^N)\setminus \{0\}}{\inf }\frac{{\int }_{\Omega }|\Delta u{|}^{p}dx}{({\int }_{\Omega }|u(x){|}^{p^{\ast }}dx{)}^{\frac{p}{p^{\ast }}}}.$$

$${\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{g(x)h(y)}{|x-y{|}^{\mu }}dxdy\le C(\mu ,N,r,s)\Vert g{\Vert }_{L^r({\mathbb{R}}^N)}\Vert h{\Vert }_{L^s({\mathbb{R}}^N)}.$$

$${\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|u(x){|}^t|u(y){|}^t}{|x-y{|}^{\mu }}dxdy$$

$$\frac{p(2N-\mu )}{2N}\le t\le \frac{p(2N-\mu )}{2(N-2p)}$$

$${\left({\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|u(x){|}^{p_{\mu }^{\ast }}|u(y){|}^{p_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dxdy\right)}^{\frac{1}{p_{\mu }^{\ast }}}\le C(N,\mu {)}^{\frac{1}{p_{\mu }^{\ast }}}|u{|}_{p^{\ast }}^2.$$

$$S_{H,L}:=\underset{u\in D^{2,p}({\mathbb{R}}^N)\setminus \{0\}}{\inf }\frac{{\int }_{{\mathbb{R}}^N}|\Delta u{|}^{p}dx}{{\left({\int }_{{\mathbb{R}}^N}{\int }_{{\mathbb{R}}^N}\frac{|u(x){|}^{p_{\mu }^{\ast }}|u(y){|}^{p_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dxdy\right)}^{\frac{p}{2p_{\mu }^{\ast }}}}.$$

$$C(N,\mu {)}^{\frac{p}{2p_{\mu }^{\ast }}}{S}_{H,L}\ge S>0,$$

$$\left({\int }_{{\mathbb{R}}^N}\frac{|u_k(y){|}^{p_{\mu }^{\ast }}|u_k(x){|}^{p_{\mu }^{\ast }-2}}{|x-y{|}^{\mu }}dy\right)u_k(x)\rightharpoonup \left({\int }_{{\mathbb{R}}^N}\frac{|u(y){|}^{p_{\mu }^{\ast }}|u(x){|}^{p_{\mu }^{\ast }-2}}{|x-y{|}^{\mu }}dy\right)u(x)\text{weakly in}{L}^{\frac{Np}{Np-N+2p}}({\mathbb{R}}^N).$$

$$|u_k{|}^{p_{\mu }^{\ast }}\rightharpoonup |u{|}^{p_{\mu }^{\ast }}\text{weakly in}{L}^{\frac{2N}{2N-\mu }}({\mathbb{R}}^N),$$

$$|u_k{|}^{p_{\mu }^{\ast }-2}{u}_k\rightharpoonup |u{|}^{p_{\mu }^{\ast }-2}u\text{weakly in}{L}^{\frac{2Np}{2Np-\mu p-2N+4p}}({\mathbb{R}}^N),$$

$${\int }_{{\mathbb{R}}^N}\frac{|u_k(y){|}^{p_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dy\rightharpoonup {\int }_{{\mathbb{R}}^N}\frac{|u(y){|}^{p_{\mu }^{\ast }}}{|x-y{|}^{\mu }}dy\text{weakly in}{L}^{\frac{2N}{\mu }}({\mathbb{R}}^N),$$

$$\left({\int }_{{\mathbb{R}}^N}\frac{|u_k(y){|}^{p_{\mu }^{\ast }}|u_k(x){|}^{p_{\mu }^{\ast }-2}}{|x-y{|}^{\mu }}dy\right)u_k(x)\rightharpoonup \left({\int }_{{\mathbb{R}}^N}\frac{|u(y){|}^{p_{\mu }^{\ast }}|u(x){|}^{p_{\mu }^{\ast }-2}}{|x-y{|}^{\mu }}dy\right)u(x)\text{weakly in}{L}^{\frac{Np}{Np-N+2p}}({\mathbb{R}}^N)$$

$$\underset{k\to \infty }{\lim }\left({\int }_{{\mathbb{R}}^N}(|x{|}^{-\mu }\ast |u_k{|}^{p_{\mu }^{\ast }})|u_k{|}^{p_{\mu }^{\ast }}-{\int }_{{\mathbb{R}}^N}(|x{|}^{-\mu }\ast |u_k-u{|}^{p_{\mu }^{\ast }})|u_k-u{|}^{p_{\mu }^{\ast }}\right)={\int }_{{\mathbb{R}}^N}(|x{|}^{-\mu }\ast |u{|}^{p_{\mu }^{\ast }})|u{|}^{p_{\mu }^{\ast }}.$$

$${\int }_{{\mathbb{R}}^N}$$

$$={\int }_{{\mathbb{R}}^N}(|x{|}^{-\mu }\ast (|u_k{|}^{p_{\mu }^{\ast }}-|u_k-u{|}^{p_{\mu }^{\ast }}))(|u_k{|}^{p_{\mu }^{\ast }}-|u_k-u{|}^{p_{\mu }^{\ast }})$$

$$+2{\int }_{{\mathbb{R}}^N}(|x{|}^{-\mu }\ast (|u_k{|}^{p_{\mu }^{\ast }}-|u_k-u{|}^{p_{\mu }^{\ast }}))|u_k-u{|}^{p_{\mu }^{\ast }}.$$

$$|u_k{|}^{p_{\mu }^{\ast }}-|u_k-u{|}^{p_{\mu }^{\ast }}\to |u{|}^{p_{\mu }^{\ast }}\text{in}{L}^{\frac{2N}{2N-\mu }}({\mathbb{R}}^N)\text{as}k\to \infty .$$

$$|x{|}^{-\mu }\ast (|u_k{|}^{p_{\mu }^{\ast }}-|u_k-u{|}^{p_{\mu }^{\ast }})\to |x{|}^{-\mu }\ast |u{|}^{p_{\mu }^{\ast }}\text{in}{L}^{\frac{2N}{\mu }}({\mathbb{R}}^N)\text{as}k\to \infty .$$

$$\zeta =|u{|}^{p^{\ast }}+\sum _{i\in J}{\zeta }_i{\delta }_{z_i},{\zeta }_i>0,$$

Full text

[TABLE]

where $a\geq 0$, $b>0$, $0<\mu<N$, $N>2p$, $p\geq 2$, $\theta\geq 1$, $\alpha$ and $\lambda$ are positive real parameters, $p_{\mu}^{*}=\frac{p(2N-\mu)}{2(N-2p)}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. The function $f\in L^{t}(\mathbb{R}^{N})$ with $t=\frac{p^{*}}{(p^{*}-r)}$ if $p<r<p^{*}:=\frac{Np}{N-2p}$ and $t=\infty$ if $r\geq p^{*}$. We first prove the concentration compactness principle for the $p$-biharmonic Choquard-type equation. Then using the variational method together with the concentration-compactness, we established the existence and multiplicity of solutions to the above problem with respect to parameters $\lambda$ and $\alpha$ for different values of $r$. The results obtained here are new even for $p-$Laplacian.

1. Introduction

This paper is concerned with the existence of solutions of the following Kirchhoff $p$-biharmonic equation involving Choquard nonlinearity

[TABLE]

where $N>2p$, $p\geq 2$, $a\geq 0$, $b>0$, $\theta\geq 1$, $\|u\|=\left(\int_{\mathbb{R}^{N}}|\Delta u|^{p}dx\right)^{1/p}$, $0<\mu<N$, $f\in L^{t}(\mathbb{R}^{N})$ with $t=\frac{p^{*}}{(p^{*}-r)}$ if $p<r<p^{*}:=\frac{Np}{N-2p}$ and $t=\infty$ if $r\geq p^{*}$. Here $p_{\mu}^{*}=\frac{p(2N-\mu)}{2(N-2p)}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality.

In recent years, mathematicians have been studying non-local problems, the existence and multiplicity of solutions, due to their vast applications. One of the non-local problems involves Choquard-type nonlinearity. The nonlinear Choquard equation

[TABLE]

where $\alpha\in(0,N)$, $f=F^{\prime}\in C(\mathbb{R},\mathbb{R})$, $g\in C(\mathbb{R}^{N}\times\mathbb{R},\mathbb{R})$ and $I_{\alpha}$ is a Riesz potential with

[TABLE]

was first studied by S. Pekar [38] in 1954, for $V(x)=0$, $F(u)=|u|^{2}$, $\alpha=2,~g(x,u)=0$ and $N=3$, to describe the quantum theory of the polaron at rest. Later, Choquard [26] used equation (1.1) to study an electron trapped in its hole, and Penrose used equation (1.1) as a model for self-gravitating matter in [39], respectively. In the last decade, Moroz and Schaftingen [34, 35, 36] studied the Choquard equations and proved the existence of ground state solutions. They further proved the regularity, positivity and radial symmetry of the solutions. Carvalho, Silva, and Goulart [10] studied (1.1) with convex-concave type of growth and established the multiplicity of solutions using the Nehari method with a fine analysis on the nonlinear Rayleigh quotient. For a more detailed overview of the Choquard-type equations, we refer readers to [3, 16, 37] and references therein.

Another type of non-local operator which researchers are studying currently is the Kirchhoff operator. In [21] Kirchhoff introduced the following equation

[TABLE]

where $\rho,\rho_{0},h,E,L$ are constants, to extend the classical D’Alembert’s wave equation by incorporating the changes in the length of the string produced by transverse vibrations. To know more about Kirchhoff operator, one can see [2, 12, 15] and references therein. On the other hand, many authors considered the Choquard-Kirchhoff type problems with Hardy-Littlewood-Sobolev critical nonlinearities. Due to the presence of Kirchhoff operator and Choquard type nonlinearity, these type of problems are called doubly non-local. As an example, we cite [25] in which Liang, Pucci and Zhang considered the following problem

[TABLE]

where $\lambda,\gamma>0$ are parameters. They showed the existence and multiplicity of solutions for $1<q\leq 2$. With no offense of providing the full list, for more details, please refer [24, 19, 41] and references therein.

Problems involving quasilinear problems have been the focus of intensive research in recent years. Problems involving $p$-biharmonic operators are of great importance as they appear in many applications, such as in elasticity and plate theory, Quantum mechanics, Astrophysics, Material sciences, see [23, 32, 33].

[TABLE]

where $\Omega$ is a bounded domain in $\mathbb{R}^{N}$, $p<r\leq p^{*}=\frac{Np}{N-2p}$. The authors showed that the problem (1.2) possesses infinitely many solutions for $q<p^{*}$, using Fountain’s theorem and for the case $q=p^{*}$, the existence result is obtained using abstract critical point theory based on a pseudo-index related to the cohomological index. While in [13], Chung and Minh investigated the $p$-biharmonic Kirchhoff type problem for the bounded domain and proved the existence of a non-trivial solution by the Mountain Pass theorem. On the other hand, there are very few articles available on problems involving the $p$-biharmonic operator over $\mathbb{R}^{N}$. In [30], Liu and Chen considered the following $p$-biharmonic elliptic equation

[TABLE]

where $2<2p<N$, $1<r<p<q<p^{*}=\frac{Np}{N-2p}$ and the potential function $V(x)\in C(\mathbb{R}^{N})$ satisfies $\displaystyle\inf_{x\in\mathbb{R}^{N}}V(x)>0$. Here, they established the existence of infinitely many high-energy solutions to equation (1.3), using the variational methods. In [5], Bae, Kim, Lee, and Park examined the following $p$-biharmonic Kirchhoff type equation

[TABLE]

where the function $\phi(x,v)$ is of type $|v|^{p-2}v$, $\phi(x,v)=\frac{d}{dv}\phi_{0}(x,v)$, the potential function $V(x)\in C(\mathbb{R}^{N})$ and $g:\mathbb{R}^{N}\times\mathbb{R}\rightarrow\mathbb{R}$ satisfies the Carathéodory condition. For $1<p<p^{*}=\frac{Np}{N-2p}$, using Mountain Pass theorem and Fountain theorem, the authors proved the existence of a non-trivial weak solution and multiplicity of weak solutions. For more results on problems involving $p$-biharmonic operators, we refer [6, 7, 8, 1], see also the references therein.

To the best of our knowledge, there is no result that takes into account the problem involving the Kirchhoff operator, $p-$biharmonic operator, and the convolution term. Precisely, we will study a new class of equations with a physical, chemical, and biological background. In this paper, we consider the critical Choquard-Kirchhoff equation involving $p$-biharmonic operator in the whole space $\mathbb{R}^{N}$. Motivated by the results introduced in [25, 18] and a few papers on the $p$-biharmonic operator, we established the existence and multiplicity of solutions to the problem $(\mathcal{P}_{\alpha,\lambda})$ for the non-degenerate and degenerate case. In this article, we established the existence and multiplicity of solutions depending on a different range of $r$ and $\theta$. The main difficulty here is the lack of compactness of the embedding. To solve this issue, we established the concentration-compactness lemma for the biharmonic operator with Choquard nonlinearity. Furthermore, we used variational methods to prove the existence and multiplicity of solutions.

Before stating our main result, we assume the following condition on the function $f:\mathbb{R}^{N}\rightarrow\mathbb{R}$,

1. $(f_{1})$

$f\geq 0$ with $f\not\equiv 0$ in $\mathbb{R}^{N}$, $f\in L^{\frac{p^{*}}{p^{*}-r}}$. 2. $(f_{2})$

Let $\Omega:=\{x\in\mathbb{R}^{N}:f(x)>0\}$ be an open subset of $\mathbb{R}^{N}$ with $0<|\Omega|<\infty.$

The main results of this article are as follows:

Theorem 1.1**.**

Let $1<r<p$, $p\theta<2p_{\mu}^{*}$. Suppose $f$ satisfies $(f_{1})$ and $(f_{2})$. Then

1. (i)

For each $\lambda>0$ there exists $\overline{\Lambda}>0$ such that for all $\alpha\in(0,\overline{\Lambda})$, the problem $(\mathcal{P}_{\alpha,\lambda})$ has a sequence of non-trivial solutions $\{u_{n}\}$ with $\mathcal{E}_{\alpha,\lambda}\leq 0$ and $u_{n}\rightarrow 0$, as $n\rightarrow\infty$. 2. (ii)

For each $\alpha>0$ there exists $\underline{\Lambda}>0$ such that for all $\lambda\in(0,\underline{\Lambda})$, the problem $(\mathcal{P}_{\alpha,\lambda})$ has a sequence of non-trivial solutions $\{u_{n}\}$ with $\mathcal{E}_{\alpha,\lambda}\leq 0$ and $u_{n}\rightarrow 0$, as $n\rightarrow\infty$.

Theorem 1.2**.**

Let $r=p$, $p\theta<2p_{\mu}^{*}$. Suppose $f$ satisfy $(f_{1})$. Then there exists a positive constant $\hat{a}$ such that for each $a>\hat{a}$ and $\lambda\in(0,aS\|f\|^{-1}_{\frac{}{}})$, the problem $(\mathcal{P}_{\alpha,\lambda})$ has at least $n$ pairs of non-trivial solutions.

Theorem 1.3**.**

Let $p\theta\leq r<p^{*}$, $p\theta<2p_{\mu}^{*}$. Suppose $f$ satisfy $(f_{1})$. Then there exists a positive constant $\tilde{a}$ such that for each $a>\tilde{a}$ and for all $\lambda>0$, the problem $(\mathcal{P}_{\alpha,\lambda})$ has infinitely many non-trivial solutions.

Theorem 1.4**.**

Let $p\geq 2$, $p<r<p^{*}$, $f\geq 0$ with $f\not\equiv 0$ in $\mathbb{R}^{N}$, and $p\theta\geq 2p_{\mu}^{*}$ . If either $p\theta=2p_{\mu}^{*}$, $a>0$ and $b>2^{p}S_{H,L}^{-\frac{2p_{\mu}^{*}}{p}}$ or $p\theta>2p_{\mu}^{*}$, $a>0$ and

[TABLE]

then there exists $\lambda^{*}>0$ such that the problem $(\mathcal{P}_{\alpha,\lambda})$ admits at least two non-trivial solutions in $D^{2,p}(\mathbb{R}^{N})$ for all $\lambda>\lambda^{*}$.

Theorem 1.5**.**

(Degenerate Case) Let $1<r<p^{*}$, $f$ satisfies $(f_{1})$ and $p\theta\geq 2p_{\mu}^{*}$. If $a=0$ and $b>2^{p}S_{H,L}^{-\frac{2p_{\mu}^{*}}{p}}$, then for all $\lambda>0$, the problem $\mathcal{(}P_{\alpha,\lambda})$ admits infinitely many pairs of distinct solutions in $D^{2,p}(\mathbb{R}^{N})$ for all $\lambda>\lambda^{*}$. Moreover, any solution $u\in D^{2,p}(\mathbb{R}^{N})\setminus\{0\}$ satisfies

[TABLE]

Remark 1.6**.**

The case when $p<r<p\theta$ with $p\theta<2p^{*}_{\mu}$ is an open case due to lack of minimizers.

Remark 1.7**.**

One can generalized these result to the following $p$-biharmonic Kirchhoff equation with critical Stein-Weiss type nonlinearity

[TABLE]

where $a\geq 0$, $b\geq 0$, $0<\mu+2\beta<N$, $N>2p$, $p\geq 2$, $\theta\geq 1$ and $\alpha$, $\lambda$ are real positive parameters, $p_{\mu,\beta}^{*}=\frac{p(2N-2\beta-\mu)}{2(N-2p)}$ is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality. The function $f\in L^{t}(\mathbb{R}^{N})$ with $t=\frac{p^{*}}{(p^{*}-r)}$ if $p<r<p^{*}:=\frac{Np}{N-2p}$ and $t=\infty$ if $r\geq p^{*}$.

Organization of article: In section 2, we enlist the variational framework and preliminary results. We demonstrate the proof of the concentration compactness principle for the $p$-biharmonic operator with critical Choquard-type nonlinearity. In section 3, we discuss the fundamental results for Palais-Smale sequence, for different ranges of $r$, in both the cases $p\theta<2p^{*}_{\mu}$ and $p\theta\geq 2p^{*}_{\mu}$. In section 4, we consider the case $p\theta\geq 2p^{*}_{\mu}$ and prove the Theorems 1.4, 1.5. The case $p\theta<2p^{*}_{\mu}$ is discussed section 5, which is further divided into three subsections $5.1$, $5.2$ and $5.3$ followed by the proofs of Theorems 1.1, 1.2, 1.3 respectively.

2. Variational framework and Preliminary results

In this section we recall the main notations and tools that will be needed in the sequel. Define the space $D^{2,p}(\mathbb{R}^{N})=\left\{u\in L^{p^{*}}(\mathbb{R}^{N}):\int_{\mathbb{R}^{N}}|\Delta u|^{p}dx<\infty\right\},$ equipped with the norm

[TABLE]

is a Banach space. We define $S$ to be the best Sobolev constant for the embedding of $D^{2,p}(\mathbb{R}^{N})$ into $L^{p^{*}}(\mathbb{R}^{N})$ with $p^{*}=\frac{Np}{N-2p}$ as

[TABLE]

To handle the convolution-type nonlinearity, we need the well-known Hardy-Littlewood-Sobolev inequality, which is stated as:

Proposition 2.1**.**

(Hardy-Littlewood-Sobolev inequality [27, Theorem 4.3]) Let $r$, $s>1$ and $0<\mu<N$ with $1/r+\mu/N+1/s=2$, $g\in L^{r}(\mathbb{R}^{N})$ and $h\in L^{s}(\mathbb{R}^{N})$. Then there exists a sharp constant $C(\mu,N,r,s)$, independent of $g,$ $h$ such that

[TABLE]

Using the Hardy-Littlewood-Sobolev inequality (2.2), for $s=r=t$, the integral is

[TABLE]

is well-defined if $|u|^{t}\in L^{q}(\mathbb{R}^{N})$ for some $q>1$ satisfying $\frac{2}{q}+\frac{\mu}{N}=2$. For $u\in W^{2,p}(\mathbb{R}^{N})$, by the Sobolev embedding theorem, we have $p\leq tq\leq\frac{Np}{N-2p}$. Thus

[TABLE]

In this sense, we call $p_{*\mu}=\frac{p(2N-\mu)}{2N}$ the lower critical exponent and $p^{*}_{\mu}=\frac{p(2N-\mu)}{2(N-2p)}$ the upper critical exponent in the sense of the Hardy Littlewood-Sobolev inequality. Infact, for $u\in D^{2,p}(\mathbb{R}^{N})$, the Hardy Liitlewood Sobolev inequality for the upper critical exponent leads to the following

[TABLE]

We define $S_{H,L}$ to be the best constant as

[TABLE]

Then it is easy to see that

[TABLE]

where $S$ is defined in (2.1).

Lemma 2.2**.**

Let $N>2p$, $0<\mu<N$ and $\{u_{k}\}$ be a bounded sequence in $L^{p^{*}}(\mathbb{R}^{N})$, then the following result holds as $k\rightarrow\infty$

[TABLE]

Proof.

Let $\{u_{k}\}$ be a bounded sequence in $L^{p^{*}}(\mathbb{R}^{N})$, then one can easily verify that

[TABLE]

as $k\rightarrow\infty$. The Riesz potential defines a continuous map from $L^{\frac{2N}{2N-\mu}}(\mathbb{R}^{N})$ to $L^{\frac{2N}{\mu}}(\mathbb{R}^{N})$ by Hardy-Littlewood-Sobolev inequality. Thus, we have

[TABLE]

as $k\rightarrow\infty$. Then, on combining (2.4) and (2.5), we obtain

[TABLE]

as $k\rightarrow\infty$, which is the required result. ∎

In order to prove the Palais-Smale condition, we need the following Lemma which is inspired by the Brézis-Lieb convergence Lemma (see [9]).

Lemma 2.3**.**

Let $N>2p$, $0<\mu<N$ and $\{u_{k}\}$ be a bounded sequence in $L^{\frac{pN}{N-2p}}(\mathbb{R}^{N})$. If $u_{k}\rightarrow u$ a.e. in $\mathbb{R}^{N}$ as $k\rightarrow\infty$, then

[TABLE]

Proof.

The proof is similar to the proof of the Brézis-Lieb Lemma (see [9]) or Lemma 2.2 [17]. But for completeness, we give the detail. Consider

[TABLE]

Now by using [34] $(\text{lemma}\;2.5)$, for $q=p^{*}_{\mu}=\frac{p(2N-\mu)}{2(N-2p)}$ and $r=\frac{pN}{2N-\mu}p^{*}_{\mu}$, then we obtain

[TABLE]

Also the Hardy-Littlewood-Sobolev inequality implies that

[TABLE]

Hence, with the help of [42] $(\text{Prop.}\;5.4.7)$, we obtain $|u_{k}-u|^{p^{*}_{\mu}}\rightharpoonup 0$ weakly in $L^{\frac{pN}{2N-\mu}}(\mathbb{R}^{N})$ as $k\rightarrow\infty$. So using this together with (2.7), (2.8), in (2), we obtain the required result. ∎

We recall the concentration compactness Lemmas given by P. L. Lions [28, 29].

Lemma 2.4**.**

Let $\{u_{k}\}$ be a bounded sequence in $D^{2,p}(\mathbb{R}^{N})$ converging weakly and a.e. to $u\in D^{2,p}(\mathbb{R}^{N})$ such that $|\Delta u_{k}|^{p}\rightharpoonup\omega$, $|u_{k}|^{p^{*}}\rightharpoonup\zeta$ in the sense of measure. Then, for at most countable set $J$, there exist families of distinct points $\{\omega_{i}:i\in J\}$ and $\{\zeta_{i}:i\in J\}$ in $\mathbb{R}^{N}$ satisfying

[TABLE]

where $\zeta$, $\omega$ are bounded and non negative measures on $\mathbb{R}^{N}$ and $\delta_{z_{i}}$ is the Dirac mass at $z_{i}$. In particular, $\displaystyle\sum_{i\in J}(\zeta_{i})^{\frac{p}{p^{*}}}<\infty$.

Lemma 2.5**.**

Let $\{u_{k}\}\subset D^{2,p}(\mathbb{R}^{N})$ be a sequence in Lemma 2.4 and defined

[TABLE]

Then it follows that

[TABLE]

and

Now, we prove the following concentration compactness Lemma for our problem $(\mathcal{P}_{\alpha,\lambda})$.

Lemma 2.6**.**

Let $N>2p$, $0<\mu<N$, and $1\leq p\leq\frac{p(2N-\mu)}{2(N-2p)}$. If $\{u_{k}\}$ is a bounded sequence in $D^{2,p}(\mathbb{R}^{N})$ converges weakly, to $u$ as $k\rightarrow\infty$ and such that $|u_{k}|^{p^{*}}\rightharpoonup\zeta$ and $|\Delta u_{k}|^{p}\rightharpoonup\omega$ in the sense of measure. Assume that

[TABLE]

weakly in the sense of measure where $\nu$ is a bounded positive measure on $\mathbb{R}^{N}$ and define

[TABLE]

[TABLE]

Then there exists a countable sequence of points $\{z_{i}\}_{i\in J}\subset\mathbb{R}^{N}$ and families of positive numbers $\{\omega_{i}:i\in J\}$, $\{\zeta_{i}:i\in J\}$ and $\{\nu_{i}:i\in J\}$ such that

[TABLE]

and

[TABLE]

where $\delta_{x}$ is the Dirac-mass of mass $1$ concentrated at $x\in\mathbb{R}^{N}$.

For the energy at infinity, we have

[TABLE]

[TABLE]

and

[TABLE]

Proof.

Let $v_{k}=u_{k}-u$. Then $\{v_{k}\}$ converging weakly to [math] in $D^{2,p}(\mathbb{R}^{N})$ and $v_{k}(x)\rightarrow 0$ a.e. in $\mathbb{R}^{N}$ as the bounded sequence $\{u_{k}\}$ converging weakly to $u$ in $D^{2,p}(\mathbb{R}^{N})$. Lemma 2.4 yields

[TABLE]

Firstly, we show that for every $\phi\in C_{c}^{\infty}(\mathbb{R}^{N})$,

[TABLE]

For this, we denote

[TABLE]

Since $\phi\in C_{c}^{\infty}(\mathbb{R}^{N})$, we have for every $\delta>0$, there exists $K>0$ such that

[TABLE]

As we know that Riesz potential defines a linear operator and using $v_{k}(x)\rightarrow 0$ a.e. in $\mathbb{R}^{N}$, so we obtain

[TABLE]

Thus $\Psi_{k}(x)\rightarrow 0$ a.e. in $\mathbb{R}^{N}$. We note that

[TABLE]

where $\Phi(x,y)=\frac{|\phi(y)|^{p_{\mu}^{*}}-|\phi(x)|^{p_{\mu}^{*}}}{|x-y|^{\mu}}$. Moreover, for almost all $x$, there exists some $R>0$ large enough such that

[TABLE]

Using the mean value theorem and the fact that $p\geq 2$, $\Phi(x,y)\in L^{q}(B_{R})$ for each $x$, where $q<\frac{N}{\mu-1}$ if $\mu>1$, $q\leq+\infty$ if $0<\mu\leq 1$. With the help of Young’s inequality, there exists $t>\frac{2N}{\mu}$ such that

[TABLE]

where $K$ is same as in $\eqref{n31}$. Moreover, one can easily see that for $R>0$ large enough

[TABLE]

and so, we have

[TABLE]

Thus for $s>0$ small enough, we obtain

[TABLE]

Using this together with $\Psi_{k}(x)\rightarrow 0$ a.e. in $\mathbb{R}^{N}$, we have

[TABLE]

Combining this with (2.14), we have

[TABLE]

Now, for every $\phi\in C_{c}^{\infty}(\mathbb{R}^{N})$, by Hardy-Littlewood-Sobolev inequality, we obtain

[TABLE]

Equation (2.13) yields

[TABLE]

On taking the limit as $k\rightarrow\infty$, we obtain

[TABLE]

Employing Lemma $1.2$ in [28], one can directly obtain (2.11).

Further, let $\phi=\chi_{\{z_{i}\}}$, $i\in J$ and using this in (2.15), we obtain

[TABLE]

Definition of $S_{H,L}$ yields

[TABLE]

Also equation (2.13) and $v_{k}\rightarrow 0$ in $L^{p}_{loc}(\mathbb{R}^{N})$ give

[TABLE]

On passing the limit as $k\rightarrow\infty$, we have

[TABLE]

Let $\phi=\chi_{\{z_{i}\}}$, $i\in J$ and applying this in (2.16), we have

[TABLE]

This completes the proof of (2.9) and (2.12).

Now, we prove the possible loss of mass at infinity. For $R>1$, let $\psi_{R}\in C^{\infty}(\mathbb{R}^{N})$ be such that $\psi_{R}=1$ for $|x|>R+1$, $\psi_{R}(x)=0$ for $|x|<R$ and $0\leq\psi_{R}(x)\leq 1$ on $\mathbb{R}^{N}$. For every $R$, we have

[TABLE]

Taking $R\rightarrow\infty$, by Lebesgue’s dominated convergent theorem, we deduce

[TABLE]

By the Hardy-Littlewood-Sobolev inequatlity (2.2), we obtain

[TABLE]

This implies

[TABLE]

Similarly, by the Hardy-Littlewood-Sobolev inequatlity (2.2), we obtain

[TABLE]

This implies

[TABLE]

This completes the proof. ∎

We state the general version of the mountain pass Lemma which will be used to prove the Theorem.

Theorem 2.7**.**

Let $I$ be a functional on a Banach space $X$ and $I\in C^{1}(E,\mathbb{R})$. If there exists $\alpha$, $\rho$ such that

1. (1)

$I(u)\geq\alpha$, $u\in X$ with $\|u\|=\rho$; 2. (2)

$I(u)=0$* and $I(e)<0$ for some $e\in E$ with $\|e\|>\rho$.*

Define

[TABLE]

and

[TABLE]

Then there exists a sequence $\{u_{k}\}_{k}\subset X$ such that $I(u_{k})\rightarrow c$ and $I^{\prime}(u_{k})\rightarrow 0$ in $X^{\prime}$.

Definition 2.8**.**

Let $I:X\rightarrow\mathbb{R}$ be a $C^{1}$ functional on a Banach space $X$.

1. (1)

For $c\in\mathbb{R}$, a sequence $\{u_{k}\}\subset X$ is a $(PS)_{c}$ (Palais-Smale sequence at level $c$) in $X$ for $I$ if $I(u_{k})=c+o_{k}(1)$ and $I^{\prime}(u_{k})\rightarrow 0$ in $X^{-1}$ as $k\rightarrow\infty.$ 2. (2)

We say $I$ satisfies $(PS)_{c}$ condition if for any Palais-Smale sequence $\{u_{k}\}$ in $X$ for $I$ has a convergent subsequence in X.

We define the energy functional $\mathcal{E}_{\alpha,\lambda}$ corresponding to the problem $(\mathcal{P}_{\alpha,\lambda})$ as

[TABLE]

Then, by Hardy-Littlewood-Sobolev inequality, one can easily see that $\mathcal{E}_{\alpha,\lambda}\in C^{1}(D^{2,p}(\mathbb{R}^{N}),\mathbb{R})$. Moreover, $u$ is a weak solution of the problem $(\mathcal{P}_{\alpha,\lambda})$ if and only if $u$ is a critical point of the functional $\mathcal{E}_{\alpha,\lambda}$. A function $u\in D^{2,p}(\mathbb{R}^{N})$ is said to be a weak solution of $(\mathcal{P}_{\alpha,\lambda})$ if, for all $\phi\in D^{2,p}(\mathbb{R}^{N})$,

[TABLE]

Throughout the article, for $\alpha=1$, we denote the problem $(\mathcal{P}_{\alpha,\lambda})$ by $(\mathcal{P}_{\lambda})$ and energy functional $\mathcal{E}_{\alpha,\lambda}(u)$ by $\mathcal{E}_{\lambda}(u)$.

3. The Palais-Smale condition

This section is divided into two subsections $3.1$ and $3.2$ in which we discuss how the $(PS)_{c}$ sequence satisfies the Palais-Smale condition in the cases $p\theta<2p_{\mu}^{*}$ and $p\theta\geq 2p_{\mu}^{*}$ respectively.

3.1. Case 1: $p\theta<2p_{\mu}^{*}$

In this subsection, we use concentration compactness principle which we proved in section $2$ to show the $(PS)_{c}$ condition for different range of $r$.

Lemma 3.1**.**

Let $p\theta<2p^{*}_{\mu}$ and $1<r<p^{*}$. Then any $(PS)_{c}$ sequence $\{u_{k}\}$ of $\mathcal{E}_{\alpha,\lambda}$ is bounded in $D^{2,p}(\mathbb{R}^{N})$.

Proof.

Let $\{u_{k}\}$ be a $(PS)_{c}$ sequence in $D^{2,p}(\mathbb{R}^{N})$. Then

[TABLE]

and for all $\phi\in C_{c}^{\infty}(\mathbb{R}^{N})$,

[TABLE]

Now using Hölder inequality and Sobolev embedding Theorem, we can easily deduce that

[TABLE]

Case 1: $1<r<p$. Equations (3.1), (3.1) and (3.3) yield, as $k\rightarrow\infty$

[TABLE]

since $p\theta<2p^{*}_{\mu}$, and ${1<r<p}$. This implies that $\{u_{k}\}$ is bounded in $D^{2,p}(\mathbb{R}^{N})$.

Case 2: $r=p$. Again as in case 1, we have, as $k\rightarrow\infty$

[TABLE]

since $0<\lambda<\frac{a}{\lambda S^{-1}\|f\|_{\frac{p^{*}}{p^{*}-r}}}$, $\{u_{k}\}$ is bounded in $D^{2,p}(\mathbb{R}^{N})$.

Case 3: $p<r<p\theta$. Same as in case 1, we have, as $k\rightarrow\infty$

[TABLE]

since $p\theta<2p^{*}_{\mu}$, and ${p<r<p\theta}$. This implies that $\{u_{k}\}$ is bounded in $D^{2,p}(\mathbb{R}^{N})$.

Case 3: $p\theta\leq r<p^{*}$. Using (3.1), (3.1) and (3.3), as $k\rightarrow\infty$, we deduce that

[TABLE]

using the fact that $p\theta\leq r<p^{*}<2p^{*}_{\mu}$. Therefore, $\{u_{k}\}$ is bounded in $D^{2,p}(\mathbb{R}^{N}).$ This complete the proof of Lemma.∎∎

Lemma 3.2**.**

Let $\{u_{k}\}\subset D^{2,p}(\mathbb{R}^{N})$ be a Palais-Smale sequence of functional $\mathcal{E}_{\alpha,\lambda}$. If $p\theta<2p_{\mu}^{*}$, $1<r<p$ and $c<0$. Then the following two properties holds:

1. 1

For each $\lambda>0$ there exists $\overline{\Lambda}>0$ such that $\mathcal{E}_{\alpha,\lambda}$ satisfies the $(PS)_{c}$ condition for all $\alpha\in(0,\overline{\Lambda})$. 2. 2

For each $\alpha>0$ there exists $\underline{\Lambda}>0$ such that $\mathcal{E}_{\alpha,\lambda}$ satisfies the $(PS)_{c}$ condition for all $\lambda\in(0,\underline{\Lambda})$.

This means there exists a subsequence of $\{u_{k}\}$ which converges strongly in $D^{2,p}(\mathbb{R}^{N})$.

Proof.

Let $\{u_{k}\}\subset D^{2,p}(\mathbb{R}^{N})$ be a $(PS)_{c}$ sequence. Then by Lemma 3.1, $\{u_{k}\}$ is a bounded in $D^{2,p}(\mathbb{R}^{N})$. Therefore we can assume that $u_{k}\rightharpoonup u$ in $D^{2,p}(\mathbb{R}^{N})$, $u_{k}\rightarrow u$ a.e in $\mathbb{R}^{N}$.

By Lemma 2.9, there exists at most countable set $J$, sequence of points $\{z_{i}\}_{i\in J}\subset\mathbb{R}^{N}$ and families of positive numbers $\{\mu_{i}:i\in J\}$, $\{\zeta_{i}:i\in J\}$ and $\{\omega_{i}:i\in J\}$ such that

[TABLE]

and

[TABLE]

where $\delta_{z}$ is the Dirac-mass of mass $1$ concentrated at $z\in\mathbb{R}^{N}$.

Moreover, we can construct a smooth cut-off function $\psi_{\epsilon,i}$ centered at $z_{i}$ such that

[TABLE]

for any $\epsilon>0$ small. Since $\{\psi_{\epsilon,i}u_{k}\}$ is a bounded sequence in $D^{2,p}(\mathbb{R}^{N})$, so we have

[TABLE]

One can easily see that

[TABLE]

Now consider

[TABLE]

Also using the same idea as above, we obtain

[TABLE]

Notice that

[TABLE]

Thus

[TABLE]

Now, combining (3.1)-(3.5), we deduce

[TABLE]

Therefore, $a\omega_{i}\leq\alpha\nu_{i}$. Together this with the fact that $S_{H,L}\nu_{i}^{\frac{p}{2p_{\mu}^{*}}}\leq\omega_{i}$, we obtain

[TABLE]

Now, we claim that the first case can not occur. Suppose not, then there exists $i_{0}\in J$ such that $\omega_{i_{0}}\geq\left(a\alpha^{-1}S_{H,L}^{\frac{2N-\mu}{N-2p}}\right)^{\frac{N-2p}{N-\mu+2p}}$. Equation (3.3), the Hölder inequality, the Sobolev inequality and the Young inequality imply that

[TABLE]

Using this fact, we have

[TABLE]

Thus, for any $\alpha>0$, we choose $\lambda_{1}>0$ so small that for every $\lambda\in(0,\lambda_{1})$, the right hand side of (3.1) is greater than zero, which gives a contradiction.

Similarly, if for any $\lambda>0$, we choose $\alpha_{1}>0$ so small that for every $\alpha\in(0,\alpha_{1})$, the right hand side of (3.1) is greater than zero, which gives a required contradiction. Consequently, $\omega_{i}=0$ for all $i\in J$.

To obtain the possible concentration of mass at infinity, we can define a cut-off function $\psi_{R}\in C^{\infty}(\mathbb{R}^{N})$ such that $\psi_{R}(x)=0$ on $|x|<R$, $\psi_{R}(x)=1$ on $|x|>R+1$, $|\nabla\psi_{R}|\leq\frac{2}{R}$ and $|\Delta\psi_{R}|\leq\frac{2}{R^{2}}$.

Now applying the Hardy-Littlewood-Sobolev, Hölder’s inequalitity, we have

[TABLE]

Using the relation $\langle\mathcal{E}_{\alpha,\lambda}^{\prime}(u_{k}),u_{k}\psi_{R}\rangle\rightarrow 0$, we obtain

[TABLE]

Therefore, $a\omega_{\infty}\leq\alpha K\zeta_{\infty}^{\frac{p_{\mu}^{*}}{p^{*}}}$. Combining this with Lemma 2.5, we obtain

[TABLE]

Now,

[TABLE]

Thus, for any $\alpha>0$, we choose $\lambda_{2}>0$ so small that for every $\lambda\in(0,\lambda_{2})$, the right hand side of (3.1) is greater than zero, which gives a contradiction. Similarly, if for any $\lambda>0$, we choose $\alpha_{2}>0$ so small that for every $\alpha\in(0,\alpha_{2})$, the right hand side of (3.1) is greater than zero, which gives a required contradiction. Consequently, $\omega_{\infty}=0$.

From the above arguments, Take $\overline{\Lambda}=\min\{\lambda_{1},\lambda_{2}\}$ and $\underline{\Lambda}=\min\{\alpha_{1},\alpha_{2}\}$.

Then for any $c<0$, $\alpha>0$, we have $\omega_{i}=0$ for all $i\in J$ and $\omega_{\infty}=0$ for all $\lambda\in(0,\overline{\Lambda})$.

Similarly, for any $c<0$, $\lambda>0$, we have $\omega_{i}=0$ for all $i\in J$ and $\omega_{\infty}=0$ for all $\alpha\in(0,\underline{\Lambda})$.

Hence,

[TABLE]

[TABLE]

Since $(\|u_{k}\|)_{k}$ is bounded and $\mathcal{E}^{\prime}_{\alpha,\lambda}(u)=0$, the weak lower semicontinuity of the norm and the Brézis-Lieb Lemma yield as $(k\rightarrow\infty)$

[TABLE]

Thus $\{u_{k}\}$ converges strongly to $u$ in $D^{2,p}(\mathbb{R}^{N})$. This completes the proof of the Lemma.∎∎

Lemma 3.3**.**

Let $r=p$ and $p\theta<2p_{\mu}^{*}$. Suppose that $\{u_{k}\}$ is a $(PS)_{c}$ sequence for $\mathcal{E}_{\alpha,\lambda}$ in $D^{2,p}(\mathbb{R}^{N})$, with

[TABLE]

*where $c_{1}:=\left(\frac{1}{p}-\frac{1}{2p^{*}_{\mu}}\right)\left(aS_{H,L}\alpha^{-\frac{N-2p}{2N-\mu}}\right)^{\frac{2N-\mu}{N-\mu+2p}}$ and $c_{2}:=\left(\frac{1}{p}-\frac{1}{2p_{\mu}^{*}}\right)\left(aS\right)^{\frac{p^{*}_{\mu}}{p^{*}_{\mu}-p}}(\alpha K)^{-\frac{p}{p^{*}_{\mu}-p}}$.

Then for all $\lambda\in\left(0,aS\|f\|_{\frac{p^{*}}{p^{*}-p}}^{-1}\right)$, $\{u_{k}\}$ satisfies the $(PS)_{c}$ condition.*

Proof.

For $u\in D^{2,p}(\mathbb{R}^{N})$ and $r=p$, the Hölder inequality and Sobolev inequality imply that

[TABLE]

Let $\{u_{k}\}$ be a $(PS)_{c}$ for $\mathcal{E}_{\alpha,\lambda}$ for $c<c^{*}$. Then $\{u_{k}\}$ is bounded from Lemma 3.1. Now using the last estimate, for all $\lambda\in\left(0,aS\|f\|_{\frac{p^{*}}{p^{*}-p}}^{-1}\right)$, arguing similarly as in Lemma 3.2, in substitute of (3.1), we obtain

[TABLE]

Following the same argument as in Lemma 3.2 for concentration of mass at infinity and using equation (3.1), we obtain

[TABLE]

which is absurd since $c<c^{*}:=min\{c_{1},c_{2}\}$. Now the rest of the proof follows in similar manner as in the proof of Lemma 2.10. ∎

Lemma 3.4**.**

Let $p\theta\leq r<p^{*}$ and $p\theta<2p_{\mu}^{*}$. Suppose that $\{u_{k}\}$ be a $(PS)_{c}$ sequence for $\mathcal{E}_{\alpha,\lambda}$ in $D^{2,p}(\mathbb{R}^{N})$ with $c<c^{*},$ where $c^{*}$ is defined same as in Lemma 3.3. Then $\{u_{k}\}$ satisfies $(PS)_{c}$ condition.

Proof.

Let $\{u_{k}\}$ be a $(PS)_{c}$ for $\mathcal{E}_{\alpha,\lambda}$ for $c<c^{*}$. Then by Lemma 3.1, we have $\{u_{k}\}$ is bounded. Now following the similar arguments as in Lemma 3.2, we get

[TABLE]

Following the same as in Lemma 3.2, for mass at infinity and using (3.1), we obtain

[TABLE]

which is contradiction since $c<c^{*}:=min\{c_{1},c_{2}\}$. The rest of the proof follows as in the proof of Lemma 2.10. ∎

3.2. Case 2: $p\theta\geq 2p_{\mu}^{*}$

In this subsection, we prove the strong convergence of the $(PS)_{c}$ sequence in both the cases non-degenerate and degenerate.

To start with the energy functional $\mathcal{E}_{\lambda}$ corresponding to the problem $(\mathcal{P}_{\lambda})$ when $\alpha=1$

[TABLE]

Lemma 3.5**.**

Let $p\geq 2$, $f$ satisfies $(f_{1})-(f_{2})$ and $p\theta\geq 2p_{\mu}^{*}$ . Then $\mathcal{E}_{\lambda}$ satisfies the $(PS)_{c}$ condition in $D^{2,p}(\mathbb{R}^{N})$ for all $\lambda>0$, in the following cases:

1. (1)

$p<r<p^{*}$. If either $p\theta=2p_{\mu}^{*}$, $a>0$ and $b>2^{p}S_{H,L}^{-\frac{2p_{\mu}^{*}}{p}}$ or $p\theta>2p_{\mu}^{*}$, $a>0$ and $b>b^{1}$, where $b^{1}$ is defined in (1.4). 2. (2)

$1\leq r<p^{*}$, $a=0$, $b>2^{p}S_{H,L}^{-\frac{2p_{\mu}^{*}}{p}}$.

Proof.

For all $\lambda>0$, suppose $\{u_{k}\}$ be a Palais-Smale sequence of $\mathcal{E}_{\lambda}$ in $D^{2,p}(\mathbb{R}^{N})$ at any level $c<0$. Then by Lemma 3.1, $\{u_{k}\}$ is a bounded in $D^{2,p}(\mathbb{R}^{N})$. Therefore as $k\rightarrow\infty$ we can assume up to a subsequence still denoted by $\{u_{k}\}$ such that $u_{k}\rightharpoonup u$ weakly in $D^{2,p}(\mathbb{R}^{N})$, $u_{k}\rightarrow u$ a.e in $\mathbb{R}^{N}$, $u_{k}\rightarrow u$ strongly in $L^{q}_{loc}(\mathbb{R}^{N})$ for $1\leq q<p^{*}$ and $|u_{k}|^{p^{*}-2}u_{k}\rightharpoonup|u|^{p^{*}-2}u$ weakly in $L^{\frac{p^{*}}{p^{*}-1}}(\mathbb{R}^{N})$. We claim that

[TABLE]

As $f\in L^{\frac{p^{*}}{p^{*}-r}}(\mathbb{R}^{N})$, for any $\epsilon>0$, there exist $R_{\epsilon}>0$ such that

[TABLE]

Then Hölder inequality and above inequality yield

[TABLE]

Now, for any non-empty measurable subset $\Omega\subset B_{R_{\epsilon}}$ and boundedness of $\{u_{k}\}$ give

[TABLE]

This implies the sequence $\{f(x)(|u_{k}|^{r}-|u|^{r})\}$ is equi-integrable in $B_{R_{\epsilon}}(0)$. Hence Vitali convergence Theorem implies

[TABLE]

Combining (3.10) and (3.11), conclude the claim (3.9). Moreover, one can easily deduce that

[TABLE]

Then, we may assume that $\displaystyle\lim_{k\rightarrow\infty}\|u_{k}-u\|=l$

[TABLE]

Also, the definition of weak convergence in $D^{2,p}(\mathbb{R}^{N})$,

[TABLE]

Since $\{u_{k}\}$ is a $(PS)_{c}$ sequence, by the boundedness of $\{u_{k}\}$, (3.12), we have

[TABLE]

Now using the following inequality, for any $p\geq 2$,

[TABLE]

together with the definition of $S_{H,L}$ yield

[TABLE]

Taking limit $k\rightarrow\infty$, we have

[TABLE]

which imply

[TABLE]

Case 1 : When $a>0$, $p\theta=2p_{\mu}^{*}$, and $2^{p}S_{H,L}^{-\frac{2p_{\mu}^{*}}{p}}<b$, from (3.13), one can easily deduce that $l=0$. Thus $u_{k}\rightarrow u$ in $D^{2,p}(\mathbb{R}^{N})$.

Case 2: When $a=0$, $p\theta=2p_{\mu}^{*}$, and $2^{p}S_{H,L}^{-\frac{2p_{\mu}^{*}}{p}}<b$, equation (3.13) yield that $l=0$. Thus $u_{k}\rightarrow u$ in $D^{2,p}(\mathbb{R}^{N})$.

Case 3: When $a>0$, $p\theta>2p_{\mu}^{*}$ and $b>b^{1}$.

Applying Young’s inequality in the right hand side of (3.13), we obtain

[TABLE]

where $b^{1}$ is given in (1.4). Thus $(b-b^{*})l^{p\theta}\leq 0$. In view of (1.4), we deduce $l=0$. Hence, we obtain that $u_{k}\rightarrow u$ strongly in $D^{2,p}(\mathbb{R}^{N})$, as required. ∎

4. Case: $p\theta\geq 2p_{\mu}^{*}$

In this section, first we show that the functional is coercive and bounded below and then we prove the Theorems 1.4 and 1.5.

Lemma 4.1**.**

Let $p\theta\geq 2p_{\mu}^{*}$, $\alpha=1$ and $1<r<p^{*}$. Then show that $\mathcal{E}_{\lambda}$ is coercive and bounded below for all $a\geq 0$ and $b>\begin{cases}&S^{-\frac{2p^{*}_{\mu}}{p}}_{H,L}\mbox{\;\; if}\ p\theta=p^{*}_{\mu}\\ &0\quad\quad\ \mbox{\;\; if}\ p\theta>p^{*}_{\mu}\end{cases}$.

Proof.

Let $u\in D^{2,p}(\mathbb{R}^{N})$. Then equations (2.3) and (3.3) yield

[TABLE]

Case 1: $a\geq 0$, $p<r<p^{*}$ and $p\theta>2p_{\mu}^{*}$,

[TABLE]

Case 2: $a=0$, $1<r<p^{*}$ and $p\theta=2p_{\mu}^{*}$,

[TABLE]

Case 2: $a>0$, $p<r<p^{*}$ and $p\theta=2p_{\mu}^{*}$,

[TABLE]

In all the cases, we conclude that $\mathcal{E}_{\lambda}$ is bounded below and coercive. ∎

So, we define $m:=\displaystyle\inf_{u\in D^{2,p}(\mathbb{R}^{N})}\mathcal{E}_{\lambda}(u)$, which is well defined by Lemma 4.1.

Proof of Theorem 1.4: We first show that the problem $(\mathcal{P}_{\lambda})$ has a non-trivial least energy solution.

We claim that there exists $\lambda^{1}>0$ such that $m<0$ for all $\lambda>\lambda^{1}$.

Choose a function $\Phi\in D^{2,p}(\mathbb{R}^{N})$ with $\|\Phi\|=1$ and $\int_{\mathbb{R}^{N}}f(x)|\Phi|^{r}dx>0$, which is possible due to $f\geq 0$ and $f\not\equiv 0$ in $\mathbb{R}^{N}$. Then

[TABLE]

for all $\lambda>\lambda^{1}$ with $\lambda^{1}=\frac{r\left(\frac{a}{p}+\frac{b}{p\theta}\right)}{\int_{\mathbb{R}^{N}}f(x)|\Phi|^{r}dx}$.

Hence, by Lemma 3.5 and [[31], Theorem 4.4], there exists $u_{1}\in D^{2,p}(\mathbb{R}^{N})$ such that $\mathcal{E}_{\lambda}(u_{1})=m$ and $\mathcal{E}^{\prime}_{\lambda}(u_{1})=0$. Thus $u_{1}$ is a non-trivial solution of $(\mathcal{P}_{\lambda})$ with $\mathcal{E}_{\lambda}(u_{1})<0$.

Next, we show that $(\mathcal{P}_{\lambda})$ has a mountain pass solution. For all $u\in D^{2,p}(\mathbb{R}^{N})$, using (3.3) and (2.3), we obtain

[TABLE]

Since ${p<r<p^{*}}$, there exists $\rho>0$ small enough and $\eta>0$ such that $\mathcal{E}_{\lambda}(u)>\eta$ with $\|u\|=\rho$ for all $u\in D^{2,p}(\mathbb{R}^{N})$. Define

[TABLE]

where $\Gamma(t)=\{\gamma:\gamma\in(C[0,1],D^{2,p}(\mathbb{R}^{N})):\gamma(0)=0,\gamma(1)=u_{1}\}.$ Then $c>0$. Lemma 3.5 yields that $\mathcal{E}_{\lambda}$ satisfies the assumption of the mountain pass Lemma, see [4, Theorem 2.1]. Thus there exists $u_{2}\in D^{2,p}(\mathbb{R}^{N})$ such that $\mathcal{E}_{\lambda}(u_{2})=c>0$ and $\mathcal{E}^{\prime}_{\lambda}(u_{2})=0$. Hence, $u_{2}$ is a non-trivial solution of $(\mathcal{P}_{\lambda})$ different from $u_{1}$.∎

To prove Theorem 1.5: We will use Kranoselskii’s genus theory [22]. Let $X$ be a real Banach space and $\sum$ the family of the set $E\subset X\setminus\{0\}$ such that $E$ is closed in $X$ and symmetric with respect to [math], i.e.

[TABLE]

For each $E\in\sum$, we say genus of $E$ is a number $k$ denoted by $\gamma(E)=k$ if there is an odd map $h\in C(E,\mathbb{R}^{N}\setminus\{0\})$ and $k$ is the smallest integer with this property.

Lemma 4.2**.**

([11]) Let $X=\mathbb{R}^{N}$ and $\partial\Omega$ be the boundary of an open, symmetric, and bounded subset $\Omega\subset\mathbb{R}^{N}$ with $0\in\Omega$. Then $\gamma(\partial\Omega)=N$.

It follows from Lemma 4.2, $\gamma(\mathbb{S}^{N-1})=N$, where $\mathbb{S}^{N-1}$ the surface of the unit sphere in $\mathbb{R}^{N}$.

Now, we will use the following Theorem to obtain the existence of infinitely many solutions of $(\mathcal{P}_{\lambda})$.

Theorem 4.3**.**

([11], [14]) Let $I\in C^{1}(X,\mathbb{R})$ be an even functional satisfying $(PS)$ condition. Furthermore

1. (1)

$I$* is bounded from below and even.* 2. (2)

There is a compact set $E\in\sum$ such that $\gamma(E)=n$ and $\displaystyle\sup_{u\in\sum}I(u)<I(0)$,

then $I$ has at least $n$ pairs of distinct critical points and their corresponding critical values are less than $I(0)$.

Proof of Theorem 1.5: Let $\{e_{1},e_{2},\cdots\}$ be a Schauder basis of $D^{2,p}(\mathbb{R}^{N})$. Foe each $n\in\mathbb{N}$, define $E_{n}=span\{e_{1},e_{2},\cdots,e_{n}\}$, the subspace of $D^{2,p}(\mathbb{R}^{N})$ generated by $e_{1},e_{2},\cdot\cdot\cdot,e_{n}$. Define $L^{r}(\mathbb{R}^{N},f)=\{u:\mathbb{R}^{N}\rightarrow\mathbb{R}|\int_{\mathbb{R}^{N}}f(x)|u(x)|^{r}dx<\infty\}$, endowed with the norm

[TABLE]

By assumption $(f_{1})$, one can easily see that $E_{n}$ can be continuously embedded into $L^{r}(\mathbb{R}^{N},f)$. As we know that all the norms are equivalent on a finite dimensional Banach space. Thus there exist a constant $C(n)$ depending on $n$ such that for all $u\in E_{n},$

[TABLE]

Then

[TABLE]

Choose $R>0$ be a constant such that $\frac{b}{p\theta}R^{p\theta-r}<\lambda C(n).$ Hence for all $0<t<R$,

[TABLE]

for all $u\in K:=\{u\in E_{n}:\|u\|=t\}$. Then it follows that

[TABLE]

Clearly, $E_{n}$ and $\mathbb{R}^{N}$ are isomorphic and $K$ and $\mathbb{S}^{N-1}$ are homeomorphic. Therefore, we conclude that $\gamma(K)=n$ by Lemma 4.2. Moreover, $\mathcal{E}_{\lambda}$ is bounded below and satisfies $(PS)_{c}$ condition by Lemmas 4.1 and 3.5. Thus, Theorem 4.3 give $\mathcal{E}_{\lambda}$ has at least $n$ pair of distinct critical points. The arbitrariness of $n$ yields that $\mathcal{E}_{\lambda}$ has infinitely many pairs of distinct solutions.

Let $u\in D^{2,p}(\mathbb{R}^{N})\setminus\{0\}$ be a solution of $(\mathcal{P}_{\lambda})$. Then

[TABLE]

Using this together with (3.3) and (2.3) yield,

[TABLE]

Since $b>2^{p}S_{H,L}^{-\frac{2p_{\mu}^{*}}{p}}$, $p\theta=2p_{\mu}^{*}$, we obtain

[TABLE]

implies

[TABLE]

Hence the proof is complete.∎

5. Case: $p\theta<2p_{\mu}^{*}$

This section is divided into three subsections $5.1$, $5.2$ and $5.3$. In subsections $5.1$, $5.2$ and $5.3$, we give the proofs of our main Theorems for $1<r<p$, $r=p$ and $p\theta\leq r<p^{*}$ respectively.

5.1. $1<r<p$

In this subsection, we first recall the definition of genus and then to prove Theorem 1.1, we use a result by Kajikiya see [[20], Theorem 1], which is an extension of the symmetric mountain pass theorem.

Definition 5.1**.**

Let $X$ be a Banach space, and $A$ be a subset of $X$. The set $A$ is said to be symmetric if $u\in A$ implies $-u\in A.$ For a closed symmetric set $A$ which does not contain the origin, we define a genus $\gamma(A)$ of $A$ by the smallest integer $k$ such that there exists an odd continuous mapping from $A$ to $\mathbb{R}^{k}\setminus\{0\}$. If there does not exist such $k$, we define $\gamma(A)=\infty$ . Moreover, we set $\gamma(\emptyset)=0$ .

For any $n\in\mathbb{N}$, let us define the set $\Sigma_{n}$ as

[TABLE]

Theorem 5.2**.**

Let $X$ be an infinite dimensional Banach space and $I\in C^{1}(X,\mathbb{R})$. Suppose that the following hypotheses hold.

• $(\mathcal{A}_{1})$

The functional $I$ is even and bounded from below in $X$, $I(0)=0$ and $I$ satisfies the local Palais-Smale condition.

• $(\mathcal{A}_{2})$

For each $n\in\mathbb{N},$ there exists $A_{n}\in\Sigma_{n}$ such that

[TABLE]

Then $I$ admits a sequence of critical points $\{u_{n}\}$ in $X$ such that $u_{n}\not=0$, $I(u_{n})\leq 0$ for each $n$ and $u_{n}\to 0$ in $X$ as $n\to\infty$.

Proof of Theorem 1.1 : From the hypotheses, it follows that $\mathcal{E}_{\alpha,\lambda}$ is even and $\mathcal{\mathcal{E}}_{\alpha,\lambda}(0)=0.$ Also Lemma 3.4 ensures that $\mathcal{E}_{\alpha,\lambda}$ satisfies the $(PS)_{c}$ condition for all $c<0$. But observe that, $\mathcal{E}_{\alpha,\lambda}$ is not bounded from below in $D^{2,p}(\mathbb{R}^{N})$. So, for applying Theorem 5.2, we use a truncation technique.

Let $w\in D^{2,p}(\mathbb{R}^{N})$. Then equations (3.3) and (2.3) yield

[TABLE]

Define the function $\mathcal{K}:\mathbb{[}0,\infty)\to\mathbb{R}$ as

[TABLE]

Since $1<r<p$, for any $\alpha>0$ we can choose $\lambda_{0}$ sufficiently small such that for all $\lambda\in(0,\lambda_{0})$ there exist $0<t_{1}<t_{2}$ so that $\mathcal{K}<0$ in $(0,t_{1})$, $\mathcal{K}>0$ in $(t_{1},t_{2})$ and $\mathcal{K}<0$ in $(t_{2},\infty)$.

Similarly, for any $\lambda>0$ we can choose $\alpha_{0}$ sufficiently small such that for all $\alpha\in(0,\alpha_{0})$ there exist $0<t_{1}^{\prime}<t_{2}^{\prime}$ so that $\mathcal{K}<0$ in $(0,t_{1}^{\prime})$, $\mathcal{K}>0$ in $(t_{1}^{\prime},t_{2}^{\prime})$ and $\mathcal{K}<0$ in $(t_{2}^{\prime},\infty)$. Therefore $\mathcal{K}(t_{1})=0=\mathcal{K}(t_{2})$ and $\mathcal{K}(t_{1}^{\prime})=0=\mathcal{K}(t_{2}^{\prime})$. Next, we choose a non-increasing function $\mathcal{I}\in C^{\infty}([0,\infty),[0,1])$ such that

[TABLE]

and set $\Phi(u):=\mathcal{I}(\|u\|).$ Now we define the truncated functional $\hat{\mathcal{E}}_{\alpha,\lambda}:D^{2,p}(\mathbb{R}^{N})\to\mathbb{R}$ of $\mathcal{E}_{\alpha,\lambda}$ as

[TABLE]

Then, it can easily seen that $\hat{\mathcal{E}}_{\alpha,\lambda}$ satisfies the following:

1. (1)

$\hat{\mathcal{E}}_{\alpha,\lambda}\in C^{1}(D^{2,p}(\mathbb{R}^{N}),\mathbb{R})$, $\hat{\mathcal{E}}_{\alpha,\lambda}(0)=0$. 2. (2)

$\hat{\mathcal{E}}_{\alpha,\lambda}$ is even, coercive and bounded from below in $D^{2,p}(\mathbb{R}^{N})$. 3. (3)

Let $c<0$, then for any $\lambda>0$ there exists $\overline{\Lambda}>0$ such that for all $\alpha\in(0,\overline{\Lambda})$, $\hat{\mathcal{E}}_{\alpha,\lambda}$ satisfies the Palais-Smale condition, by Lemma 2.10. 4. (4)

Let $c<0$, then for any $\alpha>0$ there exists $\underline{\Lambda}>0$ such that $\hat{\mathcal{E}}_{\alpha,\lambda}$ satisfies the Palais-Smale condition for all $\alpha\in(0,\underline{\Lambda})$, by Lemma 2.10. 5. (5)

If $\hat{\mathcal{E}}_{\alpha,\lambda}(u)<0,$ then $\|u\|\leq t_{1}$ and $\hat{\mathcal{E}}_{\alpha,\lambda}(u)=\mathcal{E}_{\alpha,\lambda}(u)$.

For any $n\in\mathbb{N}$, we consider $n$ numbers of disjoint open sets denoted by $V_{j}$, $j=1,2,\cdots n$ with $\cup_{j=1}^{n}V_{j}\subset\Omega$, where $\Omega\not=\emptyset$ is given as in Theorem 1.1. Now we choose $u_{j}\in D^{2,p}(\mathbb{R}^{N})\cap C_{0}^{\infty}(V_{j})\setminus\{0\}$, with $\|u_{j}\|=1$ for each $j=1,2,\cdots,n$. Set

[TABLE]

Now we claim that there exists $0<\varrho_{n}<t_{1},$ sufficiently small such that

[TABLE]

Suppose that (5.3) does not hold. Then there exists a sequence $\{u_{k}\}:=\{u_{k}^{(n)}\}$ in $X_{n}$ such that

[TABLE]

Let’s set

[TABLE]

Then $w_{k}\in D^{2,p}(\mathbb{R}^{N})$ and $\|w_{k}\|=1.$ Since $X_{n}$ is finite dimensional, there exists $w\in X_{n}\setminus\{0\}$ such that

[TABLE]

As $w\not\equiv 0$, we get $|u_{k}(x)|\to\infty$ as $k\to\infty$. Thus, as $k\to\infty$,

[TABLE]

Using this together with (5.2), we obtain

[TABLE]

as $k\to\infty.$ This contradicts (5.4). Thus, the claim is proved.

Now choose $A_{n}:=\{u\in X_{n}\;:\;\|u\|=\varrho_{n}\}$. Clearly $\gamma(A_{n})=n$ and $A_{n}$ is closed and symmetric, and hence $A_{n}\in\Sigma_{n}$ and also from (5.3), $\sup_{u\in A_{n}}\hat{\mathcal{E}}_{\alpha,\lambda}(u)<0.$ Therefore, $\hat{\mathcal{E}}_{\alpha,\lambda}$ satisfies all the assumption in Theorem 5.2. Thus, $\hat{\mathcal{E}}_{\alpha,\lambda}$ admits a sequence of critical points $\{u_{n}\}$ in $D^{2,p}(\mathbb{R}^{N})$ such that $u_{n}\not=0$, $\hat{\mathcal{E}}_{\alpha,\lambda}(u_{n})\leq 0$ for each $n\in\mathbb{N}$ and $\|u_{n}\|\to 0$ as $n\to\infty$. So, for $t_{1}>0$, there exists $n_{0}\in\mathbb{N}$ such that for all $n\geq n_{0}$ it follows that $\|u\|<t_{1}$ which yields that $\hat{\mathcal{E}}_{\alpha,\lambda}(u_{n})={\mathcal{E}}_{\alpha,\lambda}(u_{n})$ for all $n>n_{0}.$ This concludes the proof of the theorem.∎

5.2. $r=p$ and $\alpha=1$

In this subsection, we use the following $\mathbb{Z}_{2}$-symmetric version of mountain pass theorem due to [40], to prove Theorem 1.2.

Theorem 5.3**.**

Let $X$ be an infinite dimensional Banach space with $X=Y\oplus Z$, where $Y$ is finite dimensional and let $I\in C^{1}(X,\mathbb{R})$ be an even functional with $I(0)=0$ such that the following conditions hold.

1. $(\mathcal{I}_{1})$

There exist positive constants $\rho,\alpha>0$ such that $I(u)\geq\alpha$ for all $u\in\partial B_{\rho}(0)\cap Z$; 2. $(\mathcal{I}_{2})$

There exists $c^{*}>0$ such that $I$ satisfies the $(PS)_{c}$ condition for $0<c<c^{*}$; 3. $(\mathcal{I}_{3})$

For any finite dimensional subspace $\tilde{X}\subset X$, there exists $R=R(\tilde{X})>0$ such that $I(u)\leq 0$ for all $u\in\tilde{X}\setminus B_{R}(0)$.

Assume that $Y$ is $k-$dimensional and $Y=span\{v_{1},v_{2},\cdots,v_{k}\}$. For $n\geq k$, inductively choose $v_{n+1}\not\in Y_{n}:=span\{v_{1},v_{2},\cdots,v_{n}\}.$ Let $R_{n}=R(Y_{n})$ and $D_{n}=B_{R_{n}}(0)\cap Y_{k}$. Define

[TABLE]

and

[TABLE]

[TABLE]

where $\gamma(E)$ is Krasnoselskii’s genus of $E$. For each $j\in\mathbb{N}$, set

[TABLE]

Then $0<\alpha\leq c_{j}\leq c_{j+1}$ for $j>k$ and if $j>k$ and $c_{j}<c^{*}$, then $c_{j}$ is a critical value of $I$. Furthermore, if $c_{j}=c_{j+1}=\cdots=c_{j+m}=c<c^{*}$ for $j>k$, then $\gamma(K_{c})\geq m+1$, where

[TABLE]

Now we show that $\mathcal{E}_{\lambda}$ satisfies all the hypotheses of Theorem 5.3, when $r=p$.

Lemma 5.4**.**

Let $\alpha=1$ and $r=p$. Then $\mathcal{E}_{\lambda}$ satisfies the conditions $(\mathcal{I}_{1})$-$(\mathcal{I}_{3})$ of Theorem 5.3 for all $\lambda\in(0,\,aS\|f\|^{-1}_{\frac{p^{*}}{p^{*}-p}}).$

Proof.

We first show the hypotheses $(\mathcal{I}_{1})-(\mathcal{I}_{3})$ of Theorem 5.3.

$(\mathcal{I}_{1}):$ For $u\in D^{2,p}(\mathbb{R}^{N}),$ arguing similarly as in (5.1), we have

[TABLE]

Now for $\lambda<aS\|f\|^{-1}_{\frac{p^{*}}{p^{*}-p}}$, we can choose $\|u\|=l<<1$ so that $\mathcal{E}_{\lambda}(u)\geq\mathcal{K}>0.$

$(\mathcal{I}_{2}):$ It follows from Lemma 3.3.

$(\mathcal{I}_{3}):$ To show this, first claim that for any finite dimensional subspace $Y$ of $D^{2,p}(\mathbb{R}^{N})$ there exists $R_{0}=R_{0}(Y)$ such that $\mathcal{E}_{\lambda}(u)<0$ for all $u\in D^{2,p}(\mathbb{R}^{N})\setminus B_{R_{0}}(Y),$ where $B_{R_{0}}(Y)=\{u\in D^{2,p}(\mathbb{R}^{N}):\|u\|\leq R_{0}\}.$ Fix $\phi\in D^{2,p}(\mathbb{R}^{N}),\;\|\phi\|=1.$ For $t>1$, we get

[TABLE]

Since $Y$ is finite dimensional, all norms are equivalent on $Y$, which yields that there exists some constant $C(Y)>0$ such that $C(Y)\|\phi\|\leq\|\phi\|_{\mu}.$ Therefore from (5.2), we obtain

[TABLE]

as $t\to\infty$. Hence, there exists $R_{0}>0$ large enough such that $\mathcal{E}_{\lambda}(u)<0$ for all $u\in D^{2,p}(\mathbb{R}^{N})$ with $\|u\|={R}$ and $R\geq R_{0}$. Therefore, $\mathcal{E}_{\lambda}$ satisfies the assertion $(\mathcal{I}_{3})$. ∎

Lemma 5.5**.**

There exists a non-decreasing sequence $\{S_{n}\}$ of positive real numbers, independent of $\lambda$ such that for any $\lambda>0$, we have

[TABLE]

where $\Gamma_{n}$ is defined in (5.5).

Proof.

Recalling the definition of $c_{n}^{\lambda}$ and (5.2), we get

[TABLE]

Then clearly from the definition of $\Gamma_{n}$, it follows that $S_{n}<\infty$ and $S_{n}\leq S_{n+1}$. ∎

Proof of Theorem 1.2: From the hypotheses of the theorem it follows that $\mathcal{E}_{\alpha,\lambda}$ is even and we have $\mathcal{E}_{\alpha,\lambda}(0)=0.$ From the Lemma 5.5, we can choose, $\hat{a}>0$ sufficiently large such that for any $a>\hat{a}$,

[TABLE]

that is,

[TABLE]

Hence, for all $\lambda\in(0,\,aS\|f\|^{-1}_{\frac{p^{*}}{p^{*}-p}})$ and $a>\hat{a}$, we have

[TABLE]

Now by Theorem 5.3, we infer that the levels $c_{1}^{\lambda}\leq c_{2}^{\lambda}\leq\cdots\leq c_{n}^{\lambda}$ are critical values of $\mathcal{I}_{\lambda}.$ Therefore, if $c_{1}^{\lambda}<c_{2}^{\lambda}<\cdots<c_{n}^{\lambda}$, then $\mathcal{E}_{\lambda}$ has at least $n$ number of critical points. Furthermore, if $c_{j}^{\lambda}=c_{j+1}^{\lambda}$ for some $j=1,2,\cdots,k-1$, then again Theorem 5.3 yields that $A_{c_{j}^{\lambda}}$ is an infinite set. Hence, the problem $(\mathcal{P}_{\lambda})$ has infinitely many solutions. Consequently, the problem $(\mathcal{P}_{\lambda})$ has at least $n$ pairs of solutions in $D^{2,p}(\mathbb{R}^{N})$.∎

5.3. $p\theta\leq r<p^{*}$ and $\alpha=1$

In this subsection, we prove Theorem 1.3 using Theorem 5.3. For that, first we show $\mathcal{E}_{\lambda}$ verifies all the hypotheses of Theorem 5.3, when $p\theta\leq r<p^{*}$.

Lemma 5.6**.**

Let $\alpha=1$ and $p\theta\leq r<p^{*}$. Then $\mathcal{E}_{\lambda}$ satisfies the conditions $(\mathcal{I}_{1})$-$(\mathcal{I}_{3})$ of Theorem 5.3 in the following cases:

1. (1)

If $r=p\theta$ and $\lambda\in\left(0,bS^{\theta}||f||_{\frac{p^{*}}{p^{*}-p\theta}}^{-1}\right)$. 2. (2)

If $p\theta<r<p^{*}$ and $\lambda>0$.

Proof.

Let $u\in D^{2,p}(\mathbb{R}^{N})$ with $\|u\|<1$. Using the similar arguments as in (5.1), we get

[TABLE]

If $r=p\theta$

[TABLE]

In the view of (5.7), (5.3), $p<r$ and $p<p_{\mu}^{*}$, we can choose $0<\rho<1$ sufficiently small so that, we obtain for all $u\in D^{2,p}(\mathbb{R}^{N})$ with $\|u\|=\rho$, $\mathcal{E}_{\lambda}(u)\geq\beta>0$ for some $\beta>0$ depending on $\rho$. Thus $(\mathcal{I}_{1})$ holds.

$(\mathcal{I}_{2})$ follows from Lemma 3.4, since $c^{**}>0$ and for $(\mathcal{I}_{3})$, the argument follows similarly to that of Lemma 5.4. ∎

Proof of Theorem 1.3 Using Lemma 5.6 and proceeding in a way similar to Lemma 5.5 and in Theorem 5.3, we can conclude that the problem $(\mathcal{P}_{\lambda})$ has at least $n$ pairs of solutions for all $\lambda>0$.∎

Acknowledgements : The first and second authors would like to thank the Science and Engineering Research Board, Department of Science and Technology, Government of India, for the financial support under the grant SRG/2022/001946 and SPG/2022/002068, respectively.