# Infinitely many solutions for p-harmonic equation with singular term

## Abstract

In this paper, we study the following p-harmonic problem involving the Hardy term:

where Ω is an open bounded domain containing the origin in ${\mathbb{R}}^{N}$, $1 and $0\le \lambda <{\left[N\left(p-1\right)\left(N-2p\right)/{p}^{2}\right]}^{p}$. By using the variational method, we prove that the above problem has infinitely many solutions with positive energy levels.

AMS Subject Classification:35J60, 35J65.

## 1 Introduction

The main purpose of this paper is to show the existence of infinitely many solutions for the following p-harmonic equation:

$\left\{\begin{array}{cc}\mathrm{\Delta }\left({|\mathrm{\Delta }u|}^{p-2}\mathrm{\Delta }u\right)-\lambda \frac{{|u|}^{p-2}u}{{|x|}^{2p}}=f\left(x,u\right),\hfill & x\in \mathrm{\Omega },\hfill \\ u=\frac{\partial u}{\partial n}=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}$
(1.1)

where Ω is an open bounded domain containing the origin in ${\mathbb{R}}^{N}$, the boundary Ω is smooth. $1, $0\le \lambda <\overline{\lambda }={\left[N\left(p-1\right)\left(N-2p\right)/{p}^{2}\right]}^{p}$. $\frac{\partial }{\partial n}$ is the outer normal derivative. Nonlinearity $f\left(x,u\right)$ satisfies the following conditions:

(f1) $f\left(x,u\right)$ is continuous on $\mathrm{\Omega }×\mathbb{R}$ and limits subcritical growing at infinity; that is,

(1.2)

where ${p}^{\ast }=Np/\left(N-2p\right)$ is the critical exponent of Sobolev’s embedding ${W}_{0}^{2,p}\left(\mathrm{\Omega }\right)↪{L}^{s}\left(\mathrm{\Omega }\right)$.

(f2) For any $x\in \mathrm{\Omega }$, $f\left(x,u\right)$ satisfies

$\underset{u\to 0}{lim}\frac{f\left(x,u\right)u}{{|u|}^{p}}=0\phantom{\rule{1em}{0ex}}\text{and}\phantom{\rule{1em}{0ex}}\underset{u\to \mathrm{\infty }}{lim}\frac{f\left(x,u\right)u}{{|u|}^{p}}=+\mathrm{\infty }.$
(1.3)

(f3) Denote $G\left(x,t\right)=f\left(x,t\right)t-p{\int }_{0}^{t}f\left(x,s\right)\phantom{\rule{0.2em}{0ex}}ds$. For all $x\in \mathrm{\Omega }$, there exists a constant $M\ge 0$ such that

(1.4)

(f4) $f\left(x,u\right)$ is odd with respect to u.

By the Hardy-Rellich inequality (see [1, 2]), we know that

${\int }_{\mathrm{\Omega }}\frac{{|u|}^{p}}{{|x|}^{2p}}\phantom{\rule{0.2em}{0ex}}dx\le \frac{1}{\overline{\lambda }}{\int }_{\mathrm{\Omega }}{|\mathrm{\Delta }u|}^{p}\phantom{\rule{0.2em}{0ex}}dx.$
(1.5)

Obviously, for any $\lambda \in \left[0,\overline{\lambda }\right)$,

$\left(1-\frac{\lambda }{\overline{\lambda }}\right){\int }_{\mathrm{\Omega }}{|\mathrm{\Delta }u|}^{p}\phantom{\rule{0.2em}{0ex}}dx\le {\int }_{\mathrm{\Omega }}\left({|\mathrm{\Delta }u|}^{p}-\lambda \frac{{|u|}^{p}}{{|x|}^{2p}}\right)\phantom{\rule{0.2em}{0ex}}dx\le {\int }_{\mathrm{\Omega }}{|\mathrm{\Delta }u|}^{p}\phantom{\rule{0.2em}{0ex}}dx.$

In ${W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$, for$\lambda \in \left[0,\overline{\lambda }\right)$, we define

$\parallel u\parallel ={\parallel u\parallel }_{{W}_{0}^{2,p}\left(\mathrm{\Omega }\right)}:={\left({\int }_{\mathrm{\Omega }}\left({|\mathrm{\Delta }u|}^{p}-\lambda \frac{{|u|}^{p}}{{|x|}^{2p}}\right)\phantom{\rule{0.2em}{0ex}}dx\right)}^{\frac{1}{p}},$

this norm is equivalent to ${\left({\int }_{\mathrm{\Omega }}{|\mathrm{\Delta }u|}^{p}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/p}$.

A weak solution of the problem (1.1) is a critical point of the energy functional

$I\left(u\right)=\frac{1}{p}{\int }_{\mathrm{\Omega }}\left({|\mathrm{\Delta }u|}^{p}-\lambda \frac{{|u|}^{p}}{{|x|}^{2p}}\right)\phantom{\rule{0.2em}{0ex}}dx-{\int }_{\mathrm{\Omega }}F\left(x,u\right)\phantom{\rule{0.2em}{0ex}}dx,\phantom{\rule{1em}{0ex}}u\in {W}_{0}^{2,p}\left(\mathrm{\Omega }\right),$
(1.6)

where $F\left(x,u\right)={\int }_{0}^{u}f\left(x,s\right)\phantom{\rule{0.2em}{0ex}}ds$. It is easy to check that $I\left(u\right)$ is a continuous even functional.

Biharmonic equations can describe the static form change of a beam or the sport of a rigid body. For example, this type of equation furnishes a model for studying traveling wave in suspension bridges (see [3]). By using variational arguments, many authors investigated nonlinear biharmonic equations under Dirichlet boundary conditions or Navier boundary conditions and got interesting results (see [410]).

Li and Squassina in [11] considered the superlinear p-harmonic equation with Navier boundary conditions

(1.7)

where Ω is an open bounded domain in ${\mathbb{R}}^{N}$ with a smooth boundary Ω. $p>1$, $N\ge 2p+1$ and $f:\mathrm{\Omega }×\mathbb{R}\to \mathbb{R}$ is a Carathéodory function such that for some positive constant C,

(1.8)

By means of the Morse theory, they proved the existence of two nontrivial solutions to (1.7). After that, Li and Tang [12] considered a more general problem than (1.7). They got three solutions by the three critical points theorem which was obtained in [13]. If $f\left(x,u\right)=\lambda g\left(x,u\right)$, Candito and Bisci [14] established a well-determined interval of values of the parameter λ for which the problem (1.7) admits at least two distinct weak solutions. The authors in [15] considered the p-harmonic equation with Dirichlet conditions and obtained the existence of a nontrivial solution. Under the condition of (1.8), [16] proved the existence and multiplicity of weak solutions for the nonuniformly nonlinear problem

(1.9)

where $|a\left(x,t\right)|\le {c}_{0}\left({h}_{0}\left(x\right)+{h}_{1}\left(x\right){|t|}^{p-1}\right)$ with ${h}_{0}\left(x\right)\ge 0$, ${h}_{1}\left(x\right)\ge 1$. $a\left(x,t\right)$ and $f\left(x,t\right)$ are odd with respect to the second variable. Furthermore, $f\left(x,t\right)$ is subcritical and satisfies the Ambrosetti-Rabinowitz condition, that is, there exists a constant $\theta >p$ such that

(1.10)

The aim of this paper is to obtain infinitely many solutions for the problem (1.1) when $2p. The main difficulty lies in the fact that the embeddings ${W}_{0}^{2,p}\left(\mathrm{\Omega }\right)↪{L}^{{p}^{\ast }}\left(\mathrm{\Omega }\right)$ and ${W}_{0}^{2,p}\left(\mathrm{\Omega }\right)↪{L}^{p}\left(\mathrm{\Omega },{|x|}^{-2p}\phantom{\rule{0.2em}{0ex}}dx\right)$ are not compact. Furthermore, the assumption (f1) is not the usual subcritical growth (1.8). The condition (f3) is weaker than the A-R condition (1.10). We use the concentration compactness principle (see [17, 18]) to overcome those difficulties. The following theorem is our main result.

Theorem 1.1 Assume $f\left(x,u\right)$ satisfies (f1)-(f4), then the problem (1.1) possesses infinitely many weak solutions and the corresponding critical values are positive.

This paper proceeds as follows. In the next section, we prove the energy functional $I\left(u\right)$ satisfies the Palais-Smale condition. In Section 3, by using the symmetric mountain pass theorem (see [19]), we get the main result of this paper. Throughout the paper, denote

${S}_{p,\lambda }:=\underset{u\in {W}_{0}^{2,p}\left(\mathrm{\Omega }\right)\setminus \left\{0\right\}}{inf}\frac{{\parallel u\parallel }^{p}}{{\parallel u\parallel }_{{p}^{\ast }}^{p}}.$

We use ${\parallel u\parallel }_{q}={\left({\int }_{\mathrm{\Omega }}{|u|}^{q}\phantom{\rule{0.2em}{0ex}}dx\right)}^{1/q}$ to denote the norm of ${L}^{q}\left(\mathrm{\Omega }\right)$, C, ${C}_{i}$ stand for universal constants. We omit dx and Ω in the integrals if there is no other indication.

## 2 Palais-Smale condition

To show the (PS) sequence $\left\{{u}_{n}\right\}$ of the variational functional $I\left(u\right)$ is compact in ${W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$, we first prove the boundedness of $\left\{{u}_{n}\right\}$ by the analytic argument which has been used in [20]. Then, using the concentration compactness principle, we get the compactness of $\left\{{u}_{n}\right\}$.

Lemma 2.1 The condition (f1) implies that for any $\epsilon >0$, there exists a positive constant ${C}_{\epsilon }$ such that

$F\left(x,u\right)\le \epsilon {|u|}^{{p}^{\ast }}+{C}_{\epsilon }|u|,\phantom{\rule{1em}{0ex}}\left(x,u\right)\in \mathrm{\Omega }×\mathbb{R}.$
(2.1)

(f2) implies that $F\left(x,u\right)=o\left({u}^{p}\right)$ as $u\to 0$. Furthermore, for any $|u|\ge M>0$, there exists a small positive constant θ

$F\left(x,u\right)\ge C{|u|}^{p+\theta }\phantom{\rule{1em}{0ex}}\mathit{\text{and}}\phantom{\rule{1em}{0ex}}f\left(x,u\right)u>0.$
(2.2)

Lemma 2.2 Under the conditions (f1), (f2) and (f3), if the sequence $\left\{{u}_{n}\right\}\in {W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$ satisfies

(2.3)

then the sequence $\left\{{u}_{n}\right\}$ is bounded in ${W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$.

Proof We prove this lemma by contradiction. Without loss of generality, we assume that

(2.4)

Set ${w}_{n}={u}_{n}/\parallel {u}_{n}\parallel$, then $\parallel {w}_{n}\parallel =1$ for any $n\in \mathbb{N}$. As $n\to +\mathrm{\infty }$, there exists $w\in {W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$ such that

If $w\not\equiv 0$, we denote ${\mathrm{\Omega }}_{0}=\left\{x\in \mathrm{\Omega }:w=0\right\}$, set $\mathrm{\Omega }\setminus {\mathrm{\Omega }}_{0}$ is nonempty. (2.3) implies that

${\parallel {u}_{n}\parallel }^{p}-{\int }_{\mathrm{\Omega }}f\left(x,{u}_{n}\right){u}_{n}\phantom{\rule{0.2em}{0ex}}dx=o\left(1\right).$

Therefore,

$1-o\left(1\right)={\int }_{\mathrm{\Omega }}\frac{f\left(x,{u}_{n}\right){u}_{n}}{{\parallel {u}_{n}\parallel }^{p}}\phantom{\rule{0.2em}{0ex}}dx={\int }_{{\mathrm{\Omega }}_{0}}\frac{f\left(x,{u}_{n}\right){u}_{n}}{{\parallel {u}_{n}\parallel }^{p}}\phantom{\rule{0.2em}{0ex}}dx+{\int }_{\mathrm{\Omega }\setminus {\mathrm{\Omega }}_{0}}\frac{f\left(x,{u}_{n}\right){u}_{n}}{{\parallel {u}_{n}\parallel }^{p}}\phantom{\rule{0.2em}{0ex}}dx.$
(2.5)

As $n\to +\mathrm{\infty }$, ${\int }_{\mathrm{\Omega }}{|{u}_{n}|}^{p}\phantom{\rule{0.2em}{0ex}}dx\le \parallel {u}_{n}\parallel \to 0$. So, $|{u}_{n}|\to +\mathrm{\infty }$ for $x\in \mathrm{\Omega }\setminus {\mathrm{\Omega }}_{0}$. It follows from (f2) that

$\underset{{u}_{n}\to \mathrm{\infty }}{lim}\frac{f\left(x,{u}_{n}\right){u}_{n}}{{\parallel {u}_{n}\parallel }^{p}}=\underset{{u}_{n}\to \mathrm{\infty }}{lim}\frac{f\left(x,{u}_{n}\right){u}_{n}}{{|{u}_{n}|}^{p}}{|{w}_{n}|}^{p}=+\mathrm{\infty }.$

Since $|\mathrm{\Omega }\setminus {\mathrm{\Omega }}_{0}|>0$, then

(2.6)

Using (2.1) and (2.2), we derive that

(2.7)

From (2.6) and (2.7), we get

(2.8)

If $w\equiv 0$, then ${w}_{n}\to 0$ strongly in ${L}^{1}\left(\mathrm{\Omega }\right)$. Since $I\left({u}_{n}\right)\to c$, we know that

$c←\frac{1}{p}{\parallel {u}_{n}\parallel }^{p}-\int F\left(x,{u}_{n}\right)\phantom{\rule{0.2em}{0ex}}dx\ge \frac{1}{p}{\parallel {u}_{n}\parallel }^{p}-\epsilon \int {|{u}_{n}|}^{{p}^{\ast }}\phantom{\rule{0.2em}{0ex}}dx-{C}_{\epsilon }\int |{u}_{n}|\phantom{\rule{0.2em}{0ex}}dx.$
(2.9)

In (2.9), choose $\epsilon =1$, then

$\frac{1}{p}{\parallel {u}_{n}\parallel }^{p}\le {\parallel {u}_{n}\parallel }_{{p}^{\ast }}^{{p}^{\ast }}+{C}_{2}{\parallel {u}_{n}\parallel }_{1}+c\le {C}_{3}{\parallel {u}_{n}\parallel }_{{p}^{\ast }}^{{p}^{\ast }}+c.$

From (2.4), it follows that ${\parallel {u}_{n}\parallel }_{{p}^{\ast }}\to +\mathrm{\infty }$ as $n\to +\mathrm{\infty }$. Denote

${A}_{n}=M\frac{\parallel {u}_{n}\parallel }{{\parallel {u}_{n}\parallel }_{{p}^{\ast }}}\ge M{S}_{p,\lambda }^{1/p},$
(2.10)

where M is any fixed positive constant. It is clear that

Therefore, for n large enough, ${A}_{n}/\parallel {u}_{n}\parallel \in \left(0,1\right)$.

For every $n\in \mathbb{N}$, we define a sequence of ${t}_{n}$ as follows:

$I\left({t}_{n}{u}_{n}\right)=\underset{t\in \left[0,1\right]}{max}I\left(t{u}_{n}\right).$

We claim that

(2.11)

In fact, by (2.1)-(2.10) and the definition of ${t}_{n}$, we have

$\begin{array}{rcl}I\left({t}_{n}{u}_{n}\right)& \ge & I\left(\frac{{A}_{n}}{\parallel {u}_{n}\parallel }{u}_{n}\right)\ge \frac{1}{p}{A}_{n}^{p}-\epsilon \int {|\frac{{A}_{n}}{\parallel {u}_{n}\parallel }{u}_{n}|}^{{p}^{\ast }}\phantom{\rule{0.2em}{0ex}}dx-{C}_{\epsilon }{A}_{n}\int |{w}_{n}|\phantom{\rule{0.2em}{0ex}}dx\\ =& \frac{1}{p}{A}_{n}^{p}-\epsilon {M}^{{p}^{\ast }}-{C}_{\epsilon }{A}_{n}\int |{w}_{n}|\phantom{\rule{0.2em}{0ex}}dx.\end{array}$

If ${A}_{n}\to +\mathrm{\infty }$ as $n\to +\mathrm{\infty }$, then (2.11) follows from the above estimate. If ${A}_{n}$ is bounded for any n, that is, there exists a constant $K>0$ such that ${A}_{n}\le K$, then

where we use the fact that ${\parallel {w}_{n}\parallel }_{1}\to 0$ as $n\to +\mathrm{\infty }$. The positive constant ε is arbitrary, thus

(2.11) follows from the above inequality.

From (2.3), we know that

$pc+o\left(1\right)=pI\left({u}_{n}\right)-〈{I}^{\prime }\left({u}_{n}\right),{u}_{n}〉=\int \left(f\left(x,{u}_{n}\right){u}_{n}-pF\left(x,{u}_{n}\right)\right)\phantom{\rule{0.2em}{0ex}}dx.$
(2.12)

By the definition of ${t}_{n}$, we obtain $\frac{d}{dt}I\left(t{u}_{n}\right){|}_{t={t}_{n}}=0$ and ${t}_{n}\in \left[0,1\right]$, then $|{t}_{n}{u}_{n}|\le |{u}_{n}|$. The condition (f3) and (2.11) derive that

$\begin{array}{rcl}\int \left(f\left(x,{u}_{n}\right){u}_{n}-pF\left(x,{u}_{n}\right)\right)\phantom{\rule{0.2em}{0ex}}dx& \ge & \int \left(f\left(x,{t}_{n}{u}_{n}\right){t}_{n}{u}_{n}-pF\left(x,{t}_{n}{u}_{n}\right)\right)\phantom{\rule{0.2em}{0ex}}dx\\ =& pI\left({t}_{n}{u}_{n}\right)-〈{I}^{\prime }\left({t}_{n}{u}_{n}\right),{t}_{n}{u}_{n}〉\\ =& pI\left({t}_{n}{u}_{n}\right)-{t}_{n}\frac{d}{dt}I\left(t{u}_{n}\right){|}_{t={t}_{n}}\\ \to & +\mathrm{\infty }.\end{array}$

Which is a contradiction to (2.12). Therefore, the sequence $\left\{{u}_{n}\right\}$ is bounded in ${W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$. □

Under the conditions of Lemma 2.2, we know $\parallel {u}_{n}\parallel \le C$. Therefore, there exists a subsequence, still denoted by $\left\{{u}_{n}\right\}$, and some $u\in {W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$ such that

Obviously, u is a weak solution of (1.1). Now we prove that $u\not\equiv 0$. According to the concentration compactness principle (see [17, 18]), there exists a subsequence, still denoted by $\left\{{u}_{n}\right\}$, at most countable set J, a set of different points ${\left\{{x}_{j}\right\}}_{j\in J}\subset \overline{\mathrm{\Omega }}\setminus \left\{0\right\}$ and two positive number sequences ${\left\{{\mu }_{j}\right\}}_{j\in J\cup \left\{0\right\}}$, ${\left\{{\nu }_{j}\right\}}_{j\in J\cup \left\{0\right\}}$ such that

where ${\delta }_{{x}_{j}}$ is the unit Dirac measure at ${x}_{j}$.

Lemma 2.3 Assume $f\left(x,u\right)$ satisfies (f1), the sequence $\left\{{u}_{n}\right\}$ is bounded in ${W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$ and satisfies (2.3). Then the set $J=\left\{{x}_{1},{x}_{2},\dots ,{x}_{j}\right\}\subset \overline{\mathrm{\Omega }}$ is finite, ${\mu }_{0}-\lambda {\gamma }_{0}=0$.

Proof We first prove that ${\mu }_{j}=0$ for any ${x}_{j}\in J$.

Let $\epsilon >0$ be small enough such that $0\notin {B}_{\epsilon }\left({x}_{j}\right)$ and ${B}_{\epsilon }\left({x}_{i}\right)\cap {B}_{\epsilon }\left({x}_{j}\right)=\mathrm{\varnothing }$ for $i\ne j$. ${\varphi }_{j}\left(x\right)\in \left[0,1\right]$ is a cutting-off function in ${\mathcal{C}}_{0}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$. ${\varphi }_{j}\left(x\right)\equiv 1$ for $|x-{x}_{j}|<\epsilon /2$, ${\varphi }_{j}\left(x\right)\equiv 0$ for $|x-{x}_{j}|\ge \epsilon$, and $|\mathrm{\Delta }{\varphi }_{j}|\le 4/{\epsilon }^{2}$. Since

$\begin{array}{rcl}〈{I}^{\prime }\left({u}_{n}\right),{u}_{n}{\varphi }_{j}〉& =& \int {|\mathrm{\Delta }{u}_{n}|}^{p}{\varphi }_{j}\phantom{\rule{0.2em}{0ex}}dx+2\int {|\mathrm{\Delta }{u}_{n}|}^{p-2}\mathrm{\Delta }{u}_{n}\mathrm{\nabla }{u}_{n}\mathrm{\nabla }{\varphi }_{j}\phantom{\rule{0.2em}{0ex}}dx\\ +\int {u}_{n}{|\mathrm{\Delta }{u}_{n}|}^{p-2}\mathrm{\Delta }{u}_{n}\mathrm{\Delta }{\varphi }_{j}\phantom{\rule{0.2em}{0ex}}dx\\ -\int \lambda \frac{{|{u}_{n}|}^{p}}{{|x|}^{2p}}{\varphi }_{j}\phantom{\rule{0.2em}{0ex}}dx-\int f\left(x,{u}_{n}\right){u}_{n}{\varphi }_{j}\phantom{\rule{0.2em}{0ex}}dx,\end{array}$
(2.13)

we deduce that

(2.14)
(2.15)

By Hölder’s inequality,

$\begin{array}{c}\underset{\epsilon \to 0}{lim}\underset{n\to +\mathrm{\infty }}{lim}|\int {|\mathrm{\Delta }{u}_{n}|}^{p-2}\mathrm{\Delta }{u}_{n}\mathrm{\nabla }{u}_{n}\mathrm{\nabla }{\varphi }_{j}\phantom{\rule{0.2em}{0ex}}dx|\hfill \\ \phantom{\rule{1em}{0ex}}\le \underset{\epsilon \to 0}{lim}\underset{n\to +\mathrm{\infty }}{lim}{\parallel \mathrm{\Delta }{u}_{n}\parallel }_{{L}^{p}\left({B}_{\epsilon }\left({x}_{j}\right)\right)}^{p-1}{\parallel \mathrm{\nabla }{u}_{n}\parallel }_{{L}^{Np/\left(N-p\right)}\left({B}_{\epsilon }\left({x}_{j}\right)\right)}{\parallel \mathrm{\nabla }{\varphi }_{j}\parallel }_{{L}^{N}\left({B}_{\epsilon }\left({x}_{j}\right)\right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\underset{\epsilon \to 0}{lim}{\parallel \mathrm{\Delta }{u}_{n}\parallel }_{{L}^{p}\left({B}_{\epsilon }\left({x}_{j}\right)\right)}^{p}=0,\hfill \\ \underset{\epsilon \to 0}{lim}\underset{n\to +\mathrm{\infty }}{lim}|\int {|\mathrm{\Delta }{u}_{n}|}^{p-2}\mathrm{\Delta }{u}_{n}{u}_{n}\mathrm{\Delta }{\varphi }_{j}\phantom{\rule{0.2em}{0ex}}dx|\hfill \\ \phantom{\rule{1em}{0ex}}\le \underset{\epsilon \to 0}{lim}\underset{n\to +\mathrm{\infty }}{lim}{\parallel \mathrm{\Delta }{u}_{n}\parallel }_{{L}^{p}\left({B}_{\epsilon }\left({x}_{j}\right)\right)}^{p-1}{\parallel {u}_{n}\parallel }_{{L}^{{p}^{\ast }}\left({B}_{\epsilon }\left({x}_{j}\right)\right)}{\parallel \mathrm{\Delta }{\varphi }_{j}\parallel }_{{L}^{N/2}\left({B}_{\epsilon }\left({x}_{j}\right)\right)}\hfill \\ \phantom{\rule{1em}{0ex}}\le C\underset{\epsilon \to 0}{lim}{\parallel {u}_{n}\parallel }_{{L}^{{p}^{\ast }}\left({B}_{\epsilon }\left({x}_{j}\right)\right)}=0.\hfill \end{array}$

Using (f1), (2.14) and (2.15), we have

$\underset{\epsilon \to 0}{lim}\underset{n\to +\mathrm{\infty }}{lim}\int |f\left(x,{u}_{n}\right){u}_{n}{\varphi }_{j}|\phantom{\rule{0.2em}{0ex}}dx\le \underset{\epsilon \to 0}{lim}\underset{n\to +\mathrm{\infty }}{lim}\left(\epsilon \int {|{u}_{n}|}^{{p}^{\ast }}{\varphi }_{j}\phantom{\rule{0.2em}{0ex}}dx+{C}_{\epsilon }\int |{u}_{n}|{\varphi }_{j}\phantom{\rule{0.2em}{0ex}}dx\right)=0.$

By the above estimates, (2.13) becomes

$0=\underset{\epsilon \to 0}{lim}\underset{n\to +\mathrm{\infty }}{lim}〈{I}^{\prime }\left({u}_{n}\right),{u}_{n}{\varphi }_{j}〉\ge {\mu }_{j}.$
(2.16)

Therefore, for any $j\in J$, ${\mu }_{j}=0$, which implies that J is finite.

If the concentration is at the origin, let ${\varphi }_{0}\left(x\right)\in \left[0,1\right]$ be a cutting-off function in ${\mathcal{C}}_{0}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$. ${\varphi }_{0}\left(x\right)\equiv 1$ for $|x|<\epsilon /2$, ${\varphi }_{0}\left(x\right)\equiv 0$ for $|x|\ge \epsilon$, and $|\mathrm{\Delta }{\varphi }_{0}|\le 4/{\epsilon }^{2}$. Choose $\epsilon >0$ small enough such that ${x}_{j}\notin {B}_{\epsilon }\left(0\right)$ for all $j\in J$. Then

(2.17)
(2.18)

Similarly, we can get

(2.19)
(2.20)

From equalities (2.17)-(2.20), we get

$0=\underset{\epsilon \to 0}{lim}\underset{n\to +\mathrm{\infty }}{lim}〈{I}^{\prime }\left({u}_{n}\right),{u}_{n}{\varphi }_{0}〉\ge {\mu }_{0}-\lambda {\gamma }_{0}.$
(2.21)

On the other hand, ${\mu }_{0}-\lambda {\gamma }_{0}\ge {S}_{p,\lambda }{\nu }_{0}^{\frac{p}{{p}^{\ast }}}\ge 0$, thus ${\mu }_{0}-\lambda {\gamma }_{0}=0$. The proof is complete. □

Lemma 2.4 Assume $f\left(x,u\right)$ satisfies (f1), the sequence $\left\{{u}_{n}\right\}$ satisfying (2.3) is bounded in ${W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$. Then there exist some $u\in {W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$ and a subsequence, still denoted by $\left\{{u}_{n}\right\}$, such that

where ${\mathrm{\Omega }}_{ϵ}=\mathrm{\Omega }\setminus {\bigcup }_{i=0}^{j}{\overline{B}}_{ϵ}\left({x}_{i}\right)$, ${x}_{i}\in J=\left\{{x}_{1},{x}_{2},\dots ,{x}_{j}\right\}\cup \left\{0\right\}$.

Proof Lemma 2.3 implies that J is finite. Choose $\phi \in {C}_{0}^{\mathrm{\infty }}\left(\mathrm{\Omega }\right)$ with $\phi \left({x}_{i}\right)=0$ for any ${x}_{i}\in J\cup \left\{0\right\}$. Then

$\begin{array}{rcl}\int {|\phi {u}_{n}|}^{{p}^{\ast }}\phantom{\rule{0.2em}{0ex}}dx& =& \int {|\phi u|}^{{p}^{\ast }}\phantom{\rule{0.2em}{0ex}}dx+\sum _{i=1}^{j}{\nu }_{i}{\phi }^{{p}^{\ast }}\left({x}_{i}\right)+{\nu }_{0}{\phi }^{{p}^{\ast }}\left(0\right)\\ =& \int {|\phi u|}^{{p}^{\ast }}\phantom{\rule{0.2em}{0ex}}dx.\end{array}$
(2.22)

Since $\phi {u}_{n}\to \phi u$ a.e. in Ω, we get that

Similarly,

□

By using the Lebesgue decomposition theorem, we have

$d\mu -\lambda \phantom{\rule{0.2em}{0ex}}d\gamma ={|\mathrm{\Delta }u|}^{p}-\lambda \frac{{|u|}^{p}}{{|x|}^{2p}}+d\sigma ,$
(2.23)

where $\left({|\mathrm{\Delta }u|}^{p}-\lambda {|u|}^{p}{|x|}^{-2p}\right)\perp d\sigma$. Then $d\sigma \ge {\sum }_{j\in J}{\mu }_{j}{\delta }_{{x}_{j}}+\left({\mu }_{0}-\lambda {\gamma }_{0}\right){\delta }_{0}$.

Lemma 2.5 Assume $f\left(x,u\right)$ satisfies (f1)-(f3), the sequence $\left\{{u}_{n}\right\}\in {W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$ satisfies (2.3). Then there exist $u\in {W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$ and a subsequence, still denoted by $\left\{{u}_{n}\right\}$, such that ${u}_{n}$ converges to u strongly in ${W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$.

Proof We first prove that

$\left(\sigma -\sum _{j\in J}{\mu }_{j}{\delta }_{{x}_{j}}-\left({\mu }_{0}-\lambda {\gamma }_{0}\right){\delta }_{0}\right)\left(\overline{\mathrm{\Omega }}\right)=0.$
(2.24)

Claim for any closed set $F\subset \overline{\mathrm{\Omega }}\setminus \overline{J}$, $\overline{J}=J\cup \left\{0\right\}$, $\sigma \left(F\right)=0$. In fact, denote $r=dist\left(F,\overline{J}\right)>0$. Then, by using a finite covering theorem, there exist finite open balls ${B}_{r/4}\left({z}_{i}\right)$, with ${z}_{i}\in F$, $i=1,\dots ,m$, such that $F\subset {\bigcup }_{i=1}^{m}{B}_{r/4}\left({z}_{i}\right)\subset \overline{\mathrm{\Omega }}\setminus \overline{J}$. Let $\varrho \left(t\right)$ be a smooth cutting-off function. $0\le \varrho \left(t\right)\le 1$ for any $t\in \left[0,\mathrm{\infty }\right)$. $\varrho \left(x\right)\equiv 1$ for $0\le t\le 1/2$, and $\varrho \equiv 0$ for $t\ge 1$. Denote ${\eta }_{\epsilon ,y}=\varrho \left(|x-y|/\epsilon \right)$, then

(2.25)
(2.26)

The sequence $\left\{{\eta }_{\epsilon ,y}{u}_{n}\right\}$ is still bounded in ${W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$. Define

$\eta \left(x\right)=\sum _{i=1}^{m}{\eta }_{r/4,{z}_{i}}\left(x\right),$
(2.27)

then $\eta \left(x\right)\in {C}_{0}^{\mathrm{\infty }}\left(\overline{\mathrm{\Omega }}\setminus \overline{J}\right)$ and $0\le \eta \left(x\right)\le m$. Furthermore,

$F\subset \bigcup _{i=1}^{m}{B}_{\frac{r}{4}}\left({z}_{i}\right)\subset spt\eta \left(x\right)\subset {\mathrm{\Omega }}_{\frac{r}{4}},$
(2.28)

where $spt\eta \left(x\right)$ denotes the support of $\eta \left(x\right)$. As $n\to +\mathrm{\infty }$, (2.3) implies

$\begin{array}{rcl}0& ←& 〈{I}^{\prime }\left({u}_{n}\right)-{I}^{\prime }\left(u\right),\left({u}_{n}-u\right)\eta 〉\\ =& {\int }_{\mathrm{\Omega }}\eta \left({|\mathrm{\Delta }{u}_{n}|}^{p}-{|\mathrm{\Delta }{u}_{n}|}^{p-2}\mathrm{\Delta }{u}_{n}\mathrm{\Delta }u-{|\mathrm{\Delta }u|}^{p-2}\mathrm{\Delta }u\mathrm{\Delta }{u}_{n}+{|\mathrm{\Delta }u|}^{p}\right)\phantom{\rule{0.2em}{0ex}}dx\\ -\lambda {\int }_{\mathrm{\Omega }}\frac{\eta }{{|x|}^{2p}}\left({|{u}_{n}|}^{p}-{|{u}_{n}|}^{p-2}{u}_{n}u-{|u|}^{p-2}u{u}_{n}+{|u|}^{p}\right)\phantom{\rule{0.2em}{0ex}}dx\\ +2{\int }_{\mathrm{\Omega }}\mathrm{\nabla }\eta \mathrm{\nabla }\left({u}_{n}-u\right)\left({|\mathrm{\Delta }{u}_{n}|}^{p-2}\mathrm{\Delta }{u}_{n}-{|\mathrm{\Delta }u|}^{p-2}\mathrm{\Delta }u\right)\phantom{\rule{0.2em}{0ex}}dx\\ +{\int }_{\mathrm{\Omega }}\mathrm{\Delta }\eta \left({u}_{n}-u\right)\left({|\mathrm{\Delta }{u}_{n}|}^{p-2}\mathrm{\Delta }{u}_{n}-{|\mathrm{\Delta }u|}^{p-2}\mathrm{\Delta }u\right)\phantom{\rule{0.2em}{0ex}}dx\\ -{\int }_{\mathrm{\Omega }}\eta \left({u}_{n}-u\right)\left(f\left(x,{u}_{n}\right)-f\left(x,u\right)\right)\phantom{\rule{0.2em}{0ex}}dx\\ =& I+\mathit{II}+\mathit{III}+\mathit{IV}+V.\end{array}$

As $n\to +\mathrm{\infty }$,

$\begin{array}{rcl}I+\mathit{II}& =& {\int }_{\mathrm{\Omega }}\eta \left({|\mathrm{\Delta }{u}_{n}|}^{p}-{|\mathrm{\Delta }{u}_{n}|}^{p-2}\mathrm{\Delta }{u}_{n}\mathrm{\Delta }u-{|\mathrm{\Delta }u|}^{p-2}\mathrm{\Delta }u\mathrm{\Delta }{u}_{n}+{|\mathrm{\Delta }u|}^{p}\right)\phantom{\rule{0.2em}{0ex}}dx\\ -\lambda {\int }_{\mathrm{\Omega }}\frac{\eta }{{|x|}^{2p}}\left({|{u}_{n}|}^{p}-{|{u}_{n}|}^{p-2}{u}_{n}u-{|u|}^{p-2}u{u}_{n}+{|u|}^{p}\right)\phantom{\rule{0.2em}{0ex}}dx\\ =& {\int }_{\mathrm{\Omega }}\eta \left({|\mathrm{\Delta }{u}_{n}|}^{p}-{|\mathrm{\Delta }u|}^{p}-\lambda \frac{{|{u}_{n}|}^{p}-{|u|}^{p}}{{|x|}^{2p}}\right)\phantom{\rule{0.2em}{0ex}}dx+o\left(1\right)\\ =& {\int }_{\mathrm{\Omega }}\eta \phantom{\rule{0.2em}{0ex}}d\sigma +o\left(1\right),\end{array}$

where we have used (2.23). Lemma 2.2 implies that $\parallel {u}_{n}\parallel \le M$. Hölder’s inequality, (2.25)-(2.28) and Lemma 2.4 deduce that

By (2.1), for any $\epsilon >0$, we have

Therefore, as $n\to +\mathrm{\infty }$,

$0\le \sigma \left(F\right)\le \sigma \left(spt\left(\eta \right)\right)\le {\int }_{\mathrm{\Omega }}\eta \phantom{\rule{0.2em}{0ex}}d\sigma \le |\mathit{III}|+|\mathit{IV}|+|V|=0.$
(2.29)

Since the set F is arbitrary, (2.24) is obtained.

Lemma 2.3 means that ${\mu }_{j}=0$ for any $j\in J$ and ${\mu }_{0}-\lambda {\gamma }_{0}=0$, so $\sigma \left(\overline{\mathrm{\Omega }}\right)=0$, which implies that as $n\to +\mathrm{\infty }$,

${\int }_{\mathrm{\Omega }}\left({|\mathrm{\Delta }{u}_{n}|}^{p}-\lambda \frac{{|{u}_{n}|}^{p}}{{|x|}^{2p}}\right)\phantom{\rule{0.2em}{0ex}}dx\to {\int }_{\mathrm{\Omega }}\left({|\mathrm{\Delta }u|}^{p}-\lambda \frac{{|u|}^{p}}{{|x|}^{2p}}\right)\phantom{\rule{0.2em}{0ex}}dx.$
(2.30)

Together with ${u}_{n}\to u$ a.e. in Ω, we complete the proof. □

## 3 The proof of the main result

In this section, by using the following symmetric mountain pass theorem (see [19]), we give the proof of Theorem 1.1.

Lemma 3.1 (Symmetric mountain pass theorem)

Assume functional I satisfies the following conditions:

1. (1)

$I\in {\mathcal{C}}^{1}\left({W}_{0}^{2,p}\left(\mathrm{\Omega }\right),\mathbb{R}\right)$ is even and satisfies the Palais-Smale condition.

2. (2)

There exists a finite dimensional subspace $X\in {W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$ such that $I{|}_{X\cap \partial {B}_{r}}\ge a>0$.

3. (3)

There exists a sequence of the finite dimensional subspace $\left\{{X}_{j}\right\}$, $dim\left({X}_{j}\right)=j$ and ${r}_{j}>0$ such that

Then I has infinitely many different critical points, and the corresponding energy values are positive.

Proof of Theorem 1.1 We know the energy functional $I\left(u\right)$ is continuous and even. Lemma 2.5 implies that $I\left(u\right)$ satisfies the PS condition in ${W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$. In any finite dimensional subspace $X\subset {W}_{0}^{2,p}\left(\mathrm{\Omega }\right)$, all norms are equivalent. For any $u\in X$, since $F\left(x,u\right)=o\left({u}^{p}\right)$ as $u\to 0$, there exists a constant $q>p$ such that

(3.1)

Thus, there exists $r>0$ such that $I{|}_{X\cap \partial {B}_{r}}\ge a>0$. On the other hand, by using (2.2), we have

There exists a sequence of the finite dimensional subspace $\left\{{X}_{j}\right\}$, $dim\left({X}_{j}\right)=j$ and ${r}_{j}>0$ such that

Finally, all the assumptions of Lemma 3.1 are satisfied. Hence, the problem (1.1) possesses infinitely many weak solutions, and the corresponding critical values are positive. □

Remark 3.1 Under the same conditions of Theorem 1.1, we can also prove that the following p-harmonic type equation with Navier boundary conditions:

$\left\{\begin{array}{cc}\mathrm{\Delta }\left({|\mathrm{\Delta }u|}^{p-2}\mathrm{\Delta }u\right)-\lambda \frac{{|u|}^{p-2}u}{{|x|}^{2p}}=f\left(x,u\right),\hfill & x\in \mathrm{\Omega },\hfill \\ u=\mathrm{\Delta }u=0,\hfill & x\in \partial \mathrm{\Omega },\hfill \end{array}$
(3.2)

has infinitely many solutions $\left\{{u}_{n}\right\}\in {W}^{2,p}\left(\mathrm{\Omega }\right)\cap {W}_{0}^{1,p}\left(\mathrm{\Omega }\right)$, where Ω containing the origin is an open bounded domain in ${\mathbb{R}}^{N}$, Ω is smooth. $1, $0\le \lambda <\overline{\lambda }={\left[N\left(p-1\right)\left(N-2p\right)/{p}^{2}\right]}^{p}$.

Remark 3.2 All the results obtained above obviously hold if we choose $f\left(x,u\right)={|u|}^{q-2}u+{|u|}^{r-2}u$ with $p.

## References

1. Davies E, Hinz A:Explicit constants for Rellich inequalities in ${L}^{p}\left(\mathrm{\Omega }\right)$. Math. Z. 1998, 227: 511–523. 10.1007/PL00004389

2. Mitidieri E: A simple approach to Hardy’s inequalities. Math. Notes 2000, 67: 479–486. 10.1007/BF02676404

3. Lazer A, McKenna P: Large amplitude periodic oscillations in suspension bridges: some new connections with nonlinear analysis. SIAM Rev. 1990, 32: 537–578. 10.1137/1032120

4. Bernis F, Azorero JG, Peral I: Existence and multiplicity of nontrivial solutions in semilinear critical problems of fourth order. Adv. Differ. Equ. 1996, 1: 219–240.

5. Bonder J, Rossi J: A fourth order elliptic equation with nonlinear boundary conditions. Nonlinear Anal. 2002, 49: 1037–1047. 10.1016/S0362-546X(01)00718-0

6. Deng Y, Wang G: On inhomogeneous biharmonic equations involving critical exponents. Proc. R. Soc. Edinb. 1999, 129: 925–946. 10.1017/S0308210500031012

7. Deng Y, Yang J: Existence of multiple solutions and bifurcations for critical semilinear biharmonic equation. Syst. Sci. Math. Sci. 1995, 8: 319–326.

8. Liu X, Li W: Existence and multiplicity of solutions for fourth-order boundary value problems with parameters. J. Math. Anal. Appl. 2007, 327: 362–375. 10.1016/j.jmaa.2006.04.021

9. Zeng X, Deng Y: Existence of multiple solutions for a semilinear biharmonic equation with critical exponent. Acta Math. Sci. 2000, 20(4):547–554.

10. Zhang J, Li S: Multiple nontrivial solutions for some fourth-order semilinear elliptic problems. Nonlinear Anal. 2005, 60: 221–230.

11. Liu S, Squassina M: On the existence of solutions to a fourth-order quasilinear resonant problem. Abstr. Appl. Anal. 2002, 7(3):125–133. 10.1155/S1085337502000805

12. Li C, Tang C: Three solutions for a Navier boundary value problem involving the p -biharmonic. Nonlinear Anal. 2010, 72: 1339–1347. 10.1016/j.na.2009.08.011

13. Ricceri B: A three critical points theorem revisited. Nonlinear Anal. 2009, 70: 3084–3089. 10.1016/j.na.2008.04.010

14. Candito P, Bisci G: Multiple solutions for a Navier boundary value problem involving the p -biharmonic operator. Discrete Contin. Dyn. Syst. 2012, 5: 741–751.

15. Yang Z, Geng D, Yan H: Existence of nontrivial solutions in p -biharmonic problems with critical growth. Chin. Ann. Math., Ser. A 2006, 27(1):129–142.

16. Wang W, Zhao P: Nonuniformly nonlinear elliptic equations of p -biharmonic type. J. Math. Anal. Appl. 2008, 348: 730–738. 10.1016/j.jmaa.2008.07.068

17. Lions PL: The concentration-compactness principle in the calculus of variations: the limit case, part 1. Rev. Mat. Iberoam. 1985, 1(1):145–201.

18. Lions PL: The concentration-compactness principle in the calculus of variations: the limit case, part 2. Rev. Mat. Iberoam. 1985, 1(2):45–121.

19. Chang K: Critical Point Theory and Its Applications. Shanghai Sci. Technol., Shanghai; 1986. (in Chinese)

20. Geng D: Infinitely many solutions of p -Laplacian equations with limit subcritical growth. Appl. Math. Mech. 2007, 28(10):1373–1382. 10.1007/s10483-007-1010-x

## Acknowledgements

The authors are thankful to the referees for their helpful suggestions and necessary corrections in the completion of this paper. This research is supported by the Education Department of Henan Province (12B110002) and NSF of China (61143002).

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Correspondence to Huazhao Xie.

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### Authors’ contributions

HX carried out the existence of infinitely many solutions for the p-harmonic equation studies and drafted the manuscript. JW participated in its coordination. All authors read and approved the final manuscript.

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Xie, H., Wang, J. Infinitely many solutions for p-harmonic equation with singular term. J Inequal Appl 2013, 9 (2013). https://doi.org/10.1186/1029-242X-2013-9