p-Laplacian Equations in R + N with Critical Boundary Nonlinearity

Abstract

In this paper, we consider the following p-Laplacian equation in ${\mathbb{R}}_{+}^N$ with critical boundary nonlinearity. The existence of infinitely many solutions of the equation is proved via the truncation method.

1. Introduction

In this paper, we consider the following p-Laplacian equation in ${\mathbb{R}}_{+}^N$ with critical boundary nonlinearity

$$\left\{\begin{array}{c}-{\Delta }_{p}u=0,\mathrm{in}{\mathbb{R}}_{+}^N,\hfill \\ {|\nabla u|}^{p-2}\frac{\partial u}{\partial n}+{|u|}^{p-2}u={|u|}^{\overline{p}-2}u+\mu {|u|}^{q-2}u,\mathrm{on}{\mathbb{R}}^{N-1}=\partial {\mathbb{R}}_{+}^N,\hfill \end{array}\right.$$

(1)

where $1<p<N,\max \{p,\overline{p}-1\}<q<\overline{p}=\frac{(N-1)p}{N-p},\mu >0$ and ${\Delta }_p$ is the p-Laplacian operator, ${\Delta }_{p}u={\mathrm{div}(|\nabla u|}^{p-2}\nabla u)$. We are looking for axial solutions of the Equation (1) that are solutions of the form $u\left(x\right)=u(|y|,s)$, where we denote $x\in \overline{{\mathbb{R}}_{+}^N}$ by $x=(y,s)\in {\mathbb{R}}^{N-1}\times [0,\infty )$ and we identify ${\mathbb{R}}^{N-1}=\partial {\mathbb{R}}_{+}^N,y=(y,0)$ for $y\in {\mathbb{R}}^{N-1}$ if there is no confusion.

Introduce in $C_0^{\infty }\left(\overline{{\mathbb{R}}_{+}^N}\right)$ a norm by

$$\parallel \phi \parallel ={\left({\int }_{{\mathbb{R}}_{+}^N}{|\nabla \phi |}^{p}dx+{\int }_{{\mathbb{R}}^{N-1}}{|\phi |}^{p}dy\right)}^{\frac{1}{p}}.$$

Let W be the completion of $C_0^{\infty }\left(\overline{{\mathbb{R}}_{+}^N}\right)$ with respect to this norm and $W_r$ be the subspace of W of axial functions, that is,

$$W_r=\{u|u\in W,u\left(x\right)=u(|y|,s),x=(y,s)\in {\mathbb{R}}_{+}^N\}.$$

The problem (1) has a variational structure given by the functional

$$I\left(u\right)=\frac{1}{p}\left({\int }_{{\mathbb{R}}_{+}^N}{|\nabla u|}^{p}dx+{\int }_{{\mathbb{R}}^{N-1}}{|u|}^{p}dy\right)-\frac{1}{\overline{p}}{\int }_{{\mathbb{R}}^{N-1}}{|u|}^{\overline{p}}dy-\frac{\mu }{q}{\int }_{{\mathbb{R}}^{N-1}}{|u|}^{q}dy,u\in W_r.$$

Notice that $\overline{p}=\frac{(N-1)p}{N-p}$ is the critical exponent for the Sobolev imbedding from $W^{1,p}\left({\mathbb{R}}_{+}^N\right)$ to $L^q\left({\mathbb{R}}^{N-1}\right),p\le q\le \overline{p}$. Moreover, the imbedding from $W_r$ to $L^q\left({\mathbb{R}}^{N-1}\right)$ is continuous for $p\le q\le \overline{p}$ and compact only for $p<q<\overline{p}$ due to the dilations. Therefore, the Palais–Smale condition is not satisfied by the functional I and the problem (1) lacks the necessary compactness property. Since the pioneering work of Brezis and Nirenberg [1], significant progress has been made in recent decades for these kinds of problems lacking compactness. In particular, the authors of [2] dealt with the Laplacian equation with critical growth in the bounded domain

$$\left\{\begin{array}{c}-\Delta u={|u|}^{2^{*}-2}u+\lambda u,\mathrm{in}\Omega ,\hfill \\ u=0,\mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(2)

where $\Omega \subset {\mathbb{R}}^N,N\ge 3$ is a regular bounded domain, and $2^{*}=\frac{2N}{N-2}$. While the authors of [3] considered the Laplacian equation with subcritical nonlinear term in the whole space ${\mathbb{R}}^N$

$$\left\{\begin{array}{c}-\Delta u+a\left(x\right)u={|u|}^{q-2}u,\mathrm{in}{\mathbb{R}}^N,\hfill \\ u\left(x\right)\to 0,\mathrm{as}|x|\to \infty ,\hfill \end{array}\right.$$

(3)

where $2<q<2^{*}$ and $a\left(x\right)$ is the potential function. As to the p-Laplacian equation, there is a lot of significant work, whether in the field of ordinary differential equations [4,5,6] or partial differential equations [7,8,9], the authors of [7] considered

$$\left\{\begin{array}{c}-{\Delta }_{p}u+a\left(x\right)u={|u|}^{p^{*}-2}u+\lambda u,\mathrm{in}\Omega ,\hfill \\ u=0,\mathrm{on}\partial \Omega ,\hfill \end{array}\right.$$

(4)

where $p^{*}=\frac{Np}{N-p}$. All of these authors found the solutions as limits of approximated equations with subcritical growth in bounded domains. The lack of compactness due to dilations (in the case (2) and (4)) and shifts (in the case (3)) does not allow for deducing that a sequence of approximate solutions must have a convergent subsequence, but the fact that they solve the approximated problems gives, with use of a local Pohožaev identity, some extra estimates which lead to a proof of desired compactness.

In the Existing literature, some researchers considered the existence of finite multiple solutions [10,11]. While the subcritical problems in bounded domains have infinitely many solutions. In order to show the existence of multiple solutions of the original problems, we need to check that multiple solutions of approximated problems do not converge to the same solution of the limit problems. This is hard work. In both [2,3], some estimates on the Morse index are employed, which has been used as one of the possible devices to distinguish the limit of the multiple approximate solutions by their original variational characterization. For general p-Laplacian equations, we have no information on the Morse index; therefore, the approach in this last step in [2,3] can not be extended in a straightforward way to problems involving the p-Laplacian operator. Here, we will use the truncation method, as we did in [8,9]. First, we consider some truncated problems, the solutions of which will be used as approximate solutions. By a concentration–compactness analysis, similar to that in [2,3,7], in particular with use of a local Pohožaev identity, the theorem of convergence of approximate solutions is proved. We show that, by a careful choice of the approximate nonlinear terms, the approximated problems and the original problem share more and more solutions, as the approximation parameter tends to zero. For more references, we refer the readers to [12,13,14,15,16,17,18].

Let us describe the truncation method in more details. Let $\psi \in C_0^{\infty }(\mathbb{R},[0,1])$ be an even function such that $\psi \left(t\right)=1$ for $|t|\le 1$, $\psi \left(t\right)=0$ for $|t|\ge 2$ and $\psi$ is decreasing in $[1,2]$. Define the auxiliary functions for $\lambda \in (0,1],s\in \mathbb{R}$

$$\begin{gathered} b_{\lambda }\left(s\right)=\psi \left(\lambda s\right),m_{\lambda }\left(s\right)={\int }_0^s{b}_{\lambda }\left(\tau \right)d\tau \\ {F}_{\lambda }\left(s\right)=\frac{1}{p}{|s|}^q{|m_{\lambda }\left(s\right)|}^{\overline{p}-q},f_{\lambda }\left(s\right)=\frac{d}{ds}{F}_{\lambda }\left(s\right). \end{gathered}$$

(5)

Instead of the original problem (1), we consider the truncated problem

$$\left\{\begin{array}{c}-{\Delta }_{p}u=0,\mathrm{in}{\mathbb{R}}_{+}^N,\hfill \\ {|\nabla u|}^{p-2}\frac{\partial u}{\partial n}+{|u|}^{p-2}u=f_{\lambda }\left(u\right)+\mu {|u|}^{q-2}u,\mathrm{on}{\mathbb{R}}^{N-1}.\hfill \end{array}\right.$$

(6)

In addition, the problem (6) has a variational structure given by the functional

$$I_{\lambda }\left(u\right)=\frac{1}{p}\left({\int }_{{\mathbb{R}}_{+}^N}{|\nabla u|}^{p}dx+{\int }_{{\mathbb{R}}^{N-1}}{|u|}^{p}dy\right)-{\int }_{{\mathbb{R}}^{N-1}}{F}_{\lambda }\left(u\right)dy-\frac{\mu }{q}{\int }_{{\mathbb{R}}^{N-1}}{|u|}^{q}dy,u\in W_r.$$

Notice that the functional $I_{\lambda },\lambda >0$ is subcritical at the infinity and the imbedding from $W_r$ to $L^q\left({\mathbb{R}}^{N-1}\right),p<q<\overline{p}$ is compact. Therefore, the functional $I_{\lambda },\lambda >0$ satisfies the Palais–Smale condition.

Here are our main results.

Theorem 1.

Assume $max\{p,\overline{p}-1\}<q<\overline{p}$. Given $L>0,$ there exists $\nu =\nu \left(L\right)$, independent of λ, such that if $u\in W_r,D{I}_{\lambda }\left(u\right)=0$ and $I_{\lambda }\left(u\right)\le L$, then it holds that

$$|u\left(y\right)|\le \frac{1}{\nu },fory\in {\mathbb{R}}^{N-1}=\partial {\mathbb{R}}_{+}^N.$$

Consequently, if $\lambda <\nu$, then u is a solution of the problem (1).

Theorem 2.

Assume $max\{p,\overline{p}-1\}<q<\overline{p}$. Then, the problem (1) has infinitely many axial solutions.

Throughout the paper, we use the following notations: we use $\parallel ·\parallel$ and ${|·|}_q$ to denote the norms of W and $L^q\left({\mathbb{R}}^{N-1}\right)$, respectively, ⇀ and → to denote the weak and the strong convergence, respectively. In addition, we use the notations $B_{\delta }^{+}\left(x_0\right)=\{x|x\in {\mathbb{R}}_{+}^N,|x-x_0|<\delta \}$, $D_{\delta }\left(y_0\right)=\{y|y\in {\mathbb{R}}^{N-1},|y-y_0|<\delta \}$, $B_{\delta }^{+}=B_{\delta }^{+}\left(0\right),D_{\delta }=D_{\delta }\left(0\right)$.

The paper is organized as follows. In Section 2, we do the concentration–compactness analysis of the approximate solution sequence and prove the convergence Theorem 1. In Section 3, we construct a sequence of critical values of the truncated functionals by the symmetric mountain pass lemma. Finally, we prove the existence Theorem 2 by showing that approximated solutions are also solutions of the original problem for a sufficiently small parameter.

2. Concentration–Compactness Analysis

2.1. The Profile Decomposition

In this section, we analyze the concentration behavior for the solutions of the problem (6) as $\lambda \to 0$ and prove Theorem 1. First, we list the properties of the auxiliary functions, defined in (5) in the following lemma.

Lemma 1.

It holds that for $s\in \mathbb{R}$

$\left(a\right)$

$0\le b_{\lambda }\left(s\right)\le 1$.

$\left(b\right)$

$s{m}_{\lambda }\left(s\right)\ge 0,0\le \frac{s{b}_{\lambda }\left(s\right)}{m_{\lambda }\left(s\right)}\le 1$.

$\left(c\right)$

$m_{\lambda }\left(s\right)=s$ for $|s|\le \frac{1}{\lambda }$.

$\left(d\right)$

$min\{|s|,\frac{1}{\lambda }\}\le |m_{\lambda }\left(s\right)|\le min\{|s|,\frac{2}{\lambda }\}$.

$\left(e\right)$

$|f_{\lambda }{\left(s\right)|\le |s|}^{q-1}|m_{\lambda }{\left(s\right)|}^{\overline{p}-q}\le {|s|}^{\overline{p}-1}$.

$\left(f\right)$

$\frac{1}{q}s{f}_{\lambda }\left(s\right)-F_{\lambda }\left(s\right)=\left(\frac{1}{q}-\frac{1}{\overline{p}}\right){|s|}^{q+1}{|m_{\lambda }\left(s\right)|}^{\overline{p}-q-1}{b}_{\lambda }\left(s\right)\ge 0$.

$\left(g\right)$

$F_{\lambda }\left(s\right)-\frac{1}{\overline{p}}s{f}_{\lambda }\left(s\right)=\left(\frac{1}{\overline{p}}-\frac{q}{{\overline{p}}^2}\right){|s|}^q{|m_{\lambda }\left(s\right)|}^{\overline{p}-q-1}\left(1-\frac{s{b}_{\lambda }\left(s\right)}{m_{\lambda }\left(s\right)}\right)\ge 0$.

Proof. 

The proof is straightforward. We verify $\left(e\right)$–$\left(g\right)$. By the definition of $F_{\lambda }$ and $f_{\lambda }$, we have

$$f_{\lambda }\left(s\right)=\frac{d}{ds}{F}_{\lambda }\left(s\right)=\frac{q}{\overline{p}}{|s|}^{q-2}s|m_{\lambda }{\left(s\right)|}^{\overline{p}-q}+\frac{\overline{p}-q}{\overline{p}}{|s|}^q{|m_{\lambda }\left(s\right)|}^{\overline{p}-q-2}{m}_{\lambda }\left(s\right)b_{\lambda }\left(s\right).$$

(7)

$\left(f\right)$ and $\left(g\right)$ follow from (7), and $\left(e\right)$ follows from (7) and $\left(a\right)$, $\left(d\right)$ of this lemma. □

Lemma 2.

Let ${\lambda }_n\ge 0,u_n\in W_r$ such that $D{I}_{{\lambda }_n}\left(u_n\right)=0,I_{{\lambda }_n}\left(u_n\right)\le L$. Then, $\left\{u_n\right\}$ is bounded in $W_r$.

Proof. 

By Lemma 1 $\left(f\right)$, we have

$$\begin{array}{cc}\hfill L& \ge I_{{\lambda }_n}\left(u_n\right)=I_{{\lambda }_n}\left(u_n\right)-\frac{1}{q}⟨D{I}_{{\lambda }_n}\left(u_n\right),u_n⟩\hfill \\ \hfill & =\left(\frac{1}{p}-\frac{1}{q}\right)\left({\int }_{{\mathbb{R}}_{+}^N}|\nabla u_n{|}^{p}dx+{\int }_{{\mathbb{R}}^{N-1}}{|u_n|}^{p}dy\right)+{\int }_{{\mathbb{R}}^{N-1}}\left(\frac{1}{q}{u}_n{f}_{{\lambda }_n}\left(u_n\right)-F_{{\lambda }_n}\left(u_n\right)\right)dx\hfill \\ \hfill & \ge \left(\frac{1}{p}-\frac{1}{q}\right)\left({\int }_{{\mathbb{R}}_{+}^N}|\nabla u_n{|}^{p}dx+{\int }_{{\mathbb{R}}^{N-1}}{|u_n|}^{p}dy\right)=\left(\frac{1}{p}-\frac{1}{q}\right){\parallel u_n\parallel }^p.\hfill \end{array}$$

Hence, $\left\{u_n\right\}$ is bounded in $W_r$. □

Let $\mathcal{D}={\mathcal{D}}^p\left(\overline{{\mathbb{R}}_{+}^N}\right)$ be the completion of $C_0^{\infty }\left(\overline{{\mathbb{R}}_{+}^N}\right)$ with respect to the norm

$${\parallel \phi \parallel }_{\mathcal{D}}={\left({\int }_{{\mathbb{R}}_{+}^N}{|\nabla \phi |}^{p}dx\right)}^{\frac{1}{p}}$$

and ${\mathcal{D}}_r$ be the subspace of $\mathcal{D}$ of axial functions,

$${\mathcal{D}}_r=\left\{u|u\in \mathcal{D},u\left(x\right)=u(|y|,s),x=(y,s)\in \overline{{\mathbb{R}}_{+}^N}={\mathbb{R}}^{N-1}\times [0,\infty )\right\}.$$

Let D be the dilation group

$$D=\{g_{\sigma }|g_{\sigma }u\left(x\right)={\sigma }^{\frac{N-p}{p}}u\left(\sigma x\right),x\in \overline{{\mathbb{R}}_{+}^N},\sigma >0\}.$$

(8)

Notice that the operator $g_{\sigma }$ of D is an isometry in both $\mathcal{D}$ and $L^{\overline{p}}\left({\mathbb{R}}^{N-1}\right)$. The imbedding from ${\mathcal{D}}_r$ to $L^{\overline{p}}\left({\mathbb{R}}^{N-1}\right)$ is compact with respect to the group D that is a sequence $\left\{u_n\right\}$ of ${\mathcal{D}}_r$, satisfying $g_{{\sigma }_n}{u}_n\rightharpoonup 0$ in ${\mathcal{D}}_r$ for any sequence $\left\{g_{{\sigma }_n}\right\}$ of D, denoted by $u_n\stackrel{D}{\rightharpoonup }0$ in ${\mathcal{D}}_r$, must converge to zero in $L^{\overline{p}}\left({\mathbb{R}}^{N-1}\right)$.

Now, let $u_n$ be a bounded sequence of $W_r$. By [19,20], we have the following profile decomposition:

$$u_n=u+\sum _{k\in \mathrm{\Lambda }}{g}_{{\sigma }_{n,k}}{U}_k+r_n,$$

(9)

where $u\in W_r,U_k\in {\mathcal{D}}_r,r_n\in {\mathcal{D}}_r,{\sigma }_{n,k}\in (0,\infty )$ and $\mathrm{\Lambda }$ is an index set, satisfying

$\left(a\right)$

$u_n\rightharpoonup u$ in $W_r$, $g_{{\sigma }_{n,k}}^{-1}{u}_n\rightharpoonup U_k$ in ${\mathcal{D}}_r$ as $n\to \infty ,k\in \mathrm{\Lambda }$.

$\left(b\right)$

${\sigma }_{n,k}\to +\infty ,\frac{{\sigma }_{n,k}}{{\sigma }_{n,l}}+\frac{{\sigma }_{n,l}}{{\sigma }_{n,k}}\to +\infty$, as $n\to \infty ,k,l\in \mathrm{\Lambda },k\ne l$.

$\left(c\right)$

${\parallel u\parallel }_{\mathcal{D}}^p+{\displaystyle \sum _{k\in \mathrm{\Lambda }}}\parallel U_k{\parallel }_{\mathcal{D}}^p\le \underset{n\to \infty }{lim\; inf}{\parallel u_n\parallel }_{\mathcal{D}}^p$.

$\left(d\right)$

$r_n\rightharpoonup 0$ in ${\mathcal{D}}_r$ as $n\to \infty$, consequently $r_n\to 0$ in $L^{\overline{p}}\left({\mathbb{R}}^{N-1}\right)$ as $n\to \infty$.

We refer to [19,20] for general concepts of compactness and the profile decomposition and relevant results. For reader’s convenience, we consider the compactness of the imbedding from ${\mathcal{D}}_r$ to $L^{\overline{p}}\left({\mathbb{R}}^{N-1}\right)$ with respect to the dilation group D.

Lemma 3.

Assume ${\lambda }_n>0,{\lambda }_n\to 0$ as $n\to \infty$, $u_n\in W_r,D{I}_{{\lambda }_n}\left(u_n\right)=0$ and $I_{{\lambda }_n}\left(u_n\right)\le L$. Assume that the profile decomposition (9) holds. Then,

$\left(1\right)$

$v=|u|$ satisfies the inequality

$${\int }_{{\mathbb{R}}_{+}^N}{|\nabla v|}^{p-2}\nabla v\nabla \phi dx+{\int }_{{\mathbb{R}}^{N-1}}{v}^{p-1}\phi dy\le {\int }_{{\mathbb{R}}^{N-1}}{v}^{\overline{p}-1}\phi dy+\mu {\int }_{{\mathbb{R}}^{N-1}}{v}^{q-1}\phi dy,$$

(10)

for $\phi \ge 0,\phi \in W_r$. Consequently, for some $c>1,$

$${\int }_{{\mathbb{R}}_{+}^N}{|\nabla v|}^{p-2}\nabla v\nabla \phi dx\le c{\int }_{{\mathbb{R}}^{N-1}}{v}^{\overline{p}-1}\phi dy,for\phi \ge 0,\phi \in W_r.$$

(11)

$\left(2\right)$

$V_k=|U_k|$ satisfies the inequality

$${\int }_{{\mathbb{R}}_{+}^N}{|\nabla V_k|}^{p-2}\nabla V_k\nabla \phi dx\le {\int }_{{\mathbb{R}}^{N-1}}{V}_k^{\overline{p}-1}\phi dy\mathrm{for}\phi \ge 0,\phi \in {\mathcal{D}}_r.$$

(12)

Proof. 

We prove the conclusion for the function $V_k$. $u_n$ satisfies the equation in the weak form

$$\begin{gathered} {\int }_{{\mathbb{R}}_{+}^N}|\nabla u_n{|}^{p-2}\nabla u_n\nabla \phi dx+{\int }_{{\mathbb{R}}^{N-1}}|u_n{|}^{p-2}{u}_n\phi dy \\ ={\int }_{{\mathbb{R}}^{N-1}}{f}_{{\lambda }_n}\left(u_n\right)\phi dy+\mu {\int }_{{\mathbb{R}}^{N-1}}{|u_n|}^{q-2}{u}_n\phi dy,\phi \in W_r. \end{gathered}$$

(13)

Denote ${\tilde{u}}_n=g_{{\sigma }_{n,k}}^{-1}{u}_n$. For $\phi \in W_r$, take $g_{{\sigma }_{n,k}}\phi$ as a test function in (13). By a variable change, we obtain

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}_{+}^N}|\nabla {\tilde{u}}_n{|}^{p-2}\nabla {\tilde{u}}_n\nabla \phi dx+{\sigma }_{n,k}^{-(p-1)}{\int }_{{\mathbb{R}}^{N-1}}{|{\tilde{u}}_n|}^{p-2}{\tilde{u}}_n\phi dy\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^{N-1}}{f}_{{\tilde{\lambda }}_n}\left({\tilde{u}}_n\right)\phi dy+\mu {\sigma }_{n,k}^{\frac{N-p}{p}q-(N-1)}{\int }_{{\mathbb{R}}^{N-1}}{|{\tilde{u}}_n|}^{q-2}{\tilde{u}}_n\phi dy,\phi \in W_r\hfill \end{array}$$

(14)

where ${\tilde{\lambda }}_n={\lambda }_n{\sigma }_{n,k}^{\frac{N-p}{p}}$. In the above, we have used the fact that

$${\nu }^{-(\overline{p}-1)}{f}_{\lambda }\left(\nu s\right)=f_{\lambda \nu }\left(s\right),\lambda ,\nu >0,s\in \mathbb{R}$$

which can be proved by the very definition of the function $f_{\lambda }$.

Since ${\int }_{{\mathbb{R}}^{N-1}}|{\tilde{u}}_n{|}^{\overline{p}}dy={\int }_{{\mathbb{R}}^{N-1}}{|u_n|}^{\overline{p}}dy$ is bounded and ${\tilde{u}}_n$ is axial, for any $y\in {\mathbb{R}}^{N-1}\setminus \left\{0\right\}$

$$\underset{s\to 0}{lim}\underset{n\to \infty }{lim\; sup}{\int }_{D_y}{|{\tilde{u}}_n|}^{\overline{p}}dy=0.$$

Choose $\delta =\delta \left(y\right)$, independent of n, such that

$${\int }_{D_{4\delta }\left(y\right)}{|{\tilde{u}}_n|}^{\overline{p}}dy\le \frac{1}{2}{\overline{S}}_p^{\frac{N-1}{p-1}}$$

where ${\overline{S}}_p$ is the Sobolev constant of the imbedding $\mathcal{D}\hookrightarrow L^{\overline{p}}\left({\mathbb{R}}^{N-1}\right)$. By Lemma A4, ${\tilde{u}}_n$ is uniformly bounded in $D_{2\delta }\left(y\right)$. Consequently, by Equation (14) and the following elementary inequality (15), ${\tilde{u}}_n$ converges in $W^{1,p}\left(B_{\delta }\left(y\right)\right)$ and in $W_{\mathrm{loc}}^{1,p}\left({\mathbb{R}}_{+}^N\right)$. The following inequality (15) is useful for problems involving the p-Laplacian operator [21]. There exists a constant $c_p$ such that, for $\xi ,\eta \in {\mathbb{R}}^N$,

$$\begin{array}{cc}\hfill & {(|\xi |}^{p-2}\xi -{|\eta |}^{p-2}\eta ,\xi -\eta )\ge c_p{|\xi -\eta |}^p,\mathrm{if}p\ge 2,\hfill \\ \hfill & {(|\xi |}^{p-2}\xi -{|\eta |}^{p-2}\eta ,\xi -\eta )\ge c_p{|\xi -\eta |·(|\xi |}^p+{|\eta |}^p{)}^{\frac{p-2}{p}},\mathrm{if}1<p<2.\hfill \end{array}$$

(15)

Let ${\tilde{v}}_n=|{\tilde{u}}_n|$, ${\tilde{v}}_n$ converge to $V_k=|U_k|$ in $W_{\mathrm{loc}}^{1,p}\left({\mathbb{R}}_{+}^N\right)$ and satisfy the inequality

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}_{+}^N}{|\nabla {\tilde{v}}_n|}^{p-2}\nabla {\tilde{v}}_n\nabla \phi dx+{\sigma }_{n,k}^{-(p-1)}{\int }_{{\mathbb{R}}^{N-1}}{\tilde{v}}_n^{p-1}\phi dy\hfill \\ \hfill \le & {\int }_{{\mathbb{R}}^{N-1}}{\tilde{v}}_n^{\overline{p}-1}\phi dy+\mu {\sigma }_{n,k}^{\frac{N-p}{p}q-(N-1)}{\int }_{{\mathbb{R}}^{N-1}}{\tilde{v}}_n^{q-1}\phi dy\hfill \end{array}$$

(16)

for $\phi \in W_r,\phi \ge 0$. Assume $\phi \in C_0^{\infty }({\mathbb{R}}^N\setminus \left\{0\right\})\cap {\mathcal{D}}_r$. Taking the limit $n\to \infty$ in (16), we obtain

$${\int }_{{\mathbb{R}}_{+}^N}{|\nabla V_k|}^{p-2}\nabla V_k\nabla \phi dx\le {\int }_{{\mathbb{R}}^{N-1}}{V}_k^{\overline{p}-1}\phi dy.$$

(17)

By a density argument, (17) holds for $\phi \in \mathcal{D},\phi \ge 0$. □

Lemma 4.

The index set Λ in the profile decomposition (9) is finite.

Proof. 

By Lemma 3, $V_k$ satisfies the inequality (12). Choose $\phi =V_k$ in (12). By the Sobolev imbedding theorem

$${\int }_{{\mathbb{R}}_{+}^N}{|\nabla V_k|}^{p}dx\le {\int }_{{\mathbb{R}}^{N-1}}{V}_k^{\overline{p}}dy\le {\left({\overline{S_p}}^{-1}{\int }_{{\mathbb{R}}_{+}^N}{|\nabla V_k|}^{p}dx\right)}^{\frac{\overline{p}}{p}}$$

hence

$${\int }_{{\mathbb{R}}_{+}^N}|\nabla U_k{|}^{p}dx={\int }_{{\mathbb{R}}_{+}^N}{|\nabla V_k|}^{p}dx\ge {\overline{S}}_p^{\frac{N-1}{p-1}}.$$

By the property $\left(3\right)$ of the decomposition (9), $\mathrm{\Lambda }$ is a finite set. □

2.2. Safe Regions

Assume the profile decomposition (9) with a finite index set $\mathrm{\Lambda }$. Denote

$${\sigma }_n=min\{{\sigma }_{n,k}|k\in \mathrm{\Lambda }\}$$

and define the so-called safe regions [2]

$$\begin{array}{cc}\hfill & A_n^i=\{x|x\in {\mathbb{R}}_{+}^N,i{\sigma }_n^{-\frac{1}{p}}<|x|<(7-i){\sigma }_n^{-\frac{1}{p}}\},\hfill \\ \hfill & T_n^i=\{y|y\in {\mathbb{R}}^{N-1},i{\sigma }_n^{-\frac{1}{p}}<|y|<(7-i){\sigma }_n^{-\frac{1}{p}}\},i=1,2,3.\hfill \end{array}$$

(18)

For these regions, we have a good estimate.

Proposition 1.

There exists a constant c, independent of n, such that

$$|u_n\left(x\right)|\le cforx\in A_n^2\cup T_n^2.$$

Corollary 1.

There exists a constant c, independent of n, such that

$${\int }_{A_n^3}{|\nabla u_n|}^{p}dx\le c.$$

In order to prove these estimates, we start with the following definition.

Definition 1.

Suppose $1\le p_2<\overline{p}<p_1$, $\sigma >1$ and $\alpha >0$. Consider the following system of inequality

$$\left\{\begin{array}{c}|u_1{|}_{p_1}\le \alpha ,\hfill \\ |u_2{|}_{p_2}\le \alpha {\sigma }^{\frac{N-1}{\overline{p}}-\frac{N-1}{p_2}}.\hfill \end{array}A\right.$$

(19)

Define the norm ${|·|}_{p_1,p_2,\sigma }$ by

$${|u|}_{p_1,p_2,\sigma }=inf\{\alpha |\mathrm{there}\mathrm{exist}{u}_1,u_2\mathrm{such}\mathrm{that}|u|\le u_1+u_2\mathrm{and}\left(19\right)\mathrm{holds}\}.$$

Proposition 2.

Assume ${\lambda }_n>0,{\lambda }_n\to 0$ as $n\to \infty$, $u_n\in W_r,D{I}_{{\lambda }_n}\left(u_n\right)=0$ and $I_{{\lambda }_n}\left(u_n\right)\le L$. Assume the profile decomposition (9) holds. Denote ${\sigma }_n=min\{{\sigma }_{n,k}|k\in \mathrm{\Lambda }\}$.Then, for any $p_1,p_2$ satisfying

$$\left(1-\frac{1}{p}\right)\overline{p}<p_2<\overline{p}<p_1,$$

there exists a constant $c=c(p_1,p_2)$ such that

$$|u_n{|}_{p_1,p_2,{\sigma }_n}\le c.$$

Proof. 

By Lemma 3, $v=|u|$ satisfies the inequality

$${\int }_{{\mathbb{R}}_{+}^N}{|\nabla v|}^{p-2}\nabla v\nabla \phi dx\le c{\int }_{{\mathbb{R}}^{N-1}}{v}^{\overline{p}-1}\phi dy,\phi \ge 0,\phi \in W_r.$$

By Lemma A4, $u\in L^{\infty }\left({\mathbb{R}}^{N-1}\right)$, hence for $p_1>\overline{p}$,

$${|u|}_{p_1}\le {|u|}_{\infty }^{1-\frac{\overline{p}}{p_1}}{|u|}_{\overline{p}}^{\frac{\overline{p}}{p_1}}\le c.$$

(20)

By Lemma 3, $V_k=|U_k|$ satisfies the inequality

$${\int }_{{\mathbb{R}}_{+}^N}{|\nabla v|}^{p-2}\nabla v\nabla \phi dx\le {\int }_{{\mathbb{R}}^{N-1}}{v}^{\overline{p}-1}\phi dy,\phi \ge 0,\phi \in W_r.$$

By Theorem 2.2 of [22], there exists a constant c such that

$$|U_k\left(y\right)|=V_k\left(y\right)\le c{\left(1+{|y|}^{\frac{p}{p-1}}\right)}^{-\frac{N-p}{p}},y\in {\mathbb{R}}^{N-1}.$$

Hence, for $\left(1-\frac{1}{p}\right)\overline{p}<p_2<\overline{p}$, we have

$$\begin{array}{cc}\hfill |g_{{\sigma }_{n,k}}{U}_k{|}_{p_2}& \le {\left({\int }_{{\mathbb{R}}^{N-1}}{\left({\sigma }_{n,k}^{\frac{N-p}{p}}(1+|{\sigma }_{n,k}y{{|}^{\frac{p}{p-1}})}^{-\frac{N-p}{p}}\right)}^{p_2}dy\right)}^{\frac{1}{p_2}}\hfill \\ \hfill & =c{\sigma }_{n,k}^{\frac{N-p}{p}-\frac{N-1}{p_2}}{\left({\int }_{{\mathbb{R}}^{N-1}}{\left(1+{|y|}^{\frac{p}{p-1}}\right)}^{-\frac{N-p}{p}{p}_2}dy\right)}^{\frac{1}{p_2}}\hfill \\ \hfill & \le c{\sigma }_{n,k}^{\frac{N-1}{\overline{p}}-\frac{N-1}{p_2}}\le c{\sigma }_n^{\frac{N-1}{\overline{p}}-\frac{N-1}{p_2}}.\hfill \end{array}$$

(21)

By (20) and (21), we have

$${|u|}_{p_1,p_2,{\sigma }_n}+\sum _{k\in \mathrm{\Lambda }}{|g_{{\sigma }_{n,k}}{U}_k|}_{p_1,p_2,{\sigma }_n}\le c.$$

(22)

Define $w,W_k,R\in {\mathcal{D}}_r$ by

$$\left\{\begin{array}{c}-{\Delta }_{p}w=0,\mathrm{in}{\mathbb{R}}_{+}^N,\hfill \\ {|\nabla w|}^{p-2}=w^{\overline{p}-1},\mathrm{on}{\mathbb{R}}^{N-1},\hfill \end{array}\right.$$

(23)

$$\left\{\begin{array}{c}-{\Delta }_p{W}_k=0,\mathrm{in}{\mathbb{R}}_{+}^N,\hfill \\ |\nabla W_k{|}^{p-2}\frac{\partial W_k}{\partial n}=W_k^{\overline{p}-1},\mathrm{on}{\mathbb{R}}^{N-1},\hfill \end{array}\right.$$

(24)

$$\left\{\begin{array}{c}-{\Delta }_{p}R=0,\mathrm{in}{\mathbb{R}}_{+}^N,\hfill \\ {|\nabla R|}^{p-2}\frac{\partial R}{\partial n}={|r_n|}^{\overline{p}-1},\mathrm{on}{\mathbb{R}}^{N-1},\hfill \end{array}\right.$$

(25)

By the Wolff potential estimate([2], Corollary 4.13), we have

$$|u_n|=v_n\le c\left(w+\sum _{k\in \mathrm{\Lambda }}{W}_k+R\right).$$

(26)

By Lemma A3, for $\left(1-\frac{1}{p}\right)\overline{p}<p_2<\overline{p}<p_1$, we have

$$\begin{array}{cc}\hfill {|w|}_{p_1,p_2,{\sigma }_n}& \le c|v^{\overline{p}-p}{|}_{\frac{N-1}{p-1}}^{\frac{1}{p-1}}·{|v|}_{p_1,p_2,{\sigma }_n}\hfill \\ \hfill & \le {c|u|}_{\overline{p}}^{\frac{p}{N-p}}·{|u|}_{p_1,p_2,{\sigma }_n}\le c{|u|}_{p_1,p_2,{\sigma }_n}\le c\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill |W_k{|}_{p_1,p_2,{\sigma }_n}& \le c|g_{{\sigma }_{n,k}}{U}_k{|}_{\overline{p}}^{\frac{p}{N-p}}{|g_{{\sigma }_{n,k}}{U}_k|}_{p_1,p_2,{\sigma }_n}\hfill \\ \hfill & \le c|g_{{\sigma }_{n,k}}{U}_k{|}_{p_1,p_2,{\sigma }_n}\le c\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill {|R|}_{p_1,p_2,{\sigma }_n}& \le c|r_n{|}_{\overline{p}}^{\frac{p}{N-p}}{|r_n|}_{p_1,p_2,{\sigma }_n}\hfill \\ \hfill & =o\left(1\right)|r_n{|}_{p_1,p_2,{\sigma }_n}\hfill \\ \hfill & \le o\left(1\right)\left(|u_n{|}_{p_1,p_2,{\sigma }_n}+{|u|}_{p_1,p_2,{\sigma }_n}+\sum _{k\in \mathrm{\Lambda }}{|g_{{\sigma }_{n,k}}{U}_k|}_{p_1,p_2,{\sigma }_n}\right)\hfill \\ \hfill & =o\left(1\right)+o\left(1\right)|u_n{|}_{p_1,p_2,{\sigma }_n}.\hfill \end{array}$$

(29)

We have

$$\begin{array}{cc}\hfill |u_n{|}_{p_1,p_2,{\sigma }_n}& \le c\left({|w|}_{p_1,p_2,{\sigma }_n}+\sum _{k\in \mathrm{\Lambda }}|W_k{|}_{p_1,p_2,{\sigma }_n}+{|r_n|}_{p_1,p_2,{\sigma }_n}\right)\hfill \\ \hfill & \le c+o\left(1\right)|u_n{|}_{p_1,p_2,{\sigma }_n}\hfill \end{array}$$

and

$$|u_n{|}_{p_1,p_2,{\sigma }_n}\le c.$$

□

Lemma 5.

Assume ${\lambda }_n>0,u_n\in W_r,D{I}_{{\lambda }_n}\left(u_n\right)\le L$. Assume the profile decomposition (9) holds. Then, for $\gamma \in \left(p-1,\frac{(p-1)p\overline{p}}{(p-1)p+\overline{p}}\right)$ there exists $c=c\left(\gamma \right)$, independent of n, such that

$${\left({\gamma }^{-N}{\int }_{B_r^{+}}|u_n{|}^{\gamma }dx+{\gamma }^{-N+1}{\int }_{D_r}{|u_n|}^{\gamma }dy\right)}^{\frac{1}{\gamma }}\le cfor\gamma \ge {\sigma }_n^{-\frac{1}{p}}.$$

Proof. 

By Lemma A6 for $\gamma <1$, we have

$$\begin{array}{cc}\hfill & {\left({\gamma }^{-N}{\int }_{B_r^{+}}|u_n{|}^{\gamma }dx+{\gamma }^{-N+1}{\int }_{D_r}{|u_n|}^{\gamma }dy\right)}^{\frac{1}{\gamma }}\hfill \\ \hfill \le & c{\left({\int }_{B_1^{+}}|u_n{|}^{\gamma }dx+{\int }_{D_1}{|u_n|}^{\gamma }dy\right)}^{\frac{1}{\gamma }}+c{\int }_r^1{\left({\int }_{D_t}{|u_n|}^{\overline{p}-1}dy\right)}^{\frac{1}{p-1}}\frac{dt}{t^{1+\frac{N-p}{p-1}}}\hfill \\ \hfill \le & c+c{\int }_r^1{\left({\int }_{D_t}{|u_n|}^{\overline{p}-1}dy\right)}^{\frac{1}{p-1}}\frac{dt}{t^{1+\frac{N-p}{p-1}}}.\hfill \end{array}$$

(30)

By Proposition 2, we have $|u_n{|}_{p_1,p_2,{\sigma }_n}\le c$ for any $p_1,p_2$ such that $\left(1-\frac{1}{p}\right)\overline{p}<p_2<\overline{p}<p_1$. Let $p_2=\overline{p}-1,p_1=N\overline{p}$. Choose $v_1,v_2$ such that $|u_n|\le v_1+v_2,|v_1{|}_{p_1}\le c,{|v_2|}_{p_2}\le c{\sigma }_n^{\frac{N-1}{\overline{p}}-\frac{N-1}{p_2}}$. Then,

$$\begin{array}{cc}\hfill & {\int }_r^1{\left({\int }_{D_t}{v}_1^{\overline{p}-1}dt\right)}^{\frac{1}{p-1}}\frac{dt}{t^{1+\frac{N-p}{p-1}}}\hfill \\ \hfill \le & {\int }_r^1{\left({\int }_{D_t}{v}_1^{N\overline{p}}dy\right)}^{\frac{\overline{p}-1}{N\overline{p}(p-1)}}{\left({\int }_{D_t}dy\right)}^{\frac{N\overline{p}-\overline{p}+1}{N\overline{p}(p-1)}}\frac{dt}{t^{1+\frac{N-p}{p-1}}}\hfill \\ \hfill \le & c{\int }_r^1\frac{dt}{t^{\overline{p}}}\le c\hfill \end{array}$$

(31)

and

$$\begin{array}{cc}\hfill & {\int }_r^1{\left({\int }_{D_t}{v}_1^{\overline{p}-1}dt\right)}^{\frac{1}{p-1}}\frac{dt}{t^{1+\frac{N-p}{p-1}}}\hfill \\ \hfill \le & {\int }_r^1{\sigma }_n^{(\overline{p}-1)\left(\frac{N-1}{\overline{p}}-\frac{N-1}{\overline{p}-1}\right)-\frac{1}{p-1}}\frac{dt}{t^{1+\frac{N-p}{p-1}}}\hfill \\ \hfill =& c{\sigma }_n^{-\frac{N-p}{p(p-1)}}{\int }_r^1\frac{dt}{t^{1+\frac{N-p}{p-1}}}\le c{\left({\sigma }_n^{\frac{1}{p}}t\right)}^{-\frac{N-p}{p-1}}\le c\hfill \end{array}$$

(32)

provided $r\ge {\sigma }_n^{-\frac{1}{p}}$. Hence,

$$\begin{array}{cc}\hfill & {\left({\gamma }^{-N}{\int }_{B_r^{+}}|u_n{|}^{\gamma }dx+{\gamma }^{-N+1}{\int }_{D_r}{|u_n|}^{\gamma }dy\right)}^{\frac{1}{\gamma }}\hfill \\ \hfill \le & c+c{\int }_r^1{\left({\int }_{D_t}{|u_n|}^{\overline{p}-1}dy\right)}^{\frac{1}{p-1}}\frac{dt}{t^{1+\frac{N-p}{p-1}}}\hfill \\ \hfill \le & c+c{\int }_r^1{\left({\int }_{D_t}{|v_1|}^{\overline{p}-1}dy\right)}^{\frac{1}{p-1}}\frac{dt}{t^{1+\frac{N-p}{p-1}}}+c{\int }_r^1{\left({\int }_{D_t}{|v_2|}^{\overline{p}-1}dy\right)}^{\frac{1}{p-1}}\frac{dt}{t^{1+\frac{N-p}{p-1}}}\le c\mathrm{for}r\ge {\sigma }_n^{-\frac{1}{p}}.\hfill \end{array}$$

□

Proof of Proposition 1 and Corollary 1.

Let $w_n\left(x\right)={\sigma }_n^{-\frac{N-p}{p^2}}|u_n|\left({\sigma }_n^{-\frac{1}{p}}x\right)$, $w_n$ satisfy

$$\left\{\begin{array}{c}-{\Delta }_p{w}_n\le 0,\mathrm{in}{\mathbb{R}}_{+}^N,\hfill \\ |\nabla w_n{|}^{p-2}\frac{\partial w_n}{\partial n}\le c{w}_n^{\overline{p}-1},\mathrm{on}{\mathbb{R}}^{N-1}.\hfill \end{array}\right.$$

By the profile decomposition (9), we have

$$\begin{array}{cc}\hfill & {\int }_{1\le |y|\le 6}|w_n{|}^{\overline{p}}dy={\int }_{T_n^1}{|u_n|}^{\overline{p}}dy\hfill \\ \hfill \le & {\int }_{T_n^1}{|u|}^{\overline{p}}dy+c\sum _{k\in \mathrm{\Lambda }}{\int }_{T_n^1}|g_{{\sigma }_{n,k}}{U}_k{|}^{\overline{p}}dy+c{\int }_{T_n^1}{|r_n|}^{\overline{p}}dy\hfill \\ \hfill \le & c{\sigma }_n^{-\frac{N-1}{p}}+c\sum _{k\in \mathrm{\Lambda }}{\int }_{|y|\ge {{\sigma }_n}^{-\frac{1}{p}}}{\left|{\sigma }_{n,k}^{\frac{N-p}{p}}{U}_k\left({\sigma }_{n,k}y\right)\right|}^{\overline{p}}dy+o\left(1\right)\hfill \\ \hfill \le & o\left(1\right)+c\sum _{k\in \mathrm{\Lambda }}{\int }_{|y|\ge {\sigma }_n^{1-\frac{1}{p}}}{\left(1+{|y|}^{\frac{p}{p-1}}\right)}^{-\frac{N-p}{p}·\overline{p}}dy=o\left(1\right).\hfill \end{array}$$

(33)

By Lemma A4 and Lemma 5, for $2\le x\le 5,x\in {\mathbb{R}}_{+}^N\cup {\mathbb{R}}^{N-1}$, we have

$$\begin{array}{cc}\hfill w_n\left(x\right)& \le c{\left({\int }_{1\le |x|\le 6}{w}_n^{\gamma }\left(x\right)dx+{\int }_{1\le |x|\le 6}{w}_n^{\gamma }\left(y\right)dy\right)}^{\frac{1}{\gamma }}\hfill \\ \hfill & =c{\sigma }_n^{-\frac{N-p}{p^2}}{\left({\sigma }_n^{-\frac{N}{p}}{\int }_{A_n^1}|u_n{|}^{\gamma }dx+{\sigma }_n^{-\frac{N-1}{p}}{\int }_{T_n^1}{|u_n|}^{\gamma }dy\right)}^{\frac{1}{\gamma }}\hfill \\ \hfill & \le c{\sigma }_n^{-\frac{N-p}{p^2}}.\hfill \end{array}$$

Hence,

$$|u_n\left(x\right)|={\sigma }_n^{\frac{N-p}{p^2}}{w}_n\left({\sigma }_n^{\frac{1}{p}}x\right)\le c\mathrm{for}x\in A_n^2\cup T_n^2.$$

We complete the proof of Proposition 1. To prove Corollary 1, we choose a function $\phi \in C_0^{\infty }\left({\mathbb{R}}^N\right)$ such that $\phi \left(x\right)=1$ for $x\in A_n^3\cup T_n^3$ and $\phi \left(x\right)=0$ for $x\notin A_n^2\cup T_n^3$ and $|\nabla \phi |\le 2{\sigma }_n^{\frac{1}{p}}.$ Testing the Equation (13) by ${\phi }^p{u}_n$, we obtain

$$\begin{array}{cc}\hfill & {\int }_{A_n^2}|\nabla u_n{|}^p{\phi }^{p}dx={\int }_{A_n^2}|\nabla u_n{|}^{p-2}\nabla \left(u_n\phi \right)dx-p{\int }_{A_n^2}{|\nabla u_n|}^{p-2}\nabla u_n{u}_n{\phi }^{p-1}\nabla \phi dx\hfill \\ \hfill \le & {\int }_{T_n^2}(-|u_n{|}^p+|u_n{|}^{\overline{p}}+\mu |u_n{|}^q){\phi }^{p}dy+\frac{1}{2}{\int }_{A_n^2}|\nabla u_n{|}^p{\phi }^{p}dx+c{\int }_{A_n^2}|u_n{|}^p{|\nabla \phi |}^{p}dx.\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill & {\int }_{A_n^3}|\nabla u_n{|}^{p}dx\le {\int }_{A_n^2}{|\nabla u_n|}^p{\phi }^{p}dx\hfill \\ \hfill \le & c{\int }_{T_n^2}|u_n{|}^{\overline{p}}dy+c{\int }_{A_n^2}|u_n{|}^p{|\nabla \phi |}^{p}dx\hfill \\ \hfill \le & c{\sigma }_n^{-\frac{1}{p}(N-1)}+c{\sigma }_n^{-\frac{N}{p}+\frac{1}{p}·p}\le c{\sigma }_n^{1-\frac{N}{p}}.\hfill \end{array}$$

□

2.3. Pohožaev Identity

In the remainder of this section, following the idea of [2,3], we apply the local Pohožaev identity to prove the convergence Theorem 1.

Lemma 6.

(Local Pohožaev identity) Assume that $u\in W$ satisfies the equation

$$\left\{\begin{array}{c}-{\Delta }_{p}u=0,\mathrm{in}{\mathbb{R}}_{+}^N,\hfill \\ {|\nabla u|}^{p-2}\frac{\partial u}{\partial n}+{|u|}^{p-2}u=f_{\lambda }\left(u\right)+\mu {|u|}^{q-2}u,\mathrm{on}{\mathbb{R}}^{N-1}.\hfill \end{array}\right.$$

(34)

Let $\phi \in C_0^{\infty }\left({\mathbb{R}}^N\right)$, then

$$\begin{array}{cc}\hfill & -\frac{p-1}{p}{\int }_{{\mathbb{R}}^{N-1}}{|u|}^p\phi dy+{\int }_{{\mathbb{R}}^{N-1}}\left((N-1)F_{\lambda }\left(u\right)-\frac{N-p}{p}{f}_{\lambda }\left(u\right)u\right)dy\hfill \\ \hfill & +\left(\frac{N-1}{q}-\frac{N-p}{p}\right)\mu {\int }_{{\mathbb{R}}^{N-1}}{|u|}^q\phi dy\hfill \\ \hfill =& \frac{1}{p}{\int }_{{\mathbb{R}}_{+}^N}{|\nabla u|}^p(x,\nabla \phi )dx-{\int }_{{\mathbb{R}}_{+}^N}{|\nabla u|}^{p-2}(\nabla u,x)(\nabla u,\nabla \phi )dx-\frac{N-p}{p}{\int }_{{\mathbb{R}}_{+}^N}{|\nabla u|}^{p-2}u(\nabla u,\nabla \phi )dx\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^{N-1}}\left(\frac{1}{p}{|u|}^p-F_{\lambda }\left(u\right)-\frac{\mu }{q}{|u|}^q\right)(y,{\nabla }_y\phi )dy.\hfill \end{array}$$

(35)

Proof. 

Multiplying (34) by $(x,\nabla u)\phi$ and integration by parts, we obtain

$$\begin{array}{cc}\hfill & (N-1){\int }_{{\mathbb{R}}^{N-1}}\left(-\frac{1}{p}{|u|}^p+F_{\lambda }\left(u\right)+\frac{\mu }{q}{|u|}^q\right)\phi dy\hfill \\ \hfill =& \frac{N-p}{p}{\int }_{{\mathbb{R}}_{+}^N}{|\nabla u|}^p\phi dx-{\int }_{{\mathbb{R}}_{+}^N}{|\nabla u|}^{p-2}(\nabla u,x)(\nabla u,\nabla \phi )dx+\frac{1}{p}{\int }_{{\mathbb{R}}_{+}^N}{|\nabla u|}^p(x,\nabla \phi )dx\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^{N-1}}\left(\frac{1}{p}{|u|}^p-F_{\lambda }\left(u\right)-\frac{\mu }{q}{|u|}^q\right)(y,{\nabla }_y\phi )dy.\hfill \end{array}$$

(36)

Multiplying (34) by $u\phi$ and integration by parts, we obtain

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}^{N-1}}{(|u|}^p-f_{\lambda }\left(u\right)u-{\mu |u|}^q)\phi dy\hfill \\ \hfill =& -{\int }_{{\mathbb{R}}_{+}^N}{|\nabla u|}^p\phi dx-{\int }_{{\mathbb{R}}^{N-1}}{|\nabla u|}^{p-2}u(\nabla u,\nabla \phi )dx.\hfill \end{array}$$

(37)

Eliminating the term ${\int }_{{\mathbb{R}}_{+}^N}{|\nabla u|}^p\phi dx$, we obtain the local Pohožaev identity. □

Proof of the convergence Theorem 1.

We apply the local Pohožaev identity to the function $u_n$. Let

$$B_n^{+}=\{x|x\in {\mathbb{R}}_{+}^N,|x|<4{\sigma }_n^{-\frac{1}{p}}\},$$

$$D_n=\{y|y\in {\mathbb{R}}^{N-1},|y|<4{\sigma }_n^{-\frac{1}{p}}\}.$$

Choose $\phi \in C_0^{\infty }({\mathbb{R}}^N,[0,1])$ such that $\phi \left(x\right)=1$ for $|x|\le 3{\sigma }_n^{-\frac{1}{p}}$, $\phi \left(x\right)=0$ for $|x|\ge 4{\sigma }_n^{-\frac{1}{p}}$ and $|\nabla \phi |\le 2{\sigma }_n^{\frac{1}{p}}$. By Lemma 6, the local Pohožaev identity, we have

$$\begin{array}{cc}\hfill & -\frac{p-1}{p}{\int }_{D_n}{|u_n|}^p\phi dy+{\int }_{D_n}\left((N-1)F_{{\lambda }_n}\left(u_n\right)-\frac{N-p}{p}{f}_{{\lambda }_n}\left(u_n\right)u_n\right)\phi dy\hfill \\ \hfill & +\left(\frac{N-p}{q}-\frac{N-p}{p}\right)\mu {\int }_{D_n}{|u_n|}^q\phi dx\hfill \\ \hfill =& \frac{1}{p}{\int }_{B_n^{+}}|\nabla u_n{|}^p(x,\nabla \phi )dx-{\int }_{B_n^{+}}{|\nabla u_n|}^{p-2}(\nabla u_n,x)(\nabla u_n,\nabla \phi )dx\hfill \\ \hfill & -\frac{N-p}{p}{\int }_{B_n^{+}}{|\nabla u_n|}^{p-2}{u}_n(\nabla u_n,\nabla \phi )dx\hfill \\ \hfill & +{\int }_{D_n}\left(\frac{1}{p}|u_n{|}^p-F_{\lambda }\left(u_n\right)-\frac{\mu }{q}{|u_n|}^q\right)(y,{\nabla }_y\phi )dy.\hfill \end{array}$$

(38)

We estimate (38). Notice that the integrals of the right-hand side of (38) are taken over the domains $A_n^3$ and $T_n^3$. By Proposition 1 and Corollary 1, we know

$$\begin{array}{cc}\hfill \mathrm{RHS}\mathrm{of}\left(38\right)& \le c{\int }_{A_n^3}|\nabla u_n{|}^p|x||\nabla \phi |dx+c{\int }_{A_n^3}|\nabla u_n{|}^{p-1}|\nabla \phi |dx+c{\int }_{T_n^3}|y||{\nabla }_y\phi |dy\hfill \\ \hfill & \le c{\sigma }_n^{1-\frac{N}{p}}+c{\sigma }_n^{-\frac{N-1}{p}}\le c{\sigma }_n^{1-\frac{N}{p}}.\hfill \end{array}$$

(39)

On the other hand, by Lemma 1 $\left(7\right)$, we have

$$\begin{array}{cc}\hfill \mathrm{LHS}\mathrm{of}\left(38\right)& \ge \left(\frac{N-1}{q}-\frac{N-p}{p}\right)\mu {\int }_{D_n}|u_n{|}^q\phi dy-\frac{p-1}{p}{\int }_{D_n}{|u_n|}^p\phi dy\hfill \\ \hfill & \ge \frac{1}{2}\left(\frac{N-1}{q}-\frac{N-p}{p}\right){\int }_{D_n}{|u_n|}^q\phi dy-c{\int }_{D_n}dy\hfill \\ \hfill & \ge c{\int }_{D_n}{|u_n|}^q\phi dy-c{\sigma }_n^{-\frac{N-1}{p}}.\hfill \end{array}$$

(40)

Without loss of generality, assume ${\sigma }_{n,1}={\sigma }_n=min\{{\sigma }_{n,k}|k\in \mathrm{\Lambda }\}$. Choose L large enough such that

$${\int }_{D_L}{|U_1|}^{q}dy=\beta >0$$

where $D_L=\{y|y\in {\mathbb{R}}^{N-1},|y|<L\}.$ Since ${\tilde{u}}_n={\sigma }_n^{-\frac{N-p}{p}}{u}_n({\sigma }_n^{-1}·)$ weakly converges to $U_1$ in ${\mathcal{D}}_r$, we have

$$\begin{array}{cc}\hfill & {\int }_{D_n}|u_n{|}^q\phi dy\ge {\int }_{|y|\ge L{\sigma }_n^{-1}}{|u_n|}^{q}dy\hfill \\ \hfill =& {\sigma }_n^{\frac{N-p}{p}q-(N-1)}{\int }_{D_L}{|{\tilde{u}}_n|}^{q}dy\hfill \\ \hfill \sim & {\sigma }_n^{\frac{N-p}{p}q-(N-1)}·\beta \hfill \end{array}$$

(41)

we arrive at a contradiction

$${\sigma }_n^{\frac{N-p}{p}q-(N-1)}\le x{\sigma }_n^{-\frac{N-p}{p}}$$

for ${\sigma }_n$ large enough, since $q+1>\overline{p}=\frac{(N-1)p}{N-p}$. The index set $\mathrm{\Lambda }$ in the profile decomposition (9) must be empty, and (9) reduces to

$$u_n=u+r_n,\mathrm{and}{r}_n\to 0\mathrm{in}{L}^{\overline{p}}\left({\mathbb{R}}^{N-1}\right)\mathrm{as}n\to \infty .$$

(42)

That is, $u_n\to u$ in $L^{\overline{p}}\left({\mathbb{R}}^{N-1}\right)$. By Lemma A4, $u_n$ is uniformly bounded, and there exists $\nu =\nu \left(L\right)$ such that

$$|u_n\left(y\right)|\le \frac{1}{\nu }\mathrm{for}y\in {\mathbb{R}}^{N-1}.$$

□

3. Existence of Multiple Solutions

We define a sequence of critical values of the truncated functional $I_{\lambda },\lambda >0$ by the symmetric mountain pass lemma due to Ambrosetti and Rabinowitz, and prove that the corresponding critical points are solutions of the original problem (1) for sufficiently small parameter $\lambda$.

Definition 2.

Define the critical values of $I_{\lambda }$,

$$c_k\left(\lambda \right)=\underset{A\in {\Gamma }_k}{inf}\underset{u\in A}{sup}{I}_{\lambda }\left(u\right),k=1,2,\cdots$$

where

$${\Gamma }_k=\{A|A\subset W_r,Acompact,-A=A,\gamma (A\cap {\sigma }^{-1}\left(S_{\rho }\right))\ge k\}$$

$$G=\{\sigma |\sigma \in C(W_r,W_r),\sigma odd,\sigma \left(u\right)=uif{I}_1\left(u\right)\le 0\}$$

and

$$S_{\rho }=\{u|u\in W_r,\parallel u\parallel =\rho \},$$

ρ is chosen as a suitable positive constant such that

$$I\left(u\right)\ge \beta >0.$$

In fact, for $u\in W_r,\parallel u\parallel =\rho$, we have

$$\begin{array}{cc}\hfill I_{\lambda }\left(u\right)\ge I\left(u\right)& =\frac{1}{p}\left({\int }_{{\mathbb{R}}_{+}^N}{|\nabla u|}^{p}dx+{\int }_{{\mathbb{R}}^{N-1}}{|u|}^{p}dy\right)-\frac{1}{\overline{p}}{\int }_{{\mathbb{R}}^{N-1}}{|u|}^{\overline{p}}dy-\frac{\mu }{q}{\int }_{{\mathbb{R}}^{N-1}}{|u|}^{q}dy\hfill \\ \hfill & \ge \frac{1}{p}\left({\int }_{{\mathbb{R}}_{+}^N}{|\nabla u|}^{p}dx+{\int }_{{\mathbb{R}}^{N-1}}{|u|}^{p}dy\right)-{\int }_{{\mathbb{R}}^{N-1}}{(c|u|}^{\overline{p}}+{\epsilon |u|}^q)dy\hfill \\ \hfill & \ge c_0{\rho }^p-c_1{\rho }^{\overline{p}}\ge \frac{1}{2}{c}_0{\rho }^p:=\beta >0\hfill \end{array}$$

provided $\frac{1}{2}{c}_0{\rho }^p\ge c_1{\rho }^{\overline{p}}$.

Lemma 7.

The functional $I_{\lambda },\lambda >0$ satisfies the Palais–Smale condition.

Proof. 

Let $u_n$ be a Palais–Smale sequence of $I_{\lambda }$, and we have

$$\begin{array}{cc}\hfill & I_{\lambda }\left(u_n\right)-\frac{1}{q}⟨D{I}_{\lambda }\left(u_n\right),u_n⟩\hfill \\ \hfill =& \left(\frac{1}{p}-\frac{1}{q}\right)\left({\int }_{{\mathbb{R}}_{+}^N}|\nabla u_n{|}^{p}dx+{\int }_{{\mathbb{R}}^{N-1}}{|u_n|}^{p}dy\right)+{\int }_{{\mathbb{R}}^{N-1}}\left(\frac{1}{q}{f}_{\lambda }\left(u_n\right)u_n-F_{\lambda }\left(u_n\right)\right)dy\hfill \\ \hfill \ge & \left(\frac{1}{p}-\frac{1}{q}\right)\left({\int }_{{\mathbb{R}}_{+}^N}|\nabla u_n{|}^{p}dx+{\int }_{{\mathbb{R}}^{N-1}}{|u_n|}^{p}dy\right)=\left(\frac{1}{p}-\frac{1}{q}\right){\parallel u_n\parallel }^p\hfill \end{array}$$

hence $u_n$ is bounded in $W_r$. Since the imbedding from $W_r$ to $L^q\left({\mathbb{R}}^{N-1}\right)$ is compact, we assume $u_n\rightharpoonup u$ in $W_r$, $u_n\to u$ in $L^q\left({\mathbb{R}}^{N-1}\right)$. By Lemma 1, we have

$$\begin{array}{cc}\hfill & {\int }_{{\mathbb{R}}_{+}^N}(|\nabla u_n{|}^{p-2}\nabla u_n-|\nabla u_m{|}^{p-2}\nabla u_m,\nabla u_n-\nabla u_m)dx\hfill \\ \hfill & +{\int }_{{\mathbb{R}}^{N-1}}(|u_n{|}^{p-2}{u}_n-|u_m{|}^{p-2}{u}_m)(u_n-u_m)dy\hfill \\ \hfill =& {\int }_{{\mathbb{R}}^{N-1}}\left(f_{\lambda }\left(u_n\right)-f_{\lambda }\left(u_m\right)\right)(u_n-u_m)dy+\mu {\int }_{{\mathbb{R}}^{N-1}}(|u_n{|}^{q-2}{u}_n-|u_m{|}^{q-2}{u}_m)(u_n-u_m)dy\hfill \\ \hfill & +⟨D{I}_{\lambda }\left(u_n\right)-D{I}_{\lambda }\left(u_m\right),u_n-u_m⟩\hfill \\ \hfill \le & c\left({\left(\frac{2}{\lambda }\right)}^{\overline{p}-q}+1\right){\int }_{{\mathbb{R}}^{N-1}}(|u_n{|}^{q-1}+|u_m{|}^{q-1})|u_n-u_m|dy+o\left(1\right)\hfill \\ \hfill \le & c|u_n-u_m{|}_q+o\left(1\right)\to 0,\mathrm{as}n,m\to \infty .\hfill \end{array}$$

By the elementary inequalities (15), $u_n$ is a Cauchy sequence. □

The following proposition is well known ([23,24]).

Proposition 3.

(Ambrosetti–Rabinowitz) Assume $0<\lambda \le 1$. Then,

$\left(1\right)$

$c_k\left(\lambda \right)\ge \beta >0,k=1,2,\cdots$ are critical values of $I_{\lambda }$.

$\left(2\right)$

If $c_k\left(\lambda \right)=c_{k+1}\left(\lambda \right)=\cdots =c_{k+m-1}\left(\lambda \right)=c$, then $\gamma \left(K_c\left(\lambda \right)\right)\ge m$, where

$$K_c\left(\lambda \right)=\{u|u\in W_r,I_{\lambda }\left(u\right)=c,D{I}_{\lambda }\left(u\right)=0\}.$$

Proof of Theorem 2.

Given an integer k, let $u_j\left(\lambda \right)\in W_r,j=1,\cdots ,k$ such that $I_{\lambda }\left(u_j\left(\lambda \right)\right)=c_j\left(\lambda \right)$, $D{I}_{\lambda }\left(u_j\left(\lambda \right)\right)=0$. By Proposition 3 (2), we assume that $u_j\left(\lambda \right),j=1,\cdots ,k$ are different from each other. Since $I_{\lambda }\le I_1$ for $0<\lambda \le 1$, we have $c_1\left(\lambda \right)\le \cdots \le c_k\left(\lambda \right)\le c_k\left(1\right)$. By Theorem 1, there exists ${\nu }_k$ such that

$$|u_j\left(\lambda \right)\left(y\right)|\le \frac{1}{{\nu }_k},j=1,\cdots ,k,y\in {\mathbb{R}}^{N-1}.$$

Now, for $\lambda <{\nu }_k$, we have

$$|u_j\left(\lambda \right)\left(y\right)|\le \frac{1}{{\nu }_k}<\frac{1}{\lambda },j=1,\cdots ,k,y\in {\mathbb{R}}^{N-1}$$

hence $f_{\lambda }\left(u_j\left(\lambda \right)\right)=f\left(u_j\left(\lambda \right)\right),u_j\left(\lambda \right),j=1,\cdots ,k$ are solutions of the original problem (1). Since the integer k is arbitrary, the problem (1) has infinitely many solutions. □

For more details and background material, we refer the readers to the Appendix A, Appendix B and Appendix C of this paper.

4. Results

The main results of this paper are Theorem 1 and Theorem 2.