Multiplicity of nontrivial solutions for Kirchhoff type equations with zero mass and a critical term

Abstract

In this paper, a class of Kirchhoff type equations in {\mathbb{R}}^N(N⩾3) with zero mass and a critical term is studied. Under some appropriate conditions, the existence of multiple solutions is obtained by using variational methods and a variant of Symmetric Mountain Pass theorem. The Second Concentration Compactness lemma is used to overcome the lack of compactness in critical problem. Compared to the usual Kirchhoff-type problems, we only require the nonlinearity to satisfy the classical superquadratic condition (Ambrosetti-Rabinowitz condition).

Keywords

Kirchhoff type equations with a critical term / variational methods / Symmetric Mountain Pass theorem / Second Concentration Compactness lemma

1 Introduction and main result

In 1883, Kirchhoff [12] proposed a mathematical and physical model to describe the length variation caused by the lateral vibration of a telescopic rope in Physics. This equation is also named as Kirchhoff equation. It generalizes the classical d'Alembert wave equation. Since Lions gave its abstract research framework in 1987, a lot of research results on various Kirchhoff-type equations have been obtained. In the case of bounded domain,

(1.1)\{\begin{array}{ll}-\left(a+b{\displaystyle {\int }_{\mathrm{\Omega }}|\mathrm{\nabla }u{|}^2\mathrm{d}x}\right)\mathrm{\Delta }u=f(x,u),& x\in \mathrm{\Omega },\\ u=0,& x\in \mathrm{\partial }\mathrm{\Omega }.\end{array}

Ma and Rivera [16] first obtained the existence of positive solutions for equation (1.1) with subcritical nonlinear terms by variational methods. Perera and Zhang [19] constructed a sequence of eigenvalues tending to positive infinity by using Yang index, and obtained the existence of nontrivial solutions of equation (1.1) by calculating the critical groups. Mao and Zhang [17] obtained the multiplicity of nontrivial solutions and the existence of sign-changing solutions for equation (1.1) by using variational methods and the method of invariant sets of descending flow. Equation (1.1) with a critical term was studied in [1,6-8,18]. For the case that f(x,u )=g(x,u )+|u{|}^{2^{\ast }-2}u, the existence of positive solutions for equation (1.1) was obtained by using the Second Concentration Compactness lemma in [18], under some superlinear and subcritical growth conditions on the nonlinear term g. Fan and Wu [6] studied the existence of nontrivial solutions for equation (1.1) with some special forms of g by Nehari manifolds and the First Concentration Compactness principle. Furtado et al. [8] obtained the multiplicity of nontrivial solutions for a class of general Kirchhoff type equations in bounded domains by using Symmetric Mountain Pass theorem. For the case {\mathbb{R}}^N, the general type of the equation is that

(1.2)\begin{array}{r}-\left(a+b{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)\mathrm{\Delta }u+V(x)u=f(x,u),x\in {\mathbb{R}}^N,\end{array}

where a >0,b> 0, N⩾3. The existence and multiplicity of nontrivial solutions for equation (1.2) have been studied in [2,9-11,13,14]. For the case that V=b,f(x,u )=f(u), Li et al. [13] obtained the existence of positive solutions to equation (1.2) without Ambrosetti-Rabinowitz condition, depending on a parameter, by using the truncation technique and the monotonicity technique. He and Zou [10] studied the relation between the number of solutions and the collection of the minimum points of V. The behavior of concentration of solutions was also considered in [10]. The ground state solution of equation (1.2) without Ambrosetti-Rabinowitz condition was obtained by Guo [9]. In [14], Li et al. studied the case of potential function V=0 (if the nonlinear term f satisfies the superlinear growth at zero, equation (1.2) is called a Kirchhoff type equation with zero mass in this case, more studies on the zero mass problem can be found in [3] and references therein). By using monotone technique and variational methods, the existence of positive solution for the following Kirchhoff type equation with zero mass was obtained in [14]:

\{\begin{array}{l}-\left(a+\mu {\displaystyle {\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^2\mathrm{d}x}\right)\mathrm{\Delta }u=K(x)f(u),x\in {\mathbb{R}}^N,\\ u\in D_r^{1,2}({\mathbb{R}}^N),\end{array}

where N⩾3, a>0 is a constant, \mu ⩾0 is a parameter, K satisfies some conditions such as integrability.

In this paper, we are interested in a class of zero mass Kirchhoff type equation with a critical term in {\mathbb{R}}^N with N⩾3. Specifically, the following type of equation is studied:

(1.3)\{\begin{array}{l}-m\left({\displaystyle {\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^2\mathrm{d}x}\right)\mathrm{\Delta }u=\lambda K(x)f(u)+|u{|}^{2^{\ast }-2}u,x\in {\mathbb{R}}^N,\\ u\in D^{1,2}({\mathbb{R}}^N),\end{array}

where 2^{\ast }=\frac{2N}{N-2}, \lambda \in (0, +\mathrm{\infty }), m, f, K satisfy the following assumptions respectively.

(m₀) m\in C([0 ,+\mathrm{\infty } ),(0,+\mathrm{\infty })), there exists \sigma > 0 such that m is increasing in [0,\sigma ];

(f₁) f\in C(\mathbb{R},\mathbb{R}) and satisfies the quasi-critical growth condition:

\underset{|s|\to 0}{lim}\frac{f(s)s}{|s{|}^{2^{\ast }}}=\underset{|s|\to \mathrm{\infty }}{lim sup}\frac{f(s)s}{|s{|}^{2^{\ast }}}=0;

(f₂) there exists \theta \in (2,2^{\ast }) such that

(f₃) f is odd in s, that is f(-s)=-f(s), s\in \mathbb{R};

(K₁) K(x)> 0a.e.x\in {\mathbb{R}}^3, that is |\{x\in {\mathbb{R}}^N: K(x)⩽0\}|=0;

(K₂) there exists p\in (2,2^{\ast }) such that K\in L^{\mathrm{\infty }}({\mathbb{R}}^N)\cap L^{\frac{2^{\ast }}{2^{\ast }-p}}({\mathbb{R}}^N);

(K₃) there exist a_0,r>0 such that K(x)⩾a_0 for x\in B_r(0), where B_r(0) is an open sphere with the origin as its center and r as its radius.

Our main result is followed.

Theorem 1.1　 Suppose the assumptions {(\mathrm{m}}_0), {(\mathrm{f}}_1)-{(\mathrm{f}}_3), {(\mathrm{K}}_1)-{(\mathrm{K}}_3) hold. Then for any given k\in \mathbb{N}, there exists \lambda ( k)>0 such that problem (1.3) has at least k pairs of nontrivial solutions for all \lambda \in (\lambda (k), \mathrm{\infty }).

Remark 1.1　A similar assumption on m appeared in [8]. However, we only need m to satisfy a local monotonicity assumption.

Remark 1.2　In condition ( {\mathrm{K}}_3), the center of the ball B_r(0 ) may not be 0. Condition ({\mathrm{K}}_3) is only used to verify that the functional satisfies the symmetric mountain pass structure.

Throughout the paper, C,C_1, C_2, \dots represent positive (possibly different) constants. We denote weak convergence and strong convergence by u_n\rightharpoonup u and u_n\to u respectively. o_n(1) denotes an infinitesimal quantity as n\to \mathrm{\infty }.

The rest of the paper is organized as follows. In Section 2, we give the main variational framework of problem (1.3) and the abstract critical point theorem which will be used. The proof of our main result is given in Section 3. First, it is proved that the functional associated to problem (1.3) satisfies the geometric structure of the symmetric mountain pass. Then, it is proved that the functional satisfies the local compactness condition by the Second Concentration Compactness lemma, and the main result is obtained.

2 Preliminary

In this section, we give the main variational framework of problem (1.3) and the main abstract critical point theorem that will be used.

The main working space in this paper is D^{1,2}({\mathbb{R}}^N)=\{u \in L^{2^{\ast }}({\mathbb{R}}^N) :|\mathrm{\nabla } u|\in L^2({\mathbb{R}}^N )\} endowed with the norm

\Vert u\Vert :={\left({\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }u{|}^2\mathrm{d}x\right)}^{\frac{1}{2}},

where L^{2^{\ast }}({\mathbb{R}}^N ) is the normal 2^{\ast } power Lebesgue integrable space, its norm is

| u{|}_{2^{\ast }} ={\left({\int }_{{\mathbb{R}}^N}|u{|}^{2^{\ast }}\mathrm{d}x\right)}^{\frac{1}{2^{\ast }}}.

S is the best Sobolev constant, that is

S=\underset{u\in D^{1,2}({\mathbb{R}}^N)\setminus \{0\}}{inf}\frac{\Vert u{\Vert }^2}{|u{|}_{2^{\ast }}^2}.

Condition (m_0) implies that there exists \delta \in (0,\sigma ) such that , where \theta >2 is from condition ({\mathrm{f}}_2). Set

m_{\delta }(t)= \{\begin{array}{ll}m(t),& t⩽\delta ,\\ m(\delta ),& t>\delta .\end{array}

Then m_{\delta }\in C ([0,+\mathrm{\infty } ),(0,+\mathrm{\infty })). We consider the following auxiliary problem:

(2.1)\begin{array}{r}\{\begin{array}{ll}-m_{\delta }(\Vert u{\Vert }^2)\mathrm{\Delta }u=\lambda K(x)f(u)+|u{|}^{2^{\ast }-2}u,& x\in {\mathbb{R}}^N,\\ u\in D^{1,2}({\mathbb{R}}^N).\end{array}\end{array}

Let I_{\delta ,\lambda }:D^{1,2}( {\mathbb{R}}^N)\to \mathbb{R} be the energy functional associated to problem (2.1), which is given by

I_{\delta ,\lambda }(u) =\frac{1}{2}{M}_{\delta }(\Vert u{\Vert }^2)- \lambda {\int }_{{\mathbb{R}}^N}K(x )F(u)\mathrm{d}x-\frac{1}{2^{\ast }}{\int }_{{\mathbb{R}}^N}|u{|}^{2^{\ast }} \mathrm{d}x,

where M_{\delta }(s):= {\int }_0^s m_{\delta }(t)\mathrm{d}t. By condition ({\mathrm{f}}_1) and ( {\mathrm{K}}_2), I_{\delta ,\lambda } is well defined in D^{1,2}({\mathbb{R}}^N) and I_{\delta ,\lambda }\in C^1(D^{1,2}({\mathbb{R}}^N),\mathbb{R}). Furthermore, for every u,v\in D^{1,2}({\mathbb{R}}^N),

I_{\delta ,\lambda }^{\prime }(u)v=m_{\delta }( \Vert u {\Vert }^2){\int }_{{\mathbb{R}}^N}\mathrm{\nabla }u\mathrm{\nabla }v\mathrm{d}x-\lambda {\int }_{{\mathbb{R}}^N}K(x )f(u)v \mathrm{d}x-{\int }_{{\mathbb{R}}^N}|u{|}^{2^{\ast }-2}uv\mathrm{d}x.

By the definition of m_{\delta }, if u\in D^{1,2}({\mathbb{R}}^N) is a weak solution of problem (2.1) and \Vert u\Vert ⩽{\delta }^{\frac{1}{2}}, then m_{\delta }(\Vert u{\Vert }^2)=m (\Vert u{\Vert }^2) and therefore u is also a weak solution of the original problem (1.3). In this paper, we will show that this is true if the parameter \lambda is large enough.

In order to obtain our result, we need the following version of Symmetric Mountain Pass theorem.

Lemma 2.1 [4,20]　 Let E=V\oplus W be a real Banach space with . Suppose J\in C^1 (E,\mathbb{R}) is an even functional satisfying J(0) =0 and

(J₁) there exist \rho ,\alpha >0 such that

\underset{u\in \mathrm{\partial }{B}_{\rho }(0)\cap W}{inf}J(u )⩾\alpha ;

(J₂) there exists a subspace \hat{V}\subset E such that and

\underset{u\in \hat{V}}{max}J(u) ⩽M,forsomeM>0;

(J₃) J satisfies (PS){}_c condition for any c\in (0,M) with M as in ({\mathrm{J}}_2).

Then J possesses at least \dim \hat{V}-\dim V pairs of nontrivial critical points.

3 Proof of our main result

In this section, under the condition of Theorem 1.1, we prove our main result by obtaining the multiplicity of solutions of equation (2.1).

First, we prove that the functional I_{\delta ,\lambda } enjoys the geometric structure of symmetric mountain pass as shown in Lemma 2.1. In this paper, E=D^{1,2}( {\mathbb{R}}^N), V=\{0 \}.

Lemma 3.1　 Under the assumptions of Theorem 1.1, for each \lambda >0, I_{\delta ,\lambda } satisfies that

(i) there exist {\rho }_{\lambda },{\alpha }_{\lambda }>0 such that

\underset{u\in \mathrm{\partial }{B}_{{\rho }_{\lambda }(0)}}{inf}{I}_{\delta ,\lambda }(u)⩾ {\alpha }_{\lambda };

(ii) for any given k\in \mathbb{N} and M>0, there exists {\lambda }_{k,M}>0 with the following property: for any \lambda >{\lambda }_{k,M} we can find a k-dimensional subspace V_k^{\lambda }\subset D^{1,2}({\mathbb{R}}^3) such that

Proof　By ( {\mathrm{f}}_1), there exists C>0 such that

\begin{array}{r}F(s)⩽C|s{|}^{2^{\ast }},s\in \mathbb{R}.\end{array}

Since K\in L^{\mathrm{\infty }}({\mathbb{R}}^N), we can get that

\begin{array}{r}\begin{array}{rl}{I}_{\delta ,\lambda }(u)& ⩾\frac{1}{2}m(0)\Vert u{\Vert }^2-\lambda {\int }_{{\mathbb{R}}^N}CK(x)|u{|}^{2^{\ast }}\mathrm{d}x-\frac{1}{2^{\ast }}\Vert u{\Vert }_{2^{\ast }}^{2^{\ast }}\\ & ⩾\frac{1}{2}m(0)\Vert u{\Vert }^2-C\Vert K{\Vert }_{\mathrm{\infty }}\lambda |u{|}_{2^{\ast }}^{2^{\ast }}-\frac{1}{2^{\ast }}|u{|}_{2^{\ast }}^{2^{\ast }}\\ & ⩾\frac{1}{2}m(0)\Vert u{\Vert }^2-C{S}^{-\frac{2^{\ast }}{2}}\Vert K{\Vert }_{\mathrm{\infty }}\lambda \Vert u{\Vert }^{2^{\ast }}-\frac{1}{2^{\ast }}{S}^{-\frac{2^{\ast }}{2}}\Vert u{\Vert }^{2^{\ast }}\\ & =\Vert u{\Vert }^2\left(\frac{1}{2}m(0)-C{S}^{-\frac{2^{\ast }}{2}}\Vert K{\Vert }_{\mathrm{\infty }}\lambda \Vert u{\Vert }^{2^{\ast }-2}-\frac{1}{2^{\ast }}{S}^{-\frac{2^{\ast }}{2}}\Vert u{\Vert }^{2^{\ast }-2}\right).\end{array}\end{array}

Then I_{\delta ,\lambda }(u)> 0 for \Vert u\Vert >0 small enough which means that (i) is true.

Let \phi \in C_0^{\mathrm{\infty }}(B_1(0) ,\mathbb{R}), choose \{x_1,\dots ,x_k\}\subset B_1(0) and \tau >0 such that B_{\tau }( x_i)\subset B_r (0) with B_{\tau } (x_i)\cap B_{\tau }( x_j)=\mathrm{\varnothing }, if i,j\in \{1 ,\dots ,k\} and i\ne j. For each i\in \{1, \dots ,k\}, set {\phi }_i^{\tau }( x):=\phi (\frac{x-x_i}{\tau }), x\in B_{\tau }(x_i), and

(3.1)\begin{array}{r}{A}_{\tau }:=\frac{\Vert {\phi }_i^{\tau }{\Vert }^2}{|{\phi }_i^{\tau }{|}_{\theta }^2}={\tau }^{N-2-\frac{2N}{\theta }}\frac{\Vert \phi {\Vert }^2}{|\phi {|}_{\theta }^2}.\end{array}

Since {\mathbb{R}}^k is finite dimensional, there exists d_1=d_1 (k,\theta ) such that

(3.2)\sum _{i=1}^k|y_i{|}^{\theta }⩾d_1{\left(\sum _{i=1}^k|y_i{|}^2\right)}^{\frac{\theta }{2}},(y_1,\dots ,y_k)\in {\mathbb{R}}^k.

Set

V_{k,\tau }:=\text{span}\{{\phi }_1^{\tau },\dots ,{\phi }_k^{\tau }\}.

By (3.1) and (3.2), we can get that for any u= \sum _{i=1}^k{\alpha }_i {\phi }_i^{\tau }\in V_{k,\tau }, there holds

(3.3)\begin{array}{r}\begin{array}{rl}{\int }_{{\mathbb{R}}^3}|u{|}^{\theta }\mathrm{d}x& ={\int }_{{\cup }_{i=1}^k{B}_{\tau }(x_i)}{\left|\sum _{i=1}^k{\alpha }_i{\phi }_i^{\tau }\right|}^{\theta }\mathrm{d}x=\sum _{i=1}^k|{\alpha }_i{\phi }_i^{\tau }{|}_{\theta }^{\theta }\\ & ⩾d_1{\left(\sum _{i=1}^k|{\alpha }_i{\phi }_i^{\tau }{|}_{\theta }^2\right)}^{\frac{\theta }{2}}\\ & =d_1{\left(\sum _{i=1}^k{A}_{\tau }^{-1}\Vert {\alpha }_i{\phi }_i^{\tau }{\Vert }^2\right)}^{\frac{\theta }{2}}\\ & =d_2{\tau }^{-(N-2-\frac{2N}{\theta })\frac{\theta }{2}}\Vert u{\Vert }^{\theta },\end{array}\end{array}

where d_2=d_1|\phi {|}_{\theta }^{\theta }\Vert \phi {\Vert }^{-\theta }.

By ({\mathrm{f}}_2), there exist d_3,d_4>0 such that

F(s)⩾d_3|s{|}^{\theta }-d_4, \text{for}s\in \mathbb{R}.

By (3.3) and ({\mathrm{K}}_3), we have

\begin{array}{rl}{I}_{\delta ,\lambda }(u)& ⩽\frac{m(\delta )}{2}\Vert u{\Vert }^2-\lambda a_0\sum _{i=1}^k{\int }_{B_{\tau }(x_i)}F(u)\mathrm{d}x\\ & ⩽\frac{m(\delta )}{2}\Vert u{\Vert }^2-\lambda a_0{d}_2{d}_3{\tau }^{-(N-2-\frac{2N}{\theta })\frac{\theta }{2}}\Vert u{\Vert }^{\theta }+\lambda a_0{d}_{4}k{\omega }_N{\tau }^N,\end{array}

where {\omega }_N is the volume of the unit ball in {\mathbb{R}}^N. We set \gamma :=- (N-2-\frac{2N}{\theta })\frac{\theta }{2}, d_5=a_0{d}_2{d}_3,d_6 =a_0{d}_{4}k{\omega }_N. Then there holds

(3.4)\begin{array}{r}{I}_{\delta ,\lambda }(u)⩽\frac{m(\delta )}{2}\Vert u{\Vert }^2-\lambda d_5{\tau }^{\gamma }\Vert u{\Vert }^{\theta }+\lambda d_6{\tau }^N,u\in V_{k,\tau }.\end{array}

Since , we have . Therefore, we can choose {\gamma }_0 \in (\gamma ,N) and set \lambda ={\tau }^{-{\gamma }_0}. We consider the function:

h_{\tau } (t):=\frac{m(\delta )}{2}{t}^2-d_5{\tau }^{-{\gamma }_0+\gamma }{t}^{\theta }+d_6{\tau }^{-{\gamma }_0+N}, t>0.

By a direct calculation, it gets the maximum value at t_{\tau }=[m(\delta )(d_5\theta {)}^{-1}{\tau }^{{\gamma }_0-\gamma } {]}^{\frac{1}{\theta -2}} and t_{\tau }\to 0,\tau \to 0^{+}. Then, we have

h_{\tau } (t_{\tau })\to 0,\tau \to 0^{+}.

Therefore, for every given M>0, there exists {\tau }^{\ast }={\tau }^{\ast }( k,\theta ,N,\delta ,M )>0 such that

(3.5)\begin{array}{r}\underset{t⩾0}{max}{h}_{\tau }(t)⩽\frac{M}{2},\tau \in (0,{\tau }^{\ast }].\end{array}

Set {\lambda }_{k,M}:=({\tau }^{\ast }{)}^{-{\gamma }_0}, we get a k-dimensional subspace V_k^{\lambda }:=V_{k,{\lambda }^{-\frac{1}{{\gamma }_0}}} for every \lambda ⩾{\lambda }_{k,M}. Since {\tau }^{-{\gamma }_0}=\lambda ⩾{\lambda }_{k,M}=({\tau }^{\ast }{)}^{-{\gamma }_0} implies that \tau ⩽{\tau }^{\ast }, it follows from (3.4) and (3.5) that

Thus, (ii) is also true.□

For every c\in \mathbb{R}, a sequence \{u_n{\}}_{n=1}^{\mathrm{\infty }} \subset D^{1,2}({\mathbb{R}}^N) is a Palais-Smale sequence of I_{\delta ,\lambda } at level c ((PS){}_c sequence for short), if

I_{\delta ,\lambda }(u_n)\to c,I_{\delta ,\lambda }^{\prime } (u_n)\to 0,n\to \mathrm{\infty } ,

we say that I_{\delta ,\lambda } satisfies Palais-Smale condition at level c((PS){}_c condition for short), if every (PS){}_c sequence possesses a convergent subsequence.

Lemma 3.2　 Every (PS){}_c sequence \{u_n{\}}_{n=1}^{\mathrm{\infty }} of I_{\delta ,\lambda } is bounded in D^{1,2}({\mathbb{R}}^N).

Proof　For every (PS){}_c sequence \{u_n {\}}_{n=1}^{\mathrm{\infty }} of I_{\delta ,\lambda }, since I_{\delta ,\lambda }(u_n)\to c,I_{\delta ,\lambda }^{\prime } (u_n)\to 0, it follows from ({\mathrm{f}}_2) that for n large enough, there exists C>0 such that

\begin{array}{r}\begin{array}{rl}C+\Vert u_n\Vert & ⩾I_{\delta ,\lambda }(u_n)-\frac{1}{\theta }{I}_{\delta ,\lambda }^{\prime }(u_n)u_n\\ & ⩾\left(\frac{m(0)}{2}-\frac{m(\delta )}{\theta }\right)\Vert u_n{\Vert }^2+\left(\frac{1}{\theta }-\frac{1}{2^{\ast }}\right)|u_n{|}_{2^{\ast }}^{2^{\ast }}.\end{array}\end{array}

The fact that \theta \in (2,2^{\ast }) and implies that \{u_n{\}}_{n=1}^{\mathrm{\infty }} is bounded in D^{1,2}({\mathbb{R}}^N).□

We verify that the functional I_{\delta ,\lambda } enjoys some local compactness. By Lemma 3.2, there exists u\in D^{1,2}({\mathbb{R}}^N) such that

\begin{array}{rl}& u_n\rightharpoonup u\text{in}{D}^{1,2}({\mathbb{R}}^N),n\to \mathrm{\infty },\\ & u_n\to u\text{in}{L}_{loc}^q({\mathbb{R}}^N)\text{for}q\in [1,2^{\ast }),n\to \mathrm{\infty },\\ & u_n(x)\to u(x),n\to \mathrm{\infty },a.e.x\in {\mathbb{R}}^N,\\ & |\mathrm{\nabla }{u}_n{|}^2\rightharpoonup \mathrm{d}\eta ,|u_n{|}^{2^{\ast }}\rightharpoonup \mathrm{d}\nu ,\text{(weak convergence in the sense of measure)},\end{array}

where \mathrm{d}\eta and \mathrm{d}\nu are nonnegative and bounded measures defined on {\mathbb{R}}^N. By the Second Concentration Compactness lemma due to Lions [15] and the Concentration Compactness principle at infinity of Chabrowski [5], there exist a set of points \{z_j{\}}_{j\in J}\subset {\mathbb{R}}^N and two families of positive numbers \{{\eta }_j{\}}_{j\in J}, \{{\nu }_j{\}}_{j\in J}, where J is an at most countable index set, such that

\begin{array}{r}|\mathrm{\nabla }{u}_n{|}^2\rightharpoonup \mathrm{d}\eta ⩾|\mathrm{\nabla }u{|}^2+\sum _{j\in J}{\eta }_j{\delta }_{z_j},|u_n{|}^{2^{\ast }}\rightharpoonup \mathrm{d}\nu =|u{|}^{2^{\ast }}+\sum _{j\in J}{\nu }_j{\delta }_{z_j},\end{array}

with

where {\delta }_x is the Dirac mass centred at x\in {\mathbb{R}}^N. At infinity,

\underset{n\to \mathrm{\infty }}{lim sup}{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla } u_n{|}^2\mathrm{d}x={\int }_{{\mathbb{R}}^N}\mathrm{d}\eta +{\eta }_{\mathrm{\infty }},\underset{n\to \mathrm{\infty }}{lim sup}{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}\mathrm{d}x={\int }_{{\mathbb{R}}^N}\mathrm{d}\nu +{\nu }_{\mathrm{\infty }},

with S{\nu }_{\mathrm{\infty }}^{\frac{2}{2^{\ast }}}⩽{\eta }_{\mathrm{\infty }}, where

{\eta }_{\mathrm{\infty }}= \underset{R\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim sup}{\int }_{B_R^C(0)}|\mathrm{\nabla }{u}_n{|}^2\mathrm{d}x, {\nu }_{\mathrm{\infty }}=\underset{R\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim sup}{\int }_{B_R^C(0)}|u_n{|}^{2^{\ast }}\mathrm{d}x.◻

Lemma 3.3　 Define

c^{\ast }:=\left(\frac{1}{\theta }-\frac{1}{2^{\ast }}\right) (m(0)S {)}^{\frac{N}{2}}.

Then under the assumptions of theorem 1.1 the functional I_{\delta ,\lambda } satisfies (PS){}_c condition at any level .

Proof　Let \{u_n{\}}_{n=1}^{\mathrm{\infty }}\subset D^{1,2}({\mathbb{R}}^N ) such that

Lemma 3.2 implies that \{u_n{\}}_{n=1}^{\mathrm{\infty }} is bounded in D^{1,2}({\mathbb{R}}^N).

First, we can prove that the set J obtained above is empty. Indeed, suppose by contradiction that there exists some j_0\in J with {\nu }_{j_0}\ne 0. For any ϵ\in (0, 1), define {\varphi }_{ϵ}\in C^1 ({\mathbb{R}}^N, [0,1 ]) satisfying

\begin{array}{rl}& {\varphi }_{ϵ}(x)=1,x\in B(z_{j_0},ϵ),\\ & {\varphi }_{ϵ}(x)=0,x\in {\mathbb{R}}^N\setminus B(z_{j_0},2ϵ),\\ & |\mathrm{\nabla }{\varphi }_{ϵ}(x)|⩽\frac{2}{ϵ},x\in {\mathbb{R}}^N.\end{array}

It is easy to see that \{u_n{\phi }_{ϵ}{\}}_{n=1}^{\mathrm{\infty }} is bounded in D^{1,2}({\mathbb{R}}^N), then

\underset{n\to \mathrm{\infty }}{lim}{I}_{\delta ,\lambda }^{\prime } (u_n)(u_n{\phi }_{ϵ})=0.

That is,

(3.6)\begin{array}{rl}{o}_n(1)=& m_{\delta }(\Vert u_n{\Vert }^2){\int }_{{\mathbb{R}}^N}\mathrm{\nabla }{u}_n\mathrm{\nabla }(u_n{\varphi }_{ϵ})\mathrm{d}x-\lambda {\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\varphi }_{ϵ}\mathrm{d}x-{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}{\varphi }_{ϵ}\mathrm{d}x\\ =& m_{\delta }(\Vert u_n{\Vert }^2)\left({\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\varphi }_{ϵ}\mathrm{d}x+{\int }_{{\mathbb{R}}^N}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\varphi }_{ϵ}\mathrm{d}x\right)\\ & -\lambda {\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\varphi }_{ϵ}\mathrm{d}x-{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}{\varphi }_{ϵ}\mathrm{d}x.\end{array}

By the Hölder inequality, we have

\begin{array}{rl}\left|{\int }_{{\mathbb{R}}^N}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\varphi }_{ϵ}\mathrm{d}x\right|& =\left|{\int }_{B(z_{j_0},2ϵ)}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\varphi }_{ϵ}\mathrm{d}x\right|\\ & ⩽{\left({\int }_{B(z_{j_0},2ϵ)}|\mathrm{\nabla }{u}_n{|}^2\mathrm{d}x\right)}^{\frac{1}{2}}{\left({\int }_{B(z_{j_0},2ϵ)}|u_n\mathrm{\nabla }{\varphi }_{ϵ}{|}^2\mathrm{d}x\right)}^{\frac{1}{2}}\\ & ⩽C{\left({\int }_{B(z_{j_0},2ϵ)}|u_n\mathrm{\nabla }{\varphi }_{ϵ}{|}^2\mathrm{d}x\right)}^{\frac{1}{2}},\end{array}

where C>0 is independent of ϵ. Since for every fixed ϵ,

\underset{n\to \mathrm{\infty }}{lim}{\int }_{B(z_{j_0},2ϵ)}|u_n\mathrm{\nabla }{\varphi }_{ϵ}{|}^2\mathrm{d}x={\int }_{B(z_{j_0},2ϵ)}|u\mathrm{\nabla }{\varphi }_{ϵ}{|}^2\mathrm{d}x

and

\begin{array}{r}\begin{array}{rl}{\left({\int }_{B(z_{j_0},2ϵ)}|u\mathrm{\nabla }{\varphi }_{ϵ}{|}^2\mathrm{d}x\right)}^{\frac{1}{2}}& ⩽{\left({\int }_{B(z_{j_0},2ϵ)}|u{|}^{2^{\ast }}\mathrm{d}x\right)}^{\frac{1}{2^{\ast }}}{\left({\int }_{B(z_{j_0},2ϵ)}|\mathrm{\nabla }{\varphi }_{ϵ}{|}^N\mathrm{d}x\right)}^{\frac{1}{N}}\\ & ⩽C{\left({\int }_{B(z_{j_0},2ϵ)}|u{|}^{2^{\ast }}\mathrm{d}x\right)}^{\frac{1}{2^{\ast }}},\end{array}\end{array}

together with the fact that \{u_n{\}}_{n=1}^{\mathrm{\infty }} is bounded in D^{1,2}({\mathbb{R}}^N), we have

(3.7)\begin{array}{r}\underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim sup}{m}_{\delta }(\Vert u_n{\Vert }^2)\left|{\int }_{{\mathbb{R}}^N}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\varphi }_{ϵ}\mathrm{d}x\right|=0.\end{array}

By the definition of {\varphi }_{ϵ}, we have

\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim inf}{m}_{\delta }(\Vert u_n{\Vert }^2){\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\varphi }_{ϵ}\mathrm{d}x& ⩾\underset{n\to \mathrm{\infty }}{lim}m(0){\int }_{B(z_{j_0},2ϵ)}|\mathrm{\nabla }{u}_n{|}^2{\varphi }_{ϵ}\mathrm{d}x\\ & =m(0)\left({\int }_{B(z_{j_0},2ϵ)}|\mathrm{\nabla }u{|}^2{\varphi }_{ϵ}\mathrm{d}x+⟨\sum _{j\in J}{\eta }_j{\delta }_{x_j},{\varphi }_{ϵ}⟩\right)\\ & ⩾m(0)\left({\int }_{B(z_{j_0},2ϵ)}|\mathrm{\nabla }u{|}^2{\varphi }_{ϵ}\mathrm{d}x+{\eta }_{j_0}\right),\end{array}

and {\int }_{B(z_{j_0},2ϵ)}|\mathrm{\nabla }u{|}^2{\varphi }_{ϵ}\mathrm{d}x \to 0, as ϵ\to 0. Thus,

(3.8)\begin{array}{r}\underset{ϵ\to 0}{lim inf}\underset{n\to \mathrm{\infty }}{lim inf}{m}_{\delta }(\Vert u_n{\Vert }^2){\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\varphi }_{ϵ}\mathrm{d}x⩾m(0){\eta }_{j_0}.\end{array}

By ({\mathrm{f}}_1), for every \epsilon > 0, there exists C_{\epsilon }>0 such that

\begin{array}{r}|f(s)s|⩽\epsilon s^6+C_{\epsilon }|s{|}^p,s\in \mathbb{R},\end{array}

where p\in [2 ,6) is from ({\mathrm{K}}_2). Then

\begin{array}{r}\begin{array}{rl}\left|{\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\varphi }_{ϵ}\mathrm{d}x\right|& ⩽{\int }_{B(z_{j_0},2ϵ)}|K(x)f(u_n)u_n{\varphi }_{ϵ}|\mathrm{d}x\\ & ⩽\Vert K{\Vert }_{\mathrm{\infty }}{\int }_{B(z_{j_0},2ϵ)}(\epsilon |u_n{|}^{2^{\ast }}+C_{\epsilon }|u_n{|}^p)\mathrm{d}x\\ & ⩽C\Vert K{\Vert }_{\mathrm{\infty }}\epsilon +C_{\epsilon }\Vert K{\Vert }_{\mathrm{\infty }}{\int }_{B(z_{j_0},2ϵ)}|u_n{|}^p\mathrm{d}x,\end{array}\end{array}

where C is a positive constant which is independent of n and \epsilon. Since

\underset{n\to \mathrm{\infty }}{lim}{\int }_{B(z_{j_0},2ϵ)}|u_n{|}^p\mathrm{d}x={\int }_{B(z_{j_0},2ϵ)} | u{|}^p\mathrm{d}x,

we have

\begin{array}{r}\begin{array}{r}\underset{n\to \mathrm{\infty }}{lim sup}\left|{\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\varphi }_{ϵ}\mathrm{d}x\right|⩽C\Vert K{\Vert }_{\mathrm{\infty }}\epsilon +C_{\epsilon }\Vert K{\Vert }_{\mathrm{\infty }}{\int }_{B(z_{j_0},2ϵ)}|u{|}^p\mathrm{d}x.\end{array}\end{array}

Thus,

(3.9)\begin{array}{r}\underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim sup}\left|{\int }_{{\mathbb{R}}^3}K(x)f(u_n)u_n{\varphi }_{ϵ}\mathrm{d}x\right|=0.\end{array}

Finally, for the last term in (3.6), we have

(3.10)\begin{array}{r}\underset{ϵ\to 0}{lim}\underset{n\to \mathrm{\infty }}{lim inf}{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}{\varphi }_{ϵ}\mathrm{d}x=\underset{ϵ\to 0}{lim}\left({\int }_{B(z_{j_0},2ϵ)}|u{|}^{2^{\ast }}{\varphi }_{ϵ}\mathrm{d}x+⟨\sum _{j\in J}{\nu }_j{\delta }_{x_j},{\varphi }_{ϵ}⟩\right)={\nu }_{j_0}.\end{array}

By (3.6), we have

\begin{array}{r}\begin{array}{rl}{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}{\varphi }_{ϵ}\mathrm{d}x=& m_{\delta }(\Vert u_n{\Vert }^2)\left({\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\varphi }_{ϵ}\mathrm{d}x+{\int }_{{\mathbb{R}}^N}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\varphi }_{ϵ}\mathrm{d}x\right)\\ & -\lambda {\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\varphi }_{ϵ}\mathrm{d}x+o_n(1)\\ ⩾& m(0){\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\varphi }_{ϵ}\mathrm{d}x-m_{\delta }(\Vert u_n{\Vert }^2)\left|{\int }_{{\mathbb{R}}^N}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\varphi }_{ϵ}\mathrm{d}x\right|\\ & -\lambda \left|{\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\varphi }_{ϵ}\mathrm{d}x\right|+o_n(1).\end{array}\end{array}

That is,

\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim inf}{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}{\varphi }_{ϵ}\mathrm{d}x⩾& \underset{n\to \mathrm{\infty }}{lim inf}m(0){\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\varphi }_{ϵ}\mathrm{d}x\\ & +\underset{n\to \mathrm{\infty }}{lim inf}\left(-\left|m_{\delta }(\Vert u_n{\Vert }^2){\int }_{{\mathbb{R}}^N}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\varphi }_{ϵ}\mathrm{d}x\right|\right)\\ & +\underset{n\to \mathrm{\infty }}{lim inf}\left(-\lambda \left|{\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\varphi }_{ϵ}\mathrm{d}x\right|\right)\\ =& \underset{n\to \mathrm{\infty }}{lim inf}m(0){\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\varphi }_{ϵ}\mathrm{d}x\\ & -\underset{n\to \mathrm{\infty }}{lim sup}\left|m_{\delta }(\Vert u_n{\Vert }^2){\int }_{{\mathbb{R}}^N}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\varphi }_{ϵ}\mathrm{d}x\right|\\ & -\underset{n\to \mathrm{\infty }}{lim sup}\lambda \left|{\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\varphi }_{ϵ}\mathrm{d}x\right|.\end{array}

By taking limits as ϵ\to 0 on both sides of the last inequality, it follows from (3.7)−(3.10) that

{\nu }_{j_0}⩾m( 0){\eta }_{j_0}.

Because S{\nu }_j^{\frac{2}{2^{\ast }}}⩽{\eta }_j, j\in J, then

(3.11)\begin{array}{r}{\nu }_{j_0}⩾(m(0)S{)}^{\frac{N}{2}}.\end{array}

By (3.11), it follows from I_{\delta ,\lambda }(u_n)\to c and I_{\delta ,\lambda }^{\prime }( u_n)\to 0 that

\begin{array}{r}\begin{array}{rl}c& =\underset{n\to \mathrm{\infty }}{lim}\left(I_{\delta ,\lambda }(u_n)-\frac{1}{\theta }⟨I_{\delta ,\lambda }^{\prime }(u_n),u_n⟩\right)\\ & ⩾\underset{n\to \mathrm{\infty }}{lim inf}\left[\left(\frac{m(0)}{2}-\frac{m(\delta )}{\theta }\right){\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\varphi }_{ϵ}\mathrm{d}x+\left(\frac{1}{\theta }-\frac{1}{2^{\ast }}\right){\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}{\varphi }_{ϵ}\mathrm{d}x\right]\\ & ⩾\underset{n\to \mathrm{\infty }}{lim inf}\left(\frac{1}{\theta }-\frac{1}{2^{\ast }}\right){\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}{\varphi }_{ϵ}\mathrm{d}x.\end{array}\end{array}

Thus, by taking limits as ϵ\to 0 on both sides of the last inequality, we can get

c⩾\left(\frac{1}{\theta }-\frac{1}{2^{\ast }}\right) (m(0)S {)}^{\frac{N}{2}}.

This is a contradiction with that . Then J is empty.

Next, in order to get

\underset{n\to \mathrm{\infty }}{lim}{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}\mathrm{d}x={\int }_{{\mathbb{R}}^N}|u{|}^{2^{\ast }} \mathrm{d}x,

it suffices to show that {\nu }_{\mathrm{\infty }}=0. On the contrary, we assume that {\nu }_{\mathrm{\infty }}\ne 0. For every R>0, let {\psi }_R\in C^{\mathrm{\infty }}({\mathbb{R}}^N, [0,1 ]) be a cut-off function such that

\begin{array}{rl}& {\psi }_R(x)=0,x\in B_R(0),\\ & {\psi }_R(x)=1,|x|\in B_{2R}^C(0),\\ & |\mathrm{\nabla }{\psi }_R(x)|⩽\frac{2}{R},x\in {\mathbb{R}}^N.\end{array}

It is also easy to see that \{u_n{\psi }_R{\}}_{n=1}^{\mathrm{\infty }} is also bounded in D^{1,2}({\mathbb{R}}^N). That is,

\underset{n\to \mathrm{\infty }}{lim}{I}_{\delta ,\lambda }^{\prime } (u_n)(u_n{\psi }_R)=0.

We have

(3.12)\begin{array}{r}\begin{array}{rl}{o}_n(1)=& m_{\delta }(\Vert u_n{\Vert }^2){\int }_{{\mathbb{R}}^N}\mathrm{\nabla }{u}_n\mathrm{\nabla }(u_n{\psi }_R)\mathrm{d}x-\lambda {\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\psi }_R\mathrm{d}x\\ & -{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}{\psi }_R\mathrm{d}x\\ =& m_{\delta }(\Vert u_n{\Vert }^2)\left({\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\psi }_R\mathrm{d}x+{\int }_{{\mathbb{R}}^N}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\psi }_R\mathrm{d}x\right)\\ & -\lambda {\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\psi }_R\mathrm{d}x-{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}{\psi }_R\mathrm{d}x.\end{array}\end{array}

By the Hölder inequality, one has

\begin{array}{r}\begin{array}{rl}\left|{\int }_{{\mathbb{R}}^N}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\psi }_R\mathrm{d}x\right|& =\left|{\int }_{\{R⩽|x|⩽2R\}}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\psi }_R\mathrm{d}x\right|\\ & ⩽{\left({\int }_{\{R⩽|x|⩽2R\}}|\mathrm{\nabla }{u}_n{|}^2\mathrm{d}x\right)}^{\frac{1}{2}}{\left({\int }_{\{R⩽|x|⩽2R\}}|u_n\mathrm{\nabla }{\psi }_R{|}^2\mathrm{d}x\right)}^{\frac{1}{2}}\\ & ⩽C{\left({\int }_{\{R⩽|x|⩽2R\}}|u_n\mathrm{\nabla }{\psi }_R{|}^2\mathrm{d}x\right)}^{\frac{1}{2}},\end{array}\end{array}

where C>0 is independent of R and n. Since

\underset{n\to \mathrm{\infty }}{lim}{\int }_{\{R⩽|x|⩽2R\}}|u_n\mathrm{\nabla }{\psi }_R{|}^2\mathrm{d}x={\int }_{\{R⩽|x|⩽2R\}}|u\mathrm{\nabla }{\psi }_R{|}^2\mathrm{d}x

and

\begin{array}{r}\begin{array}{rl}{\left({\int }_{\{R⩽|x|⩽2R\}}|u\mathrm{\nabla }{\psi }_R{|}^2\mathrm{d}x\right)}^{\frac{1}{2}}& ⩽{\left({\int }_{\{R⩽|x|⩽2R\}}|u{|}^{2^{\ast }}\mathrm{d}x\right)}^{\frac{1}{2^{\ast }}}{\left({\int }_{\{R⩽|x|⩽2R\}}|\mathrm{\nabla }{\psi }_R{|}^N\mathrm{d}x\right)}^{\frac{1}{N}}\\ & ⩽C{\left({\int }_{\{R⩽|x|⩽2R\}}|u{|}^{2^{\ast }}\mathrm{d}x\right)}^{\frac{1}{2^{\ast }}},\end{array}\end{array}

together with the fact that \{u_n{\}}_{n=1}^{\mathrm{\infty }} is bounded in D^{1,2}({\mathbb{R}}^N), we can get

(3.13)\begin{array}{r}\underset{R\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim sup}{m}_{\delta }(\Vert u_n{\Vert }^2)\left|{\int }_{{\mathbb{R}}^N}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\psi }_R\mathrm{d}x\right|=0.\end{array}

By the definition of {\psi }_R, we have

{\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\psi }_R\mathrm{d}x⩾{\int }_{B_{2R}^C(0)}|\mathrm{\nabla }{u}_n{|}^2\mathrm{d}x.

Then

\underset{n\to \mathrm{\infty }}{lim sup}{m}_{\delta }(\Vert u_n{\Vert }^2){\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\psi }_R\mathrm{d}x⩾\underset{n\to \mathrm{\infty }}{lim sup}m(0){\int }_{B_{2R}^C(0)}|\mathrm{\nabla }{u}_n{|}^2\mathrm{d}x.

Thus,

(3.14)\begin{array}{r}\underset{R\to \mathrm{\infty }}{lim sup}\underset{n\to \mathrm{\infty }}{lim sup}{m}_{\delta }(\Vert u_n{\Vert }^2){\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\psi }_R\mathrm{d}x⩾m(0){\eta }_{\mathrm{\infty }}.\end{array}

It follows from ({\mathrm{f}}_1) that

\begin{array}{r}\begin{array}{rl}\left|{\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\psi }_R\mathrm{d}x\right|& ⩽{\int }_{B_R^C(0)}|K(x)f(u_n)u_n{\psi }_R|\mathrm{d}x\\ & ⩽\Vert K{\Vert }_{\mathrm{\infty }}\epsilon {\int }_{B_R^C(0)}|u_n{|}^{2^{\ast }}\mathrm{d}x+C_{\epsilon }{\int }_{B_R^C(0)}K(x)|u_n{|}^p\mathrm{d}x.\end{array}\end{array}

Since

\underset{n\to \mathrm{\infty }}{lim}{\int }_{B_R^C(0)} K(x)|u_n{|}^p\mathrm{d}x={\int }_{B_R^C(0)}K(x)|u{|}^p\mathrm{d}x,

we have

\begin{array}{r}\begin{array}{r}\underset{n\to \mathrm{\infty }}{lim sup}\left|{\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\psi }_R\mathrm{d}x\right|⩽C\Vert K{\Vert }_{\mathrm{\infty }}\epsilon +C_{\epsilon }{\int }_{B_R^C(0)}K(x)|u{|}^p\mathrm{d}x.\end{array}\end{array}

By \underset{R\to \mathrm{\infty }}{lim}{\int }_{B_R^C(0)} K(x)|u{|}^p\mathrm{d}x=0, for the above \epsilon >0, there exists R_{\epsilon }>0 such that for every R>R_{\epsilon },

Thus,

(3.15)\begin{array}{r}\underset{R\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim sup}\left|{\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\psi }_R\mathrm{d}x\right|=0.\end{array}

For the last term in (3.12), we have

(3.16)\begin{array}{r}\underset{R\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim sup}{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}{\psi }_R\mathrm{d}x=\underset{R\to \mathrm{\infty }}{lim}\underset{n\to \mathrm{\infty }}{lim sup}{\int }_{B_R^C(0)}|u_n{|}^{2^{\ast }}\mathrm{d}x={\nu }_{\mathrm{\infty }}.\end{array}

By (3.12), we have

\begin{array}{r}\begin{array}{rl}{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}{\psi }_R\mathrm{d}x=& m_{\delta }(\Vert u_n{\Vert }^2)\left({\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\psi }_R\mathrm{d}x+{\int }_{{\mathbb{R}}^N}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\psi }_R\mathrm{d}x\right)\\ & -\lambda {\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\psi }_R\mathrm{d}x+o_n(1)\\ ⩾& m(0){\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\psi }_R\mathrm{d}x-\left|m_{\delta }(\Vert u_n{\Vert }^2){\int }_{{\mathbb{R}}^N}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\psi }_R\mathrm{d}x\right|\\ & -\lambda \left|{\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\psi }_R\mathrm{d}x\right|+o_n(1),\end{array}\end{array}

that is

\begin{array}{r}\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim sup}{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}{\psi }_R\mathrm{d}x⩾& \underset{n\to \mathrm{\infty }}{lim sup}m(0){\int }_{{\mathbb{R}}^N}|\mathrm{\nabla }{u}_n{|}^2{\psi }_R\mathrm{d}x\\ & -\underset{n\to \mathrm{\infty }}{lim sup}\left|m_{\delta }(\Vert u_n{\Vert }^2){\int }_{{\mathbb{R}}^N}{u}_n\mathrm{\nabla }{u}_n\mathrm{\nabla }{\psi }_R\mathrm{d}x\right|\\ & -\underset{n\to \mathrm{\infty }}{lim sup}\lambda \left|{\int }_{{\mathbb{R}}^N}K(x)f(u_n)u_n{\psi }_R\mathrm{d}x\right|.\end{array}\end{array}

By taking limits as R\to +\mathrm{\infty } on both sides of the last inequality, it follows from (3.13)−(3.16) that

{\nu }_{\mathrm{\infty }}⩾m(0){\eta }_{\mathrm{\infty }}.

Since S{\nu }_{\mathrm{\infty }}^{\frac{2}{2^{\ast }}}⩽{\eta }_{\mathrm{\infty }}, we have

{\nu }_{\mathrm{\infty }}⩾(m(0)S{)}^{\frac{N}{2}}.

It follows from I_{\delta ,\lambda }(u_n)\to c and I_{\delta ,\lambda }^{\prime }( u_n)u_n\to 0, that

\begin{array}{r}\begin{array}{rl}c=& \underset{n\to \mathrm{\infty }}{lim}[\frac{M_{\delta }(\Vert u_n{\Vert }^2)}{2}-\frac{1}{\theta }{m}_{\delta }(\Vert u_n{\Vert }^2)\Vert u_n{\Vert }^2+\left(\frac{1}{\theta }-\frac{1}{2^{\ast }}\right){\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}\mathrm{d}x\\ & +\lambda {\int }_{{\mathbb{R}}^N}K(x)\left(\frac{1}{\theta }f(u_n)u_n-F(u_n)\right)\mathrm{d}x]\\ ⩾& \underset{n\to \mathrm{\infty }}{lim sup}\left[\left(\frac{m(0)}{2}-\frac{m(\delta )}{\theta }\right)\Vert u_n{\Vert }^2+\left(\frac{1}{\theta }-\frac{1}{2^{\ast }}\right){\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}\mathrm{d}x\right]\\ ⩾& \left(\frac{1}{\theta }-\frac{1}{2^{\ast }}\right)\underset{n\to \mathrm{\infty }}{lim sup}{\int }_{B_{2R}^C(0)}|u_n{|}^{2^{\ast }}\mathrm{d}x.\end{array}\end{array}

By taking limits as R\to \mathrm{\infty } on both sides of the last inequality, we have

\begin{array}{r}c⩾\left(\frac{1}{\theta }-\frac{1}{2^{\ast }}\right)(m(0)S{)}^{\frac{N}{2}}.\end{array}

Thus, we get a contradiction with the fact that . Therefore, {\nu }_{\mathrm{\infty }}=0. That is, for every c\in (0 ,c^{\ast }),

\underset{n\to \mathrm{\infty }}{lim}{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}\mathrm{d}x={\int }_{{\mathbb{R}}^N}|u{|}^{2^{\ast }} \mathrm{d}x.

Furthermore, due to the fact that L^{2^{\ast }} ({\mathbb{R}}^N) is a uniformly convex Banach space, we can get that u_n\to u in L^{2^{\ast }}({\mathbb{R}}^N ), as n\to \mathrm{\infty }.

Finally, we will get that u_n\to u in D^{1,2}({\mathbb{R}}^N). Since \{u_n-u{\}}_{n=1}^{\mathrm{\infty }} is bounded in D^{1,2}({\mathbb{R}}^N), we have

\begin{array}{r}\begin{array}{rl}\underset{n\to \mathrm{\infty }}{lim}{I}_{\delta ,\lambda }^{\prime }(u_n)(u_n-u)=& \underset{n\to \mathrm{\infty }}{lim}[m_{\delta }(\Vert u_n{\Vert }^2){\int }_{{\mathbb{R}}^N}\mathrm{\nabla }{u}_n\mathrm{\nabla }(u_n-u)\mathrm{d}x\\ & -\lambda {\int }_{{\mathbb{R}}^N}K(x)f(u_n)(u_n-u)\mathrm{d}x-{\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }-2}{u}_n(u_n-u)\mathrm{d}x]\\ =& 0.\end{array}\end{array}

By the Hölder inequality, we have

\begin{array}{r}\begin{array}{rl}\left|{\int }_{{\mathbb{R}}^N}{u}_n^{2^{\ast }-2}{u}_n(u_n-u)\mathrm{d}x\right|& ⩽{\left({\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}\mathrm{d}x\right)}^{\frac{2^{\ast }-1}{2^{\ast }}}{\left({\int }_{{\mathbb{R}}^N}|u_n-u{|}^{2^{\ast }}\mathrm{d}x\right)}^{\frac{1}{2^{\ast }}}\\ & ⩽C{\left({\int }_{{\mathbb{R}}^N}|u_n-u{|}^{2^{\ast }}\mathrm{d}x\right)}^{\frac{1}{2^{\ast }}}.\end{array}\end{array}

It follows from {(\mathrm{f}}_1) that there exists C>0 such that |f(s) | ⩽C|s{|}^{2^{\ast }-1}. By the Hölder inequality again, we have

\begin{array}{r}\begin{array}{rl}\left|{\int }_{{\mathbb{R}}^N}K(x)f(u_n)(u_n-u)\mathrm{d}x\right|& ⩽C\Vert K{\Vert }_{\mathrm{\infty }}{\left({\int }_{{\mathbb{R}}^N}|u_n{|}^{2^{\ast }}\mathrm{d}x\right)}^{\frac{2^{\ast }-1}{2^{\ast }}}{\left({\int }_{{\mathbb{R}}^N}|u_n-u{|}^{2^{\ast }}\mathrm{d}x\right)}^{\frac{1}{2^{\ast }}}\\ & ⩽C\Vert K{\Vert }_{\mathrm{\infty }}{\left({\int }_{{\mathbb{R}}^N}|u_n-u{|}^{2^{\ast }}\mathrm{d}x\right)}^{\frac{1}{2^{\ast }}}.\end{array}\end{array}

Thus, we can get that

\underset{n\to \mathrm{\infty }}{lim}{m}_{\delta }(\Vert u_n{\Vert }^2){\int }_{{\mathbb{R}}^N}\mathrm{\nabla }{u}_n\mathrm{\nabla }( u_n-u)\mathrm{d}x=0.

Since , we have

\underset{n\to \mathrm{\infty }}{lim}{\int }_{{\mathbb{R}}^N}\mathrm{\nabla } u_n\mathrm{\nabla }(u_n- u)\mathrm{d}x=0.

The definition of weak convergence in D^{1,2}({\mathbb{R}}^N) implies that

\underset{n\to \mathrm{\infty }}{lim}{\int }_{{\mathbb{R}}^N}\mathrm{\nabla } u\mathrm{\nabla }(u_n- u)\mathrm{d}x=0.

Therefore,

\underset{n\to \mathrm{\infty }}{lim}{\int }_{{\mathbb{R}}^N}(\mathrm{\nabla }{u}_n- \mathrm{\nabla }u)\mathrm{\nabla }(u_n- u)\mathrm{d}x=0.

That is, u_n\to u\text{in} D^{1,2}({\mathbb{R}}^N),n \to \mathrm{\infty }.□

Proof of Theorem 1.1　From the definition of I_{\delta ,\lambda } and ({\mathrm{f}}_3), it is easy to see that I_{\delta ,\lambda } is even. Condition ({\mathrm{f}}_1) implies that f(0) =0, thus I_{\delta ,\lambda }(0)=0. Lemmas 3.2 and 3.3 indicate that I_{\delta ,\lambda } satisfies (PS){}_c condition for c\in (0,c^{\ast }). Together with Lemma 3.1, we can get that all of the conditions in Lemma 2.1 are satisfied for M=min\{c^{\ast }, (\frac{m(0)}{2}-\frac{m(\delta )}{\theta })\delta \}. Thus, for every k\in \mathbb{N}, problem (2.1) has at least k pairs of nontrivial solutions for each \lambda >{\lambda }_{k,M}. Since M is fixed, we denote \lambda ( k):={\lambda }_{k,M}. Since I_{\delta ,\lambda }(u)⩽ M⩽(\frac{m(0)}{2}-\frac{m(\delta )}{\theta })\delta, for every solution u of problem (2.1), it follows from {(\mathrm{f}}_2) that

\left(\frac{m(0)}{2}-\frac{m(\delta )}{\theta }\right)\delta ⩾M⩾I_{\delta ,\lambda }(u)-\frac{1}{\theta }{I}_{\delta ,\lambda }^{\prime }( u)u⩾ \left(\frac{m(0)}{2}-\frac{m(\delta )}{\theta }\right)\Vert u{\Vert }^2.

Thus, \Vert u{\Vert }^2⩽\delta. The definition of m_{\delta } implies that m_{\delta }( \Vert u {\Vert }^2)=m(\Vert u{\Vert }^2). That is, u is also a solution of problem (1.3). The proof is completed.□

References

[1] Alves C O, Corrêa F J, Figueiredo G M. On a class of nonlocal elliptic problems with critical growth. Differ Equ Appl 2010; 2(3): 409–417

[2] Alves C O, Giovany M F. Nonlinear perturbations of a periodic Kirchhoff equation in RN. Nonlinear Anal 2012; 75(5): 2750–2759

[3] Alves C O, Souto M A S. Existence of solution for two classes of elliptic problems in RN with zero mass. J Differential Equations 2012; 252(10): 5735–5750

[4] Ambrosetti A, Rabinowitz P. Dual variational methods in critical point theory and applications. J Funct Anal 1973; 14(4): 349–381

[5] Chabrowski J. Concentration-compactness principle at infinity and semilinear elliptic equations involving critical and subcritical Sobolev exponents. Calc Var Partial Differential Equations 1995; 3(4): 493–512

[6] Fan Z, Wu Q. One Kirchhoff equation involving subcritical or critical Sobolev exponents and weigh function. Acta Math Appl Sin 2019; 42(2): 278–288

[7] FaraciFFarkas C. On a critical Kirchhoff-type problem. Nonlinear Anal, 2020, 192: 111679 (14pp)

[8] FurtadoM Fde Oliveira L Dda SilvaJ P P. Multiple solutions for a Kirchhoff equation with critical growth. Z Angew Math Phys. 2019, 70(1): 11 (15pp)

[9] Guo Z. Ground states for Kirchhoff equations without compact condition. J Differential Equations 2015; 259(7): 2884–2902

[10] He X, Zou W. Existence and concentration behavior of positive solutions for a Kirchhoff type equation in R3. J Differential Equations 2012; 252(2): 1813–1834

[11] Jn J H, Wu X. Infinitely many radial solutions for Kirchhoff-type problems in RN. J Math Anal Appl 2010; 369(2): 564–574

[12] KirchhoffG, Mechanik, Leipzig: Teubner, 1883

[13] Li Y, Li F, Shi J. Existence of a positive solution to Kirchhoff type problems without compactness conditions. J Differential Equations 2012; 253(7): 2285–2294

[14] Li Y, Li F, Shi J. Existence of positive solutions to Kirchhoff type problems with zero mass. J Math Anal Appl 2014; 410(1): 361–374

[15] Lions P L. The concentration-compactness principle in the calculus of variations. The limit case, I. Rev Mat Iberoam 1985; 1: 145–201

[16] Ma T F, Rivera J E. Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl Math Lett 2003; 16(2): 243–248

[17] Mao A, Zhang Z. Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition. Nonlinear Anal 2009; 70(3): 1275–1287

[18] Naimen D. Positive solutions of Kirchhoff type elliptic equations involving a critical Sobolev exponent. Nonlinear Differ Equ Appl 2014; 21(6): 885–914

[19] Perera K, Zhang Z. Nontrivial solutions of Kirchhoff-type problems via the Yang index. J Differential Equations 2006; 221(1): 246–255

[20] Silva E A B, Xavier M S. Multiplicity of solutions for quasilinear elliptic problems involving critical Sobolev exponents. Ann Inst H Poincaré Anal Non Linéaire 2003; 20(2): 341–358

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