Multiple positive solutions for semilinear elliptic systems involving subcritical nonlinearities in ${\mathbb{R}}^N$

For $N\ge 3$, $\alpha >1$, $\beta >1$ and $2<q<p=\alpha +\beta <2^{\ast }=2N/(N-2)$, we consider the semilinear elliptic systems

where $\epsilon ,\lambda ,\mu >0$.

Let f, g and h satisfy the following conditions:

(A1) f is a positive continuous function in ${\mathbb{R}}^N$ and ${lim}_{|x|\to \mathrm{\infty }}f(x)=f_{\mathrm{\infty }}>0$.

and $f_{\mathrm{\infty }}<1$.

(A3) $g,h\in L^m({\mathbb{R}}^N)\cap L^{\mathrm{\infty }}({\mathbb{R}}^N)$ where $m=(\alpha +\beta )/(\alpha +\beta -q)$, and $g,h\gneqq 0$.

System (${\overline{E}}_{\epsilon ,\lambda ,\mu }$) is transformed to

Actually, the weak solution $(u,v)\in H$ of ($E_{\epsilon ,\lambda ,\mu }$) is the critical point of the functional $J_{\epsilon ,\lambda ,\mu }$, that is, $(u,v)\in H$ satisfies

for any $({\phi }_1,{\phi }_2)\in H$.

The Nehari manifold ${\mathbf{M}}_{\epsilon ,\lambda ,\mu }$ contains all nontrivial weak solutions of ($E_{\epsilon ,\lambda ,\mu }$).

For the semilinear elliptic systems ($\lambda =\mu =0$)

where ${\mathbf{N}}_{max}=\{(u,v)\in H\mathrm{\setminus }\{(0,0)\}|〈I_{max}^{\mathrm{\prime }}(u,v),(u,v)〉=0\}$.

It is well known that this problem

This paper is organized as follows. First of all, we study the argument of the Nehari manifold ${\mathbf{M}}_{\epsilon ,\lambda ,\mu }$. Next, we prove that the existence of a positive solution $(u_0,v_0)\in {\mathbf{M}}_{\epsilon ,\lambda ,\mu }$ of ($E_{\epsilon ,\lambda ,\mu }$). Finally, in Section 4, we show that the condition (A2) affects the number of positive solutions of ($E_{\epsilon ,\lambda ,\mu }$); that is, there are at least k critical points $(u_i,v_i)\in {\mathbf{M}}_{\epsilon ,\lambda ,\mu }$ of $J_{\epsilon ,\lambda ,\mu }$ such that $J_{\epsilon ,\lambda ,\mu }(u_i,v_i)={\beta }_{\epsilon ,\lambda ,\mu }^i$ ((PS)-value) for $1\le i\le k$.

Theorem 1.1 ($E_{\epsilon ,\lambda ,\mu }$) has at least one positive solution $(u_0,v_0)$, that is, (${\overline{E}}_{\epsilon ,\lambda ,\mu }$) admits at least one positive solution.

Theorem 1.2 There exist two positive numbers ${\epsilon }_0$ and ${\mathrm{\Lambda }}^{\ast }$ such that ($E_{\epsilon ,\lambda ,\mu }$) has at least k positive solutions for any $0<\epsilon <{\epsilon }_0$ and $0<\lambda +\mu <{\mathrm{\Lambda }}^{\ast }$, that is, (${\overline{E}}_{\epsilon ,\lambda ,\mu }$) admits at least k positive solutions.

By studying the argument of Han [[7], Lemma 2.1], we obtain the following lemma.

Note that $J_{\epsilon ,\lambda ,\mu }$ is not bounded from below in H. From the following lemma, we have that $J_{\epsilon ,\lambda ,\mu }$ is bounded from below on ${\mathbf{M}}_{\epsilon ,\lambda ,\mu }$.

Lemma 2.2 The energy functional $J_{\epsilon ,\lambda ,\mu }$ is bounded from below on ${\mathbf{M}}_{\epsilon ,\lambda ,\mu }$.

where $p=\alpha +\beta$. Hence, we have that $J_{\epsilon ,\lambda ,\mu }$ is bounded from below on ${\mathbf{M}}_{\epsilon ,\lambda ,\mu }$. □

then ${lim}_{t\to \mathrm{\infty }}{J}_{\epsilon ,\lambda ,\mu }(tu,tv)=-\mathrm{\infty }$. Fix some $(u,v)\in H\mathrm{\setminus }\{(0,0)\}$, there exists $\overline{t}>0$ such that ${\parallel (\overline{t}u,\overline{t}v)\parallel }_H>\sigma$ and $J_{\epsilon ,\lambda ,\mu }(\overline{t}u,\overline{t}v)<0$. Let $(\overline{u},\overline{v})=(\overline{t}u,\overline{t}v)$. □

Lemma 2.4 For each $(u,v)\in H\mathrm{\setminus }\{(0,0)\}$, there exists a unique positive number $t_{u,v}$ such that $(t_{u,v}u,t_{u,v}v)\in {\mathbf{M}}_{\epsilon ,\lambda ,\mu }$ and $J_{\epsilon ,\lambda ,\mu }(t_{u,v}u,t_{u,v}v)={sup}_{t\ge 0}{J}_{\epsilon ,\lambda ,\mu }(tu,tv)$.

there exists a unique positive number ${\overline{t}}_{u,v}$ such that $r({\overline{t}}_{u,v})=0$. It follows that $R^{\mathrm{\prime }}({\overline{t}}_{u,v})=0$. Hence, ${\overline{t}}_{u,v}=t_{u,v}$. □

Remark 2.5 By Lemma 2.3(i) and Lemma 2.4, then ${\theta }_{\epsilon ,\lambda ,\mu }\ge d_0>0$ for some constant $d_0$.

then $(u_0,v_0)$ is a solution of ($E_{\epsilon ,\lambda ,\mu }$).

It follows that $\tau =0$ and $J_{\epsilon ,\lambda ,\mu }^{\mathrm{\prime }}(u_0,v_0)=0$ in $H^{-1}$. Therefore, $(u_0,v_0)$ is a nontrivial solution of ($E_{\epsilon ,\lambda ,\mu }$) and $J_{\epsilon ,\lambda ,\mu }(u_0,v_0)={\theta }_{\epsilon ,\lambda ,\mu }$. □

First of all, we define the Palais-Smale (denoted by (PS)) sequence and (PS)-condition in H for some functional J.

Applying Ekeland’s variational principle and using the same argument as in Cao-Zhou [8] or Tarantello [9], we have the following lemma.

Lemma 3.2 (i) There exists a (PS) -sequence $\{(u_n,v_n)\}$ in ${\mathbf{M}}_{\epsilon ,\lambda ,\mu }$ for $J_{\epsilon ,\lambda ,\mu }$.

In order to prove the existence of positive solutions, we want to prove that $J_{\epsilon ,\lambda ,\mu }$ satisfies the (PS) _γ -condition in H for $\gamma \in (0,\frac{p-2}{2p}\frac{{(S_{\alpha ,\beta })}^{p/(p-2)}}{{(f_{\mathrm{\infty }})}^{2/(p-2)}})$.

Lemma 3.3 $J_{\epsilon ,\lambda ,\mu }$ satisfies the (PS) _γ -condition in H for $\gamma \in (0,\frac{p-2}{2p}\frac{{(S_{\alpha ,\beta })}^{p/(p-2)}}{{(f_{\mathrm{\infty }})}^{2/(p-2)}})$.

where $c_n=o_n(1)$, $d_n=o_n(1)$ as $n\to \mathrm{\infty }$. It follows that $\{(u_n,v_n)\}$ is bounded in H. Hence, there exist a subsequence $\{(u_n,v_n)\}$ and $(u,v)\in H$ such that

Moreover, we have that $J_{\epsilon ,\lambda ,\mu }^{\mathrm{\prime }}(u,v)=0$ in $H^{-1}$. We use the Brézis-Lieb lemma to obtain (3.1) and (3.2) as follows:

(3.1)

(3.2)

Since $g\in L^m({\mathbb{R}}^N)$, where $m=p/(p-q)$, then for any $\sigma >0$, there exists $r>0$ such that ${\int }_{{[B_r^N(0)]}^c}g{(\epsilon z)}^{\frac{p}{p-q}}dz<\sigma$. By the Hölder inequality and the Sobolev embedding theorem, we get

which is a contradiction. Hence, $l=0$, that is, $(u_n,v_n)\to (u,v)$ strongly in H. □

Clearly, $w_{\epsilon }^i(z)\in H^1({\mathbb{R}}^N)$.

and by (4.1), then

(4.2)

Since