Existence of solutions for perturbed elliptic system with critical exponents

In this paper, the perturbed elliptic system with critical exponents $-\varepsilon^{2}\Delta u+V(x)u= f(x,u)+\frac{\alpha}{\alpha+\beta} K(x)|u|^{\alpha-2}u|v|^{\beta}$, $x\in{\mathbb{R}^{N}}$, $-\varepsilon^{2}\Delta v+V(x)v=g(x,v)+\frac{\beta}{\alpha+\beta }K(x)|u|^{\alpha}|v|^{\beta-2}v$, $x\in{\mathbb{R}^{N}}$, is considered, where $\alpha>1$, $\beta>1$ satisfy $\alpha+\beta =2^{\ast}$, and $2^{\ast}=2N/(N-2)$ ($N\geq3$) is the Sobolev critical exponent. Under proper conditions on V, f, g, and K, the existence result is obtained by using variational methods.

In this paper, we obtain the results of nontrivial solutions of the following perturbed elliptic system:

where $\alpha>1$, $\beta>1$ satisfy $\alpha+\beta=2^{\ast}$, $2^{\ast }=2N/(N-2)$ ($N\geq3$) is the critical Sobolev exponent, and $V(x)$, $K(x)$, $f(x,u)$, $g(x,v)$ satisfy the following conditions:

(V₀):

$V\in C(\mathbb{R}^{N},\mathbb{R})$, $V(0)=\inf_{x\in \mathbb{R}^{N}} V(x)=0$, and there exists $b>0$ such that the set $\nu ^{b}:=\{x\in\mathbb{R}^{N}:V(x)< b\}$ has a finite Lebesgue measure;

(K₀):

$K(x)\in C(\mathbb{R}^{N},\mathbb{R})$, $0<\inf K\leq\sup K<\infty$;

(H₁):

$f,g\in C(\mathbb{R}^{N}\times\mathbb{R})$, $f(x,u)=o(|u|)$, $g(x,v)=o(|v|)$ uniformly in x as $u\rightarrow 0$, $v\rightarrow0$;

(H₂):

there exist $2< q<2^{\ast}$ and $c_{0}>0$ such that

$$\bigl\vert f(x,u)\bigr\vert \leq c_{0}\bigl(1+|u|^{q-1} \bigr) \quad \mbox{for all } (x,u) $$

and

$$\bigl\vert g(x,v)\bigr\vert \leq c_{0}\bigl(1+|v|^{q-1} \bigr)\quad \mbox{for all } (x,v); $$

(H₃):

there exist $a_{0}>0$, $p>2 $, and $2<\mu<2^{\ast}$ such that $F(x,u)\geq a_{0}|u|^{p}$, $G(x,v)\geq a_{0}|v|^{p}$, $\mu F(x,u)\leq uf(x,u)$ for all $(x,u) $, and $\mu G(x,v)\leq vg(x,v)$ for all $(x,v)$, where $F(x,u)=\int_{0}^{u} f(x,s)\, ds$, $G(x,v)=\int_{0}^{v} g(x,s)\, ds$.

Recall that there have been many papers devoted to the study of the scalar equation

where one seeks for the standing waves solutions for the following nonlinear Schrödinger equation:

A standing wave of (1.3) is a solution of the form $\varphi (x,t)=u(x)\exp(-iEt/\hbar)$. Equation (1.2) has been studied extensively by many authors. We would like to cite the works of [1–23], and references therein.

For elliptic systems, there are a lot of works. Han [24] established the existence of positive solutions of the following elliptic system:

where $\lambda>0$, $\mu>0$ are parameters, and $\alpha>1$, $\beta>1$ satisfy $\alpha+\beta=2^{\ast}$; $2^{\ast}=2N/(N-2)$ ($N\geq3$) denotes the critical Sobolev exponent.

For a semilinear elliptic system involving subcritical exponents, there are a lot of results. Lin [25] obtained a multiplicity of positive solutions of the following semilinear elliptic system:

where $\alpha>1$, $\beta>1$, $2< q< p=\alpha+\beta<2^{\ast}=2N/(N-2)$.

However, as far as we know, there are almost no results on the problem (1.1) involving critical exponents in the whole space. In our work, the coupled terms of the system (1.1) are the critical nonlinearities $K(x)|u|^{\alpha-2}u|v|^{\beta}$ and $K(x)|u|^{\alpha}|v|^{\beta-2}v$ ($\alpha+\beta=p^{\ast}$). We consider the problem (1.1) and use variational methods to get positive solutions. The main difficulty is that the corresponding functional to the problem (1.1) lacks compactness because of the unbounded domain $\mathbb{R}^{N}$ and the critical exponent. We can prove the functional associated to (1.1) obeys the $(\mathrm{PS})_{c}$ condition at some energy level c. To overcome the difficulty, we follow some ideas explored in [15] and also use arguments developed in [26, 27].

The main result of this paper reads as follows.

Theorem 1

Assume that (V₀), (K₀), and (H₁)-(H₃) hold. Then, for any $\sigma>0$, there is $\varepsilon_{\sigma}>0$ such that if $\varepsilon\leq\varepsilon _{\sigma}$, the problem (1.1) has at least one solution $(u_{\varepsilon}, v_{\varepsilon})$ which satisfies

This paper is organized as follows. In Section 2, we describe the analytic setting where we restate the problem in an equivalent form by replacing $\varepsilon^{-2}$ with λ other than the usual scaling. In Section 3, we show the corresponding energy functional satisfies the $(\mathrm{PS})_{c}$ condition at the level c. Section 4 gives the fact that the energy functional possesses a mountain-pass geometry structure. The last section contains the proof of the main result.

Let $\lambda=\varepsilon^{-2}$. The problem (1.1) reads

for λ sufficiently large. We are going to prove the following result.

Theorem 2

Assume that (V₀), (K₀), and (H₁)-(H₃) hold. Then for any $\sigma>0$, there is $\Lambda _{\sigma}>0$ such that if $\lambda\geq\Lambda_{\sigma}$, the problem (2.1) has at least one solution $(u_{\lambda}, v_{\lambda})$ which satisfies

In order to prove Theorem 2, we introduce the necessary notations. The space

is a Hilbert space equipped with the inner product

and the associated norm $\|u\|_{\lambda}^{2}=(u,u)_{E_{\lambda}}$. Set $E=E_{\lambda} \times E_{\lambda}$, the Hilbert space with the norm

for any $(u,v)\in E$. It is obvious that, for each $2< p<2^{\ast}$, there exists $c_{p}>0$ such that if $\lambda\geq1$,

where the $L^{p}(\mathbb{R}^{N})$, $1 \leq p< \infty$, denote Lebesgue spaces and the norm for $L^{p}$ is denoted by $\|\cdot\|_{p}$ for $1 \leq p< \infty$.

We will show the existence results of nontrivial solutions of (2.1) by looking for critical points of the associated functional

In fact, the critical points of the functional $I_{\lambda}$ are the weak solutions of (2.1). By a weak solution $(u,v)$ of (2.1), we mean that $(u,v)\in E$ satisfies

for all $(\varphi,\psi)\in E$.

In this section, we will find the range of c where the $(\mathrm{PS})_{c}$ condition holds for the functional $I_{\lambda}$. For convenience, we give some notations.

Notations

• The dual space of a Banach space E will be denoted by $E^{\ast}$.

• $B_{r}:=\{x\in\mathbb{R}^{N}:|x|\leq r\}$ is the ball in $\mathbb{R}^{N}$.

• c, $c_{i}$ represent various positive constants, the exact values of which are not important.

• Let $C_{0}^{\infty}(\mathbb{R}^{N})$ denote the collection of smooth functions with compact support.

• $o(1)$ denotes $o(1)\rightarrow0$ as $n\rightarrow\infty$.

• $S_{\alpha,\beta}$ is the best Sobolev embedding constant defined by

  $$ S_{\alpha,\beta}= \inf_{u,v \in H^{1}(\mathbb{R}^{N})} \frac{\int_{\mathbb{R}^{N}}(|\nabla u|^{2}+|\nabla v|^{2})}{(\int_{\mathbb {R}^{N}}|u|^{\alpha}|v|^{\beta})^{\frac{2}{\alpha+\beta}}}. $$

  (3.1)

We can obtain

where S is the best Sobolev embedding constant defined by

Based on the assumptions of Theorem 2 in [27], we can show that $I_{\lambda}\in C^{1}(E,\mathbb{R})$ and the critical points of $I_{\lambda}$ are weak solutions of the problem (2.1).

Recall that we say that a sequence $\{(u_{n},v_{n})\} \subset E$ is a $(\mathrm{PS})$ sequence at level c ($(\mathrm{PS})_{c}$ sequence, for short) if $I_{\lambda}(u_{n},v_{n})\rightarrow c$ and $I_{\lambda}'(u_{n},v_{n})\rightarrow0$. $I_{\lambda}$ is said to satisfy the $(\mathrm{PS})_{c}$ condition if any $(\mathrm{PS})_{c}$ sequence contains a convergent subsequence.

Lemma 3.1

If the sequence $\{(u_{n},v_{n})\}\subset E$ is a $(\mathrm{PS})_{c}$ sequence for $I_{\lambda}$, then we find that $c\geq0$ and $\{(u_{n},v_{n})\}$ is bounded in the space E.

Proof

We have

Together with (K₀), (H₃), and $2<\mu<2^{\ast}$, we get

By the fact that $I_{\lambda}(u_{n},v_{n})\rightarrow c$ and $I_{\lambda }'(u_{n},v_{n})\rightarrow0$, we easily obtain the desired conclusion. □

Lemma 3.2

There exists a subsequence $\{ (u_{n_{j}},v_{n_{j}})\}$ such that for any $\varepsilon>0$, there is $r_{\varepsilon}>0$ with $r\geq r_{\varepsilon}$,

where $2\leq d<2^{\ast}$.

Proof

Together with Lemma 3.1, the $(\mathrm{PS})_{c}$ sequence $\{(u_{n},v_{n})\}$ for $I_{\lambda}$ is bounded in E. So, we assume $(u_{n},v_{n})\rightharpoonup (u,v)$ in E, $u_{n}\rightarrow u, v_{n}\rightarrow v$ a.e. in $\mathbb {R}^{N}$ and $(u_{n},v_{n})\rightarrow(u,v)$ in $L_{\mathrm{loc}}^{d}(\mathbb {R}^{N})\times L_{\mathrm{loc}}^{d}(\mathbb{R}^{N})$ for any $2\leq d<2^{\ast}$. Note that, for each $j\in\mathbb{N}$, we have

Thus, there exists $n_{0}\in\mathbb{N}$ such that

for all $n\geq n_{0}+1$. Without loss of generality, we may choose $n_{j}=n_{0}+j$ such that

It is easy to see there is $r_{\varepsilon}$ satisfying

Since

and

and the lemma follows. □

Let $\eta\in C^{\infty}(\mathbb{R}^{+},[0,1])$ be a smooth function satisfying $0\leq\eta(t)\leq1$, $t\geq0$. $\eta(t)=1$ if $t\leq1$ and $\eta(t)=0$ if $t\geq2$. Define $\tilde{u}_{j}(x)=\eta (2|x|/j)u(x)$ and $\tilde{v}_{j}(x)=\eta(2|x|/j)v(x)$, then

Lemma 3.3

One has

and

uniformly in $(\varphi,\psi)\in E$ with $\|(\varphi,\psi)\|_{E}\leq1$.

Proof

Note that (3.2) and local compactness of the Sobolev embedding imply that for any $r>0$,

uniformly in $\|\varphi\|\leq1$. For any $\varepsilon>0$, it follows from

that

By using Lemma 3.2 and the assumption (H₂), we get

which implies that

Similar to this proof, we can prove that the other result is correct. □

Lemma 3.4

Passing to a subsequence, we have

and

Proof

Together with the fact that $(u_{n},v_{n})\rightharpoonup(u,v)$, $(\tilde {u}_{n},\tilde{v}_{n})\rightarrow(u,v)$ in E, we get

Similar to the proof of the Brézis-Lieb lemma [28], we easily get

and

Observing the fact that $I_{\lambda}(u_{n},v_{n})\rightarrow c$ and $I_{\lambda}(\tilde{u}_{n},\tilde{v}_{n})\rightarrow I_{\lambda}(u,v)$, we obtain

In addition, for any $(\varphi,\psi)\in E$, we get

It is standard to check

and

uniformly in $\|(\varphi,\psi)\|_{E}\leq1$. By the fact of Lemma 3.3 and $I_{\lambda}'(u_{n},v_{n})\rightarrow0$, we complete the proof of Lemma 3.4. □

Set $u_{n}^{1}=u_{n}-\tilde{u}_{n}$ and $v_{n}^{1}=v_{n}-\tilde{v}_{n}$, then $u_{n}-u=u_{n}^{1}+(\tilde{u}_{n}-u)$ and $v_{n}-v=v_{n}^{1}+(\tilde{v}_{n}-v)$. We easily get $(u_{n},v_{n})\rightarrow(u,v)$ in E if and only if $(u_{n}^{1},v_{n}^{1})\rightarrow(0,0)$ in E.

Observe that

where $K_{0}=\inf_{x\in\mathbb{R}^{N}}K(x)>0$. In connection with $I_{\lambda}(u_{n}^{1},v_{n}^{1})\rightarrow c-I_{\lambda}(u,v)$ and $I_{\lambda}'(u_{n}^{1},v_{n}^{1})\rightarrow0$ in $E^{\ast}$, we get

In addition, by (K₀) and (H₂), for any $b>0$, there is a constant $C_{b}>0$ such that

Let $V_{b}(x):=\max\{V(x),b\}$, where b is the positive constant in the assumption (V₀). Since the set $\nu^{b}:=\{x\in\mathbb{R}^{N}:V(x)< b\}$ has a finite Lebesgue measure and $(u_{n}^{1},v_{n}^{1})\rightarrow(0,0)$ in $L_{\mathrm{loc}}^{2}(\mathbb{R}^{N})\times L_{\mathrm{loc}}^{2}(\mathbb{R}^{N})$, we have

Thus

Together with (3.3), we have

Set $\alpha_{0}=S_{\alpha,\beta}^{\frac{N}{2}}C_{b}^{-\frac {N}{2}}N^{-1}K_{0}$. This implies $\alpha_{0}\lambda^{1-\frac {N}{2}}\leq c-I_{\lambda}(u,v)+o(1)$.

Lemma 3.5

Assume that (V₀), (K₀), and (H₁)-(H₃) are satisfied. Then, for any $(\mathrm{PS})_{c}$, the sequence $\{(u_{n},v_{n})\}$ for $I_{\lambda}$, there exists a constant $\alpha _{0}>0$ (independent of λ) such that the functional $I_{\lambda}(u,v)$ satisfies the $(\mathrm{PS})_{c}$ condition for all $c< \alpha _{0}\lambda^{1-\frac{N}{2}}$.

Proof

We can check that, for any $(\mathrm{PS})_{c}$ sequence $\{(u_{n},v_{n})\}\subset E$ with $(u_{n},v_{n})\rightharpoonup(u,v)$, either $(u_{n},v_{n})\rightarrow (u,v)$ or $c-I_{\lambda}(u,v)\geq\alpha_{0}\lambda^{1-\frac{N}{2}}$.

On the contrary, if $(u_{n},v_{n})\nrightarrow(u,v)$, this shows

and

Based on the above mentioned conclusion, we easily find that the functional $I_{\lambda}(u,v)$ satisfies the $(\mathrm{PS})_{c}$ condition for all $c<\alpha_{0}\lambda^{1-\frac{N}{2}}$. □

We consider $\lambda\geq1$ and check that the functional $I_{\lambda}$ possesses the mountain-pass structure.

Lemma 4.1

Assume that (V₀), (K₀), and (H₁)-(H₃) are satisfied. There exist $\alpha_{\lambda}, \rho _{\lambda}>0$ such that

Proof

Note that

It is clear that, for each $s\in[2,2^{\ast}]$, there is $c_{s}$ such that if $\lambda\geq1$,

By the Young inequality, we have

Furthermore, we get

Combining (H₃) and (4.1), there is a constant $c_{\delta}$ such that

Setting $\rho_{\lambda}=(\frac{1}{8c_{\delta}})^{\frac{1}{2^{\ast }-2}}$ implies

The proof is completed. □

Lemma 4.2

For any finite-dimensional subspace $F\subset E$, we have

Proof

By the assumptions (K₀) and (H₃), it follows that

In connection with the fact that all norms in a finite-dimensional space are equivalent and $p>2$, we easily get the desired conclusion. □

Lemma 4.3

For any $\sigma>0$, there is $\Lambda _{\sigma}>0$ such that for each $\lambda\geq\Lambda_{\sigma}$, there exists $\bar{e}_{\lambda}\in E$ with $\|\bar{e}_{\lambda}\|_{E}>\rho _{\lambda}$, and we have

and

where $\rho_{\lambda}$ is defined in Lemma  4.1.

Proof

Define the functionals

and

We obtain $\Phi_{\lambda}\in C^{1}(E)$ and $I_{\lambda}(u,v)\leq\Phi _{\lambda}(u,v)$ for all $(u,v)\in E$.

Observe that

For any $\delta>0$, there are $\phi_{\delta}, \psi_{\delta}\in C_{0}^{\infty}(\mathbb{R}^{N},\mathbb{R})$ with $\|\phi_{\delta}\| _{p}=\|\psi_{\delta}\|_{p}=1$ such that

Let $e_{\lambda}(x)=(\phi_{\delta}(\lambda^{\frac{1}{2}}x),\psi _{\delta}(\lambda^{\frac{1}{2}}x))$, then $\operatorname{supp}e_{\lambda }\subset B_{\lambda^{-\frac{1}{2}}r_{\delta}}(0)$. Furthermore, we get

It is clear that

Combining $V(0)=0$ and $\operatorname{supp}(\phi_{\delta},\psi_{\delta})\subset B_{r_{\delta}}(0)$, there is $\Lambda_{\delta}>0$ such that, for all $\lambda\geq\Lambda_{\delta}$, we have

Thus, for all $\lambda\geq\Lambda_{\delta}$,

For any $\sigma>0$, we can choose $\delta>0$ so small that

and $e_{\lambda}(x)=(\phi_{\delta}(\lambda^{\frac{1}{2}}x),\psi _{\delta}(\lambda^{\frac{1}{2}}x))$. Taking $\Lambda_{\delta}=\Lambda _{\sigma}$, there is $\bar{t}_{\lambda}>0$ such that $\|\bar{t}_{\lambda}e_{\lambda}\|_{E}>\rho_{\lambda}$ and $I_{\lambda}(te_{\lambda})\leq0$ for all $t\geq\bar{t}_{\lambda}$. By (4.2), $\bar{e}_{\lambda}=\bar{t}_{\lambda}e_{\lambda}$ satisfies the requirements. □

Proof of Theorem 2

Define

where $\Gamma_{\lambda}=\{\gamma\in C([0,1],E):\gamma(0)=0,\gamma (1)=\bar{e}_{\lambda}\}$.

In addition, for any $\sigma>0$ with $\sigma<\alpha_{0}$, there is $\Lambda_{\sigma}>0$ such that for $\lambda\geq\Lambda_{\sigma}$, we can choose $c_{\lambda}$ which satisfies $c_{\lambda}\leq\sigma\lambda ^{1-\frac{N}{2}}$.

From the above mentioned results, the functional $I_{\lambda}$ satisfies the $(\mathrm{PS})_{c_{\lambda}}$ condition if $c_{\lambda}\leq\sigma\lambda^{1-\frac{N}{2}}$ and has the mountain-pass structure. Hence, there is $(u_{\lambda},v_{\lambda})\in E$ such that

That is to say, $(u_{\lambda},v_{\lambda})$ is a weak solution of (2.1). Similar to the arguments in [4], we also find that $(u_{\lambda},v_{\lambda})$ is a positive least energy solution.

Furthermore,

This shows that

We complete the proof of Theorem 2. □

The author would like to express appreciation of the referees for their precious comments and suggestions about the original manuscript. This research was supported by the National Natural Science Foundation of China (11271364).

Affiliations

1. Department of Mathematics, China University of Mining and Technology, Xuzhou, Jiangsu, 221116, People’s Republic of China

   • Juan Jiang
   •  & Wenbin Liu

Corresponding author

Correspondence to Juan Jiang.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally in this article. They read and approved the final manuscript.

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