Existence of solutions for a general quasilinear elliptic system via perturbation method

In this paper, we consider the following quasilinear elliptic system:

where D_{i}u=\frac{\partial u}{\partial x_i}, D_s{a}_{ij}(x,u)=\frac{\partial }{\partial u}{a}_{ij}(x,u), D_s{b}_{ij}(x,v)=\frac{\partial }{\partial v}{b}_{ij}(x,v), \alpha >2, \beta >2, , 2^{\ast }=\frac{2N}{N-2} is the critical Sobolev exponent and \mathrm{\Omega }\subset {\mathbb{R}}^N (N\ge 3) is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.

MSC:35J60, 35B33.

Let us consider the following quasilinear elliptic system:

where D_{i}u=\frac{\partial u}{\partial x_i}, D_s{a}_{ij}(x,u)=\frac{\partial }{\partial u}{a}_{ij}(x,u), D_s{b}_{ij}(x,v)=\frac{\partial }{\partial v}{b}_{ij}(x,v), \alpha >2, \beta >2, , 2^{\ast }=\frac{2N}{N-2} is the critical Sobolev exponent and \mathrm{\Omega }\subset {\mathbb{R}}^N (N\ge 3) is a bounded smooth domain. This system includes the following special class of system with a_{ij}(x,u)=(1+u^2){\delta }_{ij}, b_{ij}(x,v)=(1+v^2){\delta }_{ij}, i.e.,

which is referred to as the so-called modified nonlinear Schrödinger system.

Our assumptions on the functions a_{ij} and b_{ij} are as follows.

• (A₁) The functions a_{ij}\in C^1(\overline{\mathrm{\Omega }}\times \mathbb{R},\mathbb{R}), b_{ij}\in C^1(\overline{\mathrm{\Omega }}\times \mathbb{R},\mathbb{R}), a_{ij}=a_{ji}, b_{ij}=b_{ji}, i,j=1,2,\dots ,N.

• (A₂) There exist constants a_0, a_1, b_0, b_1 satisfying a_1\ge a_0>0, b_1\ge b_0>0, (\alpha +\beta -2)a_0>2a_1 and (\alpha +\beta -2)b_0>2b_1 such that

  \begin{array}{c}{a}_0(1+s^2){|\xi |}^2\le \sum _{i,j=1}^N{a}_{ij}(x,s){\xi }_i{\xi }_j\le a_1(1+s^2){|\xi |}^2,\hfill \\ b_0(1+s^2){|\xi |}^2\le \sum _{i,j=1}^N{b}_{ij}(x,s){\xi }_i{\xi }_j\le b_1(1+s^2){|\xi |}^2\hfill \end{array}

  for x\in \overline{\mathrm{\Omega }}, \xi \in {\mathbb{R}}^N, s\in \mathbb{R}.

• (A₃)

  \begin{array}{c}0\le \sum _{i,j=1}^N{D}_s{a}_{ij}(x,s)s{\xi }_i{\xi }_j\le 2\sum _{i,j=1}^N{a}_{ij}(x,s){\xi }_i{\xi }_j,\hfill \\ 0\le \sum _{i,j=1}^N{D}_s{b}_{ij}(x,s)s{\xi }_i{\xi }_j\le 2\sum _{i,j=1}^N{b}_{ij}(x,s){\xi }_i{\xi }_j\hfill \end{array}

  for x\in \overline{\mathrm{\Omega }}, \xi \in {\mathbb{R}}^N, s\in \mathbb{R}.

In recent years, much attention has been devoted to the quasilinear Schrödinger equation of the following form:

See, for example, [1] where Poppenberg et al. proved the existence of a positive ground state solution by using a constrained minimization argument. Using a change of variables, Liu et al. [2] used an Orlicz space to prove the existence of a soliton solution for equation (1.2) via the mountain pass theorem. Colin and Jeanjean [3] also made use of a change of variables but worked in the Sobolev space H^1({\mathbb{R}}^N). They proved the existence of a positive solution for equation (1.2) from the classical results given by Berestycki and Lions [4]. Liu et al. [5] established the existence of both one-sign and nodal ground states of soliton-type solutions for equation (1.2) by the Nehari method. By using the Nehari manifold method and the concentration compactness principle (see [6]) in the Orlicz space, Guo and Tang [7] considered the following quasilinear Schrödinger system:

with a(x)\ge 0, b(x)\ge 0 having a potential well and \alpha >2, \beta >2, , and they proved the existence of a ground state solution for system (1.3) which localizes near the potential well \text{int }{a}^{-1}(0) for λ large enough. Guo and Tang [8] considered also ground state solutions of the single quasilinear Schrödinger equation corresponding to system (1.3) by the same methods and obtained similar results. In particular, by the perturbation method, Liu et al. [9] considered the existence and multiplicity of solutions for the following quasilinear equation of the form

under suitable assumptions.

It is worth pointing out that the existence of one-bump or multi-bump bound state solutions for the related semilinear Schrödinger equation (1.2) for k=0 has been extensively studied. One can see Bartsch and Wang [10], Ambrosetti et al. [11], Ambrosetti et al. [12], Byeon and Wang [13], Cingolani and Lazzo [14], Cingolani and Nolasco [15], Del Pino and Felmer [16, 17], Floer and Weinstein [18], Oh [19, 20] and the references therein.

Motivated by the single equation (1.4), the purpose of this paper is to study the existence of both positive and negative solutions for the coupled quasilinear system (1.1). We mainly follow the idea of Liu et al. [9] to perturb the functional and obtain our main results. We point out that the procedure to system (1.1) is not trivial at all. Since the appearance of the quasilinear terms {\sum }_{i,j=1}^N{D}_j(a_{ij}(x,u)D_{i}u)-\frac{1}{2}{\sum }_{i,j=1}^N{D}_s{a}_{ij}(x,u)D_{i}u{D}_{j}u and {\sum }_{i,j=1}^N{D}_j(b_{ij}(x,v)D_{i}v)-\frac{1}{2}{\sum }_{i,j=1}^N{D}_s{b}_{ij}(x,v)D_{i}v{D}_{j}v, we need more delicate estimates.

The paper is organized as follows. In Section 2, we introduce a perturbation of the functional and give our main results (Theorem 2.1 and Theorem 2.2). In Section 3, we verify the Palais-Smale condition for the perturbed functional. Section 4 is devoted to some asymptotic behavior of the sequences \{(u_n,v_n)\}\subset W_0^{1,4}(\mathrm{\Omega })\times W_0^{1,4}(\mathrm{\Omega }) and \{{\mu }_n\}\subset (0,1] satisfying some conditions. Finally, our main results will be proved in Section 5.

Throughout this paper, we will use the same C to denote various generic positive constants, and we will use o(1) to denote quantities that tend to 0.

In order to obtain the desired existence of solutions for system (1.1), in this section, we introduce a perturbation of the functional and give our main results.

The weak form of system (1.1) is

for all (\phi ,\psi )\in C_0^{\mathrm{\infty }}(\mathrm{\Omega })\times C_0^{\mathrm{\infty }}(\mathrm{\Omega }), which is formally the variational formulation of the following functional:

We may define the derivative of I_0 at (u,v) in the direction of (\phi ,\psi )\in C_0^{\mathrm{\infty }}(\mathrm{\Omega })\times C_0^{\mathrm{\infty }}(\mathrm{\Omega }) as follows:

We call (u,v) a critical point of I_0 if (u,v)\in W_0^{1,2}(\mathrm{\Omega })\times W_0^{1,2}(\mathrm{\Omega }), , and 〈I_0^{\prime }(u,v),(\phi ,\psi )〉=0 for all (\phi ,\psi )\in C_0^{\mathrm{\infty }}(\mathrm{\Omega })\times C_0^{\mathrm{\infty }}(\mathrm{\Omega }). That is, (u,v) is a weak solution for system (1.1).

When we consider system (1.1) by using the classical critical point theory, we encounter the difficulties due to the lack of an appropriate working space. In general, it seems that there is no suitable space in which the variational functional I_0 possesses both smoothness and compactness properties. For smoothness, one would need to work in a space smaller than W_0^{1,2}(\mathrm{\Omega }) to control the term involving the quasilinear term in system (1.1), but it seems impossible to obtain bounds for {(\mathit{PS})}_c sequence in this setting. Several ideas and approaches, such as minimizations [1, 21], the Nehari method [5] and change of variables [2, 3], have been used in recent years to overcome the difficulties. In this paper, we consider the perturbed functional

where \mu \in (0,1] is a parameter. Then it is easy to see that I_{\mu } is a C^1-functional on W_0^{1,4}(\mathrm{\Omega })\times W_0^{1,4}(\mathrm{\Omega }). We can define also the derivative of I_{\mu } at (u,v) in the direction of (\phi ,\psi ) as follows:

for all (\phi ,\psi )\in C_0^{\mathrm{\infty }}(\mathrm{\Omega })\times C_0^{\mathrm{\infty }}(\mathrm{\Omega }). The idea of this paper is to obtain the existence of the critical points of I_{\mu } for \mu >0 small and establish suitable estimates for the critical points as \mu \to 0 so that we may pass to the limit to get the solutions for the original system (1.1).

Our main results are as follows.

Theorem 2.1 Assume that (A₁)-(A₃) hold, \alpha >2, \beta >2 and . Let {\mu }_n\to 0 and let \{(u_n,v_n)\}\subset W_0^{1,4}(\mathrm{\Omega })\times W_0^{1,4}(\mathrm{\Omega }) be a sequence of critical points of I_{{\mu }_n} satisfying I_{{\mu }_n}^{\prime }(u_n,v_n)=0 and I_{{\mu }_n}(u_n,v_n)\le C for some C independent of n. Then, up to a subsequence,

as n\to \mathrm{\infty }, and (u,v) is a critical point of I_0.

Theorem 2.2 Assume that (A₁)-(A₃) hold, \alpha >2, \beta >2 and . Then I_{\mu } has a positive critical point (u_{\mu },v_{\mu }) and a negative critical point ({\tilde{u}}_{\mu },{\tilde{v}}_{\mu }), and (u_{\mu },v_{\mu }) (resp.,({\tilde{u}}_{\mu },{\tilde{v}}_{\mu })) converges to a positive (resp., negative) solution for system (1.1) as \mu \to 0.

Notation We denote by \parallel \cdot \parallel the norm of W_0^{1,4}(\mathrm{\Omega }) and by {|\cdot |}_s the norm of L^s(\mathrm{\Omega }) ().

In this section, we verify the Palais-Smale condition ({(\mathit{PS})}_c condition in short) for the perturbed functional I_{\mu }(u,v). We have the following proposition.

Proposition 3.1 For \mu >0 fixed, the functional I_{\mu }(u,v) satisfies {(\mathit{PS})}_c condition for all c\in \mathbb{R}. That is, any sequence \{(u_n,v_n)\}\subset W_0^{1,4}(\mathrm{\Omega })\times W_0^{1,4}(\mathrm{\Omega }) satisfying, for c\in \mathbb{R},

has a strongly convergent subsequence in W_0^{1,4}(\mathrm{\Omega })\times W_0^{1,4}(\mathrm{\Omega }), where {(W_0^{1,4}(\mathrm{\Omega })\times W_0^{1,4}(\mathrm{\Omega }))}^{\ast } is the dual space of W_0^{1,4}(\mathrm{\Omega })\times W_0^{1,4}(\mathrm{\Omega }).

To give the proof of Proposition 3.1, we need the following lemma firstly.

Lemma 3.2 Suppose that a sequence \{(u_n,v_n)\}\subset W_0^{1,4}(\mathrm{\Omega })\times W_0^{1,4}(\mathrm{\Omega }) satisfies (3.1) and (3.2). Then

Proof It follows from (3.1) and (3.2) that

Thus we have

This completes the proof of Lemma 3.2. □

Now we give the proof of Proposition 3.1.

Proof of Proposition 3.1 From Lemma 3.2 , we know that \{(u_n,v_n)\} is bounded in W_0^{1,4}(\mathrm{\Omega })\times W_0^{1,4}(\mathrm{\Omega }). So there exists a subsequence of \{(u_n,v_n)\}, still denoted by \{(u_n,v_n)\}, such that

Now we prove that (u_n,v_n)\to (u,v) in W_0^{1,4}(\mathrm{\Omega })\times W_0^{1,4}(\mathrm{\Omega }). In (2.5), choosing (\phi ,\psi )=(u_n-u_m,v_n-v_m), we have

We may estimate the terms involved as follows:

Returning to (3.3), we have

which implies that \parallel (u_n-u_m,v_n-v_m)\parallel \to 0, i.e., (u_n,u_m)\to (u,v) in W_0^{1,4}(\mathrm{\Omega })\times W_0^{1,4}(\mathrm{\Omega }). This completes the proof of Proposition 3.1. □

Proposition 3.1 enables us to apply minimax argument to the functional I_{\mu }(u,v). In this section, we also study the behavior of the sequences \{(u_n,v_n)\}\subset W_0^{1,4}(\mathrm{\Omega })\times W_0^{1,4}(\mathrm{\Omega }) and \{{\mu }_n\}\subset (0,1] satisfying

The following proposition is the key of this section.

Proposition 4.1 Assume that the sequences \{(u_n,v_n)\}\subset W_0^{1,4}(\mathrm{\Omega })\times W_0^{1,4}(\mathrm{\Omega }) and \{{\mu }_n\}\subset (0,1] satisfy (4.1)-(4.3). Then, after extracting a sequence, still denoted by n, we have

and

as n\to \mathrm{\infty }.

Proof Similar to the proof of Lemma 3.2, by (4.1)-(4.3), we have

Thus

for some C independent of n. Then, up to a subsequence, we have

and

as n\to \mathrm{\infty }. This completes the proof of Proposition 4.1. □

In this section, we give the proof of our main results. Firstly, we prove Theorem 2.1.

Proof of Theorem 2.1 Note that (u_n,v_n) satisfies the following equation: