Multiplicity of solutions for Schrödinger systems with critical exponent via Ljusternik–Schnirelman theory

Abstract

In this paper, by exploiting the Nehari manifold method and the Ljusternik–Schnirelman category theory, we establish the existence and multiplicity results of the nonlinear Schrodinger systems of the form $$\begin{aligned} \left\{ \begin{array}{l} -\epsilon ^{p\left( x\right) }div\left( \left| \nabla u\right| ^{p\left( x\right) -2}\nabla u\right) +V\left( z\right) \left| u\right| ^{p\left( x\right) -2}u=Q_{u}\left( u,v\right) +\gamma H_{u}\left( u,v\right) \text { in } {\mathbf {R}}^N, \\ -\epsilon ^{q\left( x\right) }div\left( \left| \nabla v\right| ^{q\left( x\right) -2}\nabla v\right) +W\left( z\right) \left| v\right| ^{q\left( x\right) -2}v=Q_{v}\left( u,v\right) +\gamma H_{v}\left( u,v\right) \text { in } {\mathbf {R}}^N, \\ \left( u,v\right) \in W^{1,p\left( x\right) }\left( {\mathbf {R}}^{N}\right) \times W^{1,q\left( x\right) }\left( {\mathbf {R}}^{N}\right) ,\text { }u\left( z\right) ,v\left( z\right) >0\text { for all } z\in {\mathbf {R}}^{N}. \end{array} \right. \end{aligned}$$ .