Existence and asymptotic behavior of solutions for a class of semilinear subcritical elliptic systems

Abstract

In this paper we study the asymptotic behaviour of a family of elliptic systems, as far as the existence of solutions is concerned. We give a special attention to the asymptotic behaviour of W and V as ε goes to zero in the system $$\begin{array}{c}\left\{\begin{array}{c}-{\mathit{\epsilon }}^2\mathrm{\Delta }\mathit{u}+\mathit{W}\mathrm{(}\mathit{x}\mathrm{)}\mathit{u}={\mathit{Q}}_{\mathit{u}}\mathrm{(}\mathit{u}\mathrm{,}\mathit{v}\mathrm{)}\text{in }{\mathbb{R}}^{\mathit{N}}\mathrm{,}\hfill \\ -{\mathit{\epsilon }}^2\mathrm{\Delta }\mathit{v}+\mathit{V}\mathrm{(}\mathit{x}\mathrm{)}\mathit{v}={\mathit{Q}}_{\mathit{v}}\mathrm{(}\mathit{u}\mathrm{,}\mathit{v}\mathrm{)}\text{in }{\mathbb{R}}^{\mathit{N}}\mathrm{,}\hfill \\ \mathit{u}\mathrm{,}\mathit{v}\in {\mathit{H}}^1\mathrm{(}{\mathbb{R}}^{\mathit{N}}\mathrm{)}\mathrm{,}\mathit{u}\mathrm{(}\mathit{x}\mathrm{)}\mathrm{,}\mathit{v}\mathrm{(}\mathit{x}\mathrm{)}\mathrm{>}0\text{for each }\mathit{x}\in {\mathbb{R}}^{\mathit{N}}\mathrm{,}\hfill \end{array}\right.\end{array}$$ where $\mathit{\epsilon }\mathrm{>}0$ , W and V are positive potentials of ${\mathit{C}}^2$ class and Q is a p-homogeneous function with subcritical growth. We establish the existence of a positive solution by considering two classes of potentials W and V. Our arguments are based on penalization techniques, variational methods and the Moser iteration scheme.