Existence of positive solutions to perturbed nonlinear Dirichlet problems involving critical growth

Abstract

We consider the following perturbed nonlinear elliptic problem with critical growth $$\displaylines{ -\varepsilon^2\Delta u+V(x)u=f(x)|u|^{p-2}u +\frac{\alpha}{\alpha+\beta}K(x)|u|^{\alpha-2}u|v|^\beta,\quad x\in \mathbb{R}^N,\cr -\varepsilon^2\Delta v+V(x)v=g(x)|v|^{p-2}v +\frac{\beta}{\alpha+\beta}K(x)|u|^\alpha|v|^{\beta-2}v,\quad x\in \mathbb{R}^N,\cr u(x),\quad v(x)\to 0 \quad \text{as } |x|\to\infty. }$$ Using variational methods, we prove the existence of positive solutions.

Keywords

• Perturbed nonlinear Dirichlet problem
• critical growth
• Palais-Smale condition
• variational methods