Existence and concentration of a nonlinear biharmonic equation with sign-changing potentials and indefinite nonlinearity

Abstract

Abstract We consider the following nonlinear biharmonic equations: Δ2u−Δu+Vλ(x)u=f(x,u),in RN, $$ \Delta^{2} u-\Delta u+ V_{\lambda }(x)u=f(x,u),\quad \text{in } \mathbb{R}^{N}, $$ where Vλ(x) $V_{\lambda }(x)$ is allowed to be sign-changing and f is an indefinite function. Under some suitable assumptions, the existence of nontrivial solutions and the high energy solutions are obtained by using variational methods. Moreover, the phenomenon of concentration of solutions is explored. The results extend the main conclusions in recent literature.

Keywords

• Biharmonic equation
• Variational methods
• High energy solutions
• Concentration of solutions