Ground state solution for a class of modified nonlinear fourth-order elliptic equation with sign-changing unbounded potential

Abstract

We are concerned on the fourth-order elliptic equation \begin{equation}\tag{$P_{\lambda }$} \left\{ \begin{array}[c]{ll} Δ^2 u- Δu + V(x)u -λΔ[ρ(u^2)]ρ'(u^2)u= f(u)\, \, \mbox{in} \, \, \mathbb{R}^N, & u\in W^{2,2}(\mathbb{R}^N), \end{array} \right. \end{equation} where ${\Delta }^2=\Delta (\Delta )$ is the biharmonic operator, $3\le N\le 6$, the radially symmetric potential $V$ may change sign and ${\inf }_{{\mathbb{R}}^N}V(x)=-\infty$ is allowed. If $f$ satisfies a type of nonquadracity and monotonicity conditions and $\rho$ is a suitable smooth function, we prove, via variational approach, the existence of a radially symmetric nontrivial ground state solution $u_{\lambda }$ for problem $(P_{\lambda })$ for all $\lambda \ge 0$.

15 pages

References in corpus (1)

• Existence of ground states for a modified nonlinear Schrodinger equation