Solutions for a Schrödinger-Kirchhoff Equation with Indefinite Potentials

Abstract

We deal with the equation $$-\left( 1+\int\nolimits_{\mathbb{R}^3}|\nabla u|^2 dx\right)\Delta u + V(x)u=a(x)|u|^{p-1}u,\quad x\in {\mathbb{R}}^3,$$ with p ∈ (3, 5). Under some conditions on the sign-changing potentials V and a we obtain a nonnegative ground state solution. In the radial case we also obtain a nodal solution.