Ground states of nonlocal elliptic equations with general nonlinearities via Rayleigh quotient

Abstract

It is established ground states and multiplicity of solutions for a nonlocal Schrödinger equation $(-\Delta {)}^{s}u+V(x)u=\lambda a(x)|u{|}^{q-2}u+b(x)f(u)$ in ${\mathbb{R}}^N,$ $u\in H^s({\mathbb{R}}^N),$ where $0<s<\min \{1,N/2\},$ $1<q<2$ and $\lambda >0,$ under general conditions over the measurable functions $a,$ $b$, $V$ and $f.$ The nonlinearity $f$ is superlinear at infinity and at the origin, and does not satisfy any Ambrosetti-Rabinowitz type condition. It is considered that the weights $a$ and $b$ are not necessarily bounded and the potential $V$ can change sign. We obtained a sharp ${\lambda }^{\ast }>0$ which guarantees the existence of at least two nontrivial solutions for each $\lambda \in (0,{\lambda }^{\ast })$. Our approach is variational in its nature and is based on the nonlinear Rayleigh quotient method together with some fine estimates. Compactness of the problem is also considered.