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

TL;DR

This paper establishes the existence and multiplicity of ground state solutions for a nonlocal Schrödinger equation with general nonlinearities, using a variational approach based on the nonlinear Rayleigh quotient method.

Contribution

It introduces a novel variational method employing the nonlinear Rayleigh quotient to analyze nonlocal equations with general nonlinearities and sign-changing potentials.

Findings

Existence of a sharp threshold * for solution multiplicity.

At least two solutions for all ; * with ; in (0, *).

Handling of unbounded weights and sign-changing potentials.

Abstract

It is established ground states and multiplicity of solutions for a nonlocal Schr\"{o}dinger 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 $\lambda^*> 0$ which guarantees the existence of at least two nontrivial solutions for each $\lambda \in (0, \lambda^*)$. 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.