Existence and multiplicity of solutions for Kirchhoff elliptic problems with nondegenerate points via nonlinear Rayleigh quotient in ℝN

Abstract

In this work, we prove existence and multiplicity of solutions to a Kirchhoff elliptic problem in the whole space [Formula: see text]. More specifically, we consider the following nonlocal elliptic problem: [Formula: see text] where [Formula: see text], the parameters [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] with [Formula: see text] almost everywhere in [Formula: see text]. This type of problem contains the function [Formula: see text] known as the Kirchhoff function given by [Formula: see text] with [Formula: see text] and [Formula: see text] Under our assumptions the potential [Formula: see text] and the nonlinearities can be sign changing functions. Hence, our main objective is to prove that the above problem has at least two distinct nontrivial solutions, one of them being a ground state, whenever [Formula: see text] for some suitable [Formula: see text]. The main idea is to use the minimization method in the Nehari manifold together with the nonlinear Rayleigh quotient. In our setting, the main difficulty is ensuring the existence of nontrivial solutions by using the Nehari method considering the Lagrange Multipliers Theorem. In other words, we study the case where the fibering map admits inflections points with [Formula: see text]. Furthermore, for each [Formula: see text], we show a nonexistence result for our main problem. It is important to emphasize that [Formula: see text] is sharp in order to find the existence and multiplicity of nontrivial solutions for our main problem.