Existência e multiplicidade de soluções para uma classe de problemas elípticos não locais envolvendo equações do tipo Kirchhoff em domínios ilimitados

Abstract

In this work, we investigate the existence, nonexistence (in some cases) and multiplicity of nontrivial solutions for three nonlocal elliptic problems involving Kirchhofftype equations in RN. The first problem addressed concerns a class of Kirchhoff-type equations with a concave-convex nonlinearity. In the second problem, we explore the same class of Kirchhoff equations, introducing a parameter −θ in the superlinear part of the nonlinearity. This modification results in a new nonlinearity that does not satisfy the Ambrosetti-Rabinowitz condition. Furthermore, the third problem involves a Kirchhoff equation with critical growth. Our approach to these problems is based on the nonlinear Rayleigh quotient method, together with the minimization method on the Nehari manifold. These methods allowed us to reestablish, under specific conditions, results of compactness and strong convergence, ensuring that each problem studied has at least two nontrivial solutions. It is important to emphasize that to find such solutions, it was necessary to introduce more comprehensive assumptions about the potentials and weights involved.