Existência de soluções para uma classe de equações de Kirchhoff-Schrödinger com crescimento crítico de Sobolev

Abstract

In this work, we establish the existence of positive solutions for a class of stationary Kirchhoff-Schrödinger equations defined in the whole R3 with nonlinearity with critical growth in the Sobolev sense and nonnegative potential that can decay to zero at infinity. For this purpose, we use the variational method of critical point theory, which consists of associating the solutions of the equation to the critical points of a suitable functional. Furthermore, the nonlinearity is general and does not satisfy the well-known AmbrosettiRabinowitz condition, which makes the study of the compactness associated with the problem and the boundedness of Palais-Smale sequences more sophisticated. In this context, the main tools used to achieve our main results were the use of the mountain pass theorem and the Lions’ compactness principle, in addition to the common basis which consists of basic results of measure theory and Lebesgue integration, functional analysis and nonlinear functional analysis.