Problemas de Dirichlet para o operador de Grushin: passos da montanha locais

Abstract

In this work, we study problems involving the Grushin operator in Euclidean spaces, focusing on nonlinear elliptic problems, primarily of Schrödinger and Kirchhoff types, under critical growth conditions. The considered Grushin operator acts on functions defined on the Cartesian product of two Euclidean spaces by summing the Laplacian in the first variable and the Laplacian in the second variable, weighted by the norm of the first variable raised to a suitable power and multiplied by a constant factor. More precisely, we address three main problems involving this operator. In the first problem, we investigate a Kirchhoff-type equation with critical growth in three-dimensional Euclidean space, in which the nonlocal diffusion term depends on the norm of the adapted Grushin gradient. The potential considered is coercive and bounded below away from zero by a positive constant, while the nonlinear term satisfies the classical Ambrosetti-Rabinowitz condition. For this problem, we prove the existence of a positive, nontrivial weak solution in the Sobolev space associated with the Grushin operator. Next, we analyze a second elliptic problem in three-dimensional Euclidean space, where the potential undergoes a perturbation or scaling in one of its coordinates via a positive parameter, without requiring coercivity on the potential. We demonstrate that, as this parameter approaches zero, the problem admits a positive weak solution that concentrates around a local minimum point of the potential located within an open and bounded set. Finally, we address a third problem combining the features of the previous ones, consisting of a critical Kirchhoff-type equation featuring a perturbation parameter in both the diffusion term and the argument of the potential, again without imposing coercivity conditions on the potential. For this case, we similarly establish that, as the parameter vanishes, the problem possesses a positive weak solution displaying a concentration phenomenon around a local minimum point of the potential contained in an open and bounded domain.