Supercritical Schr\"odinger equations involving integro-differential operators and vanishing potentials

TL;DR

This paper investigates positive, bounded solutions for a Schrödinger equation with integro-differential operators and vanishing potentials, employing variational and penalization methods to address non-compactness and nonlinearity challenges.

Contribution

It introduces a novel approach using a weak Maximum Principle and a supersolution construction to handle the lack of regularity and decay estimates in this generalized framework.

Findings

Existence of nontrivial solutions under small perturbations

Solutions exhibit controlled asymptotic behavior

Method extends fractional Laplacian techniques to more general operators

Abstract

This paper is devoted to the study of the existence of positive and bounded solutions for a Schr\"odinger type equation defined on the entire Euclidean space, involving a general integro-differential operator. We consider the case where the potential is nonnegative and vanishes at infinity with a nonlinearity exhibiting critical or supercritical growth in the Sobolev sense. To overcome the lack of compactness and the difficulties imposed by the general structure of the nonlinearity, we employ variational methods combined with a penalization technique. Unlike the classical fractional Laplacian framework, where specific regularity results, decay estimates, and the $s$-harmonic extension are available, our approach relies on a weak Maximum Principle combined with the construction of a supersolution based on the truncated fundamental solution of the fractional Laplacian to control the asymptotic behavior of the solutions. We prove that, for sufficiently small perturbation parameters and under suitable decay conditions on the potential, the equation admits a nontrivial solution.