A nonhomogeneous critical Kirchhoff-Schr\"odinger type equation in $\mathbb{R}^{4}$ involving vanishing potentials

TL;DR

This paper proves the existence of solutions for a nonhomogeneous Kirchhoff-Schrödinger equation in four-dimensional space with critical nonlinearity, using variational methods and concentration compactness.

Contribution

It establishes the existence of weak solutions for a critical Kirchhoff-Schrödinger problem involving nonlocal terms and perturbations in -dimensional space.

Findings

Existence of mountain pass solutions.

Existence of negative energy solutions.

Application of variational methods and Lions' principle.

Abstract

In this paper we establish the existence of mountain pass and negative energy weak solutions for a Kirchhoff-Schr\"odinger type problem in $\mathbb R^4$ involving a critical nonlinearity and a suitable small perturbation. The arisen competition between the terms due to the nonlocal coefficient and critical nonlinearity turns out to be rather interesting. The main tools used in the present work are variational methods and the Lions' Concentration Compactness Principle.