On the Brezis-Nirenberg problem with nonhomogeneous Dirichlet boundary conditions

TL;DR

This paper investigates the Brézis-Nirenberg problem with nonhomogeneous boundary conditions, decomposing the Nehari manifold to establish the existence of multiple solutions using variational methods.

Contribution

It introduces a novel decomposition of the Nehari manifold for this problem and proves the existence of four solutions using Lusternik-Schnirelman and minimax techniques.

Findings

Four solutions to the problem are established.

Decomposition of the Nehari manifold is achieved.

Variational methods are effectively applied.

Abstract

In this paper, we study the decomposition of Nehari manifold for the Br\'ezis-Nirenberg problem with nonhomogeneous Dirichlet boundary conditions. By using this result, the Lusternik-Schnirelman category and the minimax principle, we establish a multiple result (four solutions) for the Br\'ezis{Nirenberg problem with nonhomogeneous Dirichlet boundary conditions.