A solution to a slightly subcritical elliptic problem with non-power nonlinearity

TL;DR

This paper proves the existence of positive solutions for a slightly subcritical elliptic problem with a non-power nonlinearity, using a Ljapunov-Schmidt reduction, marking the first such result for this class of problems.

Contribution

It introduces a novel existence proof for a class of subcritical elliptic problems with non-power nonlinearities, expanding the scope of solvable nonlinear PDEs.

Findings

Positive solutions concentrate at critical points of the Robin function.

First existence result for slightly subcritical problems with non-power nonlinearities.

Uses Ljapunov-Schmidt reduction to overcome lack of compactness.

Abstract

We consider a slightly subcritical Dirichlet problem with a non-power nonlinearity in a bounded smooth domain. For this problem, standard compact embeddings cannot be used to guarantee the existence of solutions as in the case of power-type nonlinearities. Instead, we use a Ljapunov-Schmidt reduction method to show that there is a positive solution which concentrates at a non-degenerate critical point of the Robin function. This is the first existence result for this type of generalized slightly subcritical problems.