On a problem of resonance with exponential non linearity

TL;DR

This paper investigates the existence of radial solutions to a semi-linear elliptic PDE with exponential nonlinearity on the unit disk, using degree theory and homotopy methods to handle resonance conditions.

Contribution

It introduces a novel approach combining degree theory and homotopy to establish existence results for PDEs with exponential nonlinearities under resonance.

Findings

Existence of radial solutions under specified conditions.

Application of degree theory to nonlinear PDEs.

Use of homotopy to connect PDEs and derive uniform estimates.

Abstract

We have considered the following semi linear elliptic problem on the unit disk $B$ $-\Delta u = \lambda_1 u+e^u+f $ in $B$ with the Dirichlet boundary condition and $f$ satisfying the following condition : $f\in L^r(B)$, for some $r>2$ and $-\int_B f\phi_1<4\pi$. Where $\phi_1$ is the eigen function of $(-\Delta)$ corresponding to the first eigenvalue $\lambda_1$ in $H_0^1(B)$. We shall find the existence of a radial solution of this PDE. We shall use degree theory to get the existence starting from a suitable with known solution with its degree. Connecting those two PDE's by homotopy and getting the uniform estimate for the connecting PDE's we shall achieve our result.