On a problem of resonance with exponential non linearity

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 $-{\int }_{B}f{\varphi }_1<4\pi$. Where ${\varphi }_1$ 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.