On a problem of resonance with exponential non linearity

We have considered the following semi linear elliptic problem on the unit disk $B$ $-Δu = λ_1 u+e^u+f$ in $B$ with the Dirichlet boundary condition and $f$ satisfying the following condition : $f\in L^r(B)$, for some $r>2$ and $-\int_B fϕ_1<4π$. Where $ϕ_1$ is the eigen function of $(-Δ)$ corresponding to the first eigenvalue $λ_1$ 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.

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