On a problem of resonance with exponential non linearity
  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. 
B. B. Manna, P. N. Srikanth
math.AP