On a problem of resonance with exponential nonlinearity

Abstract

Consider the following semilinear elliptic problem on B={x∈R2:|x|<1}{−Δu=λ1u+eu+f,in Bu=0on ∂B with f satisfying the following condition: f is smooth integrable radial and satisfies0<−∫Bfϕ1<8π, where ϕ1 is the eigenfunction of (−Δ) corresponding to the first eigenvalue λ1 in H01(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 equation with known solution with its degree. Connecting those two PDEs by homotopy and getting the uniform estimate for the connecting PDEs we shall achieve our result.