Two sequences of solutions for the semilinear elliptic equations with logarithmic nonlinearities

Abstract

We are interested in the following elliptic equation(0.1){−Δu=a(x)ulog⁡|u|,x∈Ω,u=0,on∂Ω, where Ω is a bounded domain of RN (N≥2) with smooth boundary ∂Ω, and a(x)∈C(Ω). The existence and multiplicity of solutions are obtained by using variational methods. Quite surprisingly, the existence of solutions is deeply influenced by the sign of a(x). More precisely,(i) if a(x)>0, equation (0.1) possesses a sequence of solutions whose energy and H01(Ω)-norms diverge to positive infinity;(ii) if a(x)<0, equation (0.1) possesses a sequence of solutions whose energy and H01(Ω)-norms converge to zero;(iii) if a(x) is sign-changing, equation (0.1) possesses two sequences of solutions: one sequence of solutions is with energy and H01(Ω)-norms diverging to positive infinity, while the other one is with energy and H01(Ω)-norms converging to zero.