On a class of singularly perturbed elliptic equations in divergence form: existence and multiplicity results

AbstractThe main purpose of this paper is to study the existence of single-peaked positive solutions of the singularly perturbed elliptic equation -ε2div(J(x)∇u)+V(x)u=f(u)inRN,where J is a symmetric uniformly elliptic matrix and V is a positive potential, possibly unbounded from above. If f(u)=up, then solutions concentrate at non-degenerate critical points of Γ(x)=V(x)p+1p-1-N2 (detJ(x))1/2