Singularly perturbed elliptic equations with symmetry: existence of solutions concentrating on spheres, part I

We deal with the existence of positive radial solutions concentrating on spheres to a class of singularly perturbed elliptic problems like −ε2u + V (|x|)u = up, u ∈ H1(Rn). Under suitable assumptions on the auxiliary potential M(r) = rn−1V θ (r), θ (p +1)/(p −1)−1/2, we provide necessary and sufficient conditions for concentration as well as the bifurcation of non-radial solutions.