Infinitely many solutions for the Schrödinger equations in RNRN with critical growth

We consider the following nonlinear problem in RNRNequation(0.1){−Δu+V(|y|)u=uN+2N−2,u>0, in RN;u∈H1(RN), where V(r)V(r) is a bounded non-negative function, N⩾5N⩾5. We show that if r2V(r)r2V(r) has a local maximum point, or local minimum point r0>0r0>0 with V(r0)>0V(r0)>0, then (0.1) has infinitely many non-radial solutions, whose energy can be made arbitrarily large. As an application, we show that the solution set of the following problem−Δu=λu+uN+2N−2,u>0 on SN has unbounded energy, as long as λ<−N(N−2)4, N⩾5N⩾5.