Multiple nonradial solutions for a nonlinear elliptic problem with singular and decaying radial potential

Abstract

Abstract Many existence and nonexistence results are known for nonnegative radial solutions to the equation -\triangle u+\frac{A}{|x|^{\alpha}}u=f(u)\quad\text{in }\mathbb{R}^{N},\,N\geq 3% ,\,A,\alpha>0,\,u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},|x|^{-% \alpha}\,dx), with the nonlinearities satisfying {|f(u)|\leq(\mathrm{const.})u^{p-1}} for some {p>2} . The existence of nonradial solutions, by contrast, is known only for {N\geq 4} , {\alpha=2} , {f(u)=u^{(N+2)/(N-2)}} and A large enough. Here we show that the above equation has multiple nonradial solutions as {A\rightarrow+\infty} for {N\geq 4} , {2/(N-1)<\alpha<2N-2} and {\alpha\neq 2} , with the nonlinearities satisfying suitable assumptions. Our argument essentially relies on the compact embeddings between some suitable functional spaces of symmetric functions, which yields the existence of nonnegative solutions of mountain-pass type, and the separation of the corresponding mountain-pass levels from the energy levels associated to radial solutions.