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

Abstract

Many existence and nonexistence results are known for nonnegative radial solutions $u\in D^{1,2}(\mathbb{R}^{N})\cap L^{2}(\mathbb{R}^{N},\left|x\right| ^{-\alpha }dx)$ to the equation \[ -\triangle u+\dfrac{A}{\left| x\right| ^{\alpha }}u=f\left( u\right) \quad \textrm{in }\mathbb{R}^{N},\quad N\geq 3,\quad A,\alpha >0, \] with nonlinearites satisfying $\left| f\left( u\right) \right| \leq \left(\mathrm{const.}\right) u^{p-1}$ for some $p>2$. Existence of nonradial solutions, by contrast, is known only for $N\geq 4$, $\alpha =2$, $f\left( u\right) =u^{(N+2)/(N-2)}$ and $A$ large enough. Here we show that the equation has multiple nonradial solutions as $A\rightarrow +\infty$ for $N\geq 4$, $2/(N-1)<\alpha <2N-2$, $\alpha\neq 2$, and 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.