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

TL;DR

This paper proves the existence of multiple nonradial solutions for a nonlinear elliptic PDE with singular and decaying radial potential, extending previous results to new parameter ranges and nonlinearities.

Contribution

It demonstrates the existence of multiple nonradial solutions for the PDE under broader conditions, especially for large A and specific alpha ranges, using variational methods.

Findings

Multiple nonradial solutions exist as A approaches infinity.

Solutions are of mountain-pass type and distinct from radial solutions.

Results extend known existence theorems to new parameter regimes.

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.