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 \[ -\triangle u+\dfrac{A}{\left| x\right| ^{α}}u=f\left( u\right) \quad \textrm{in }\mathbb{R}^{N},\quad N\geq 3,\quad A,α>0, \] with nonlinearites satisfying $\left|f\left(u\right)\right|\le \left(const.\right)u^{p-1}$ for some $p>2$. Existence of nonradial solutions, by contrast, is known only for $N\ge 4$, $\alpha =2$, $f\left(u\right)=u^{(N+2)/(N-2)}$ and $A$ large enough. Here we show that the equation has multiple nonradial solutions as $A\to +\infty$ for $N\ge 4$, $2/(N-1)<\alpha <2N-2$, $\alpha \ne 2$, 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.

References in corpus (2)

• Compactness and existence results for the $p$-Laplace equation
• Nonlinear Schrödinger equations without compatibility conditions on the potentials

Cited by in corpus (2)

• Superlinear perturbations of the eigenvalue problem for the Robin Laplacian plus an indefinite and unbounded potential
• Singular solutions for coercive quasilinear elliptic inequalities with nonlocal terms