Teresa Megan Tyler | Searcharxiv

arXiv2021

Groundstates and infinitely many high energy solutions to a class of nonlinear Schrödinger-Poisson systems

We study a nonlinear Schrödinger-Poisson system which reduces to the nonlinear and nonlocal equation \[- Δu+ u + λ^2 \left(\frac{1}{ω|x|^{N-2}}\star ρu^2\right) ρ(x) u = |u|^{q-1} u \quad x \in \mathbb R^N, \] where $ω= (N-2)|\mathbb{S}^{N-1}|,$ $λ>0,$ $q\in(2,2^{\ast} -1),$ $ρ:\mathbb R^N \to \mathbb R$ is nonnegative and locally bounded, $N=3,4,5$ and $2^*=2N/(N-2)$ is the critical Sobolev exponent. We prove existence and multiplicity of solutions working on a suitable finite energy space and under two separate assumptions which are compatible with instances where loss of compactness phenomena may occur.

math.AP↗

arXiv2018

On a class of nonlinear Schrödinger-Poisson systems involving a nonradial charge density

In the spirit of the classical work of P. H. Rabinowitz on nonlinear Schrödinger equations, we prove existence of mountain-pass solutions and least energy solutions to the nonlinear Schrödinger-Poisson system \begin{equation}\nonumber \left\{\begin{array}{lll} - Δu+ u + ρ(x) ϕu = |u|^{p-1} u, \qquad &x\in \mathbb R^3, \,\,\, -Δϕ=ρ(x) u^2,\ & x\in \mathbb R^3, \end{array} \right. \end{equation} under different assumptions on $ρ: \mathbb R^3\rightarrow \mathbb R_+$ at infinity. Our results cover the range $p\in(2,3)$ where the lack of compactness phenomena may be due to the combined effect of the invariance by translations of a `limiting problem' at infinity and of the possible unboundedness of the Palais-Smale sequences. Moreover, we find necessary conditions for concentration at points to occur for solutions to the singularly perturbed problem \begin{equation}\nonumber \left\{\begin{array}{lll} - ε^2Δu+ u + ρ(x) ϕu = |u|^{p-1} u, \qquad &x\in \mathbb R^3, \,\,\, -Δϕ=ρ(x) u^2,\ & x\in \mathbb R^3, \end{array} \right. \end{equation} in various functional settings which are suitable for both variational and perturbation methods.

math.AP↗