Concentrating solutions for a class of indefinite Schrödinger-Poisson systems with doubly critical growth - Shen, Liejun; Squassina, Marco (Topological Methods in Nonlinear Analysis, 2025)
We study the following Schrödinger-Poisson system involving critical nonlocal term with indefinite steep potential well$$\begin{cases} -\Delta u+(\lambda V(x)-\mu)u-\phi |u|^3u= f(u), & x\in\R^3, \\ -\Delta \phi= |u|^5, & x\in\R^3,\end{cases}$$%where $\lambda> 0$ is a parameter, $V\in \mathcal{C}\big(\R^3,\R^+\big)$ admits a potential well$\Omega \triangleq\text{int}V^{-1}(0)$,and $\mu> \mu_1$ is a constant suchthat the operator $L_\lambda \triangleq -\Delta + \lambda V-\mu$ is non-degenerate when$\lambda$ is large enoughwith $\{\mu_j\}_{j=1}^\infty$ denoting theDirichlet eigenvalues of $\big(-\Delta,H_0^1(\Omega)\big)$.If $f$ satisfies some suitable assumptions involving critical growth,with the help of a linking-type result involving the modifiedPankov-Nehari manifold procedure,we establish the existence and concentrating behaviorof positive solutions for the given system using variational methods.