Existence and asymptotic behavior of solutions to eigenvalue problems for Schrodinger-Bopp-Podolsky equations

Abstract

We study the existence and multiplicity of solutions for the Schrodinger-Bopp-Podolsky system $$\displaylines{ -\Delta u + \phi u = \omega u \quad\text{ in } \Omega \cr a^2\Delta^2\phi-\Delta \phi = u^2 \quad\text{ in } \Omega \cr u=\phi=\Delta\phi=0\quad\text{ on } \partial\Omega \cr \int_{\Omega} u^2\,dx =1 }$$ where \(\Omega\) is an open bounded and smooth domain in \(\mathbb R^{3}\), \(a>0 \) is the Bopp-Podolsky parameter. The unknowns are \(u,\phi:\Omega\to \mathbb R\) and \(\omega\in\mathbb R\). By using variational methods we show that for any \(a>0\) there are infinitely many solutions with diverging energy and divergent in norm. We show that ground states solutions converge to a ground state solution of the related classical Schrodinger-Poisson system, as \(a\to 0\). For more information see https://ejde.math.txstate.edu/Volumes/2023/66/abstr.html

Existence of solutions to quasilinear Schrodinger equations with exponential nonlinearity

“…which has been studied by several researchers, see for example [1,2,5,12,14], [16] for a problem in a bounded domain. Usually, to obtain the existence and multiplicity of solutions, the authors require a restriction on the norm of the term h(x) and thus (1.2) is regarded as a perturbation of the equation −∆u + V (x)u = f (x, u), x ∈ R 2 .…”

Critical points at prescribed energy level for Schrödinger-Bopp-Podolsky systems

“…We are just interested here in showing existence of solutions for the above system under suitable condition which were never considered before in the literature. For recent results on this kind of physical system we refer the reader to [1,4,8,11,12].…”