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arXiv · 1802.07289

Quasilinear Schr\"odinger-Poisson system under an exponential critical nonlinearity: existence and asymptotic of solutions

Giovany M. Figueiredo·Gaetano Siciliano
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Abstract

In this paper we consider the following quasilinear Schr\"odinger-Poisson system in a bounded domain in $\mathbb{R}^{2}$: $$ \left\{ \begin{array}[c]{ll} - \Delta u +\phi u = f(u) &\ \mbox{in } \Omega, -\Delta \phi - \varepsilon^{4}\Delta_4 \phi = u^{2} & \ \mbox{in } \Omega, u=\phi=0 & \ \mbox{on } \partial\Omega \end{array} \right. $$ depending on the parameter $\varepsilon>0$. The nonlinearity $f$ is assumed to have critical exponencial growth. We first prove existence of nontrivial solutions $(u_{\varepsilon}, \phi_{\varepsilon})$ and then we show that as $\varepsilon\to0^{+}$ these solutions converges to a nontrivial solution of the associated Schr\"odinger-Poisson system, that is by making $\varepsilon=0$ in the system above.

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Giovany M. Figueiredo, Gaetano Siciliano. 2018-02-20. Quasilinear Schr\"odinger-Poisson system under an exponential critical nonlinearity: existence and asymptotic of solutions. https://arxiv.org/abs/1802.07289

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