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arXiv · 1802.07289

Quasilinear Schrödinger-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 Schrödinger-Poisson system in a bounded domain in $\mathbb{R}^{2}$: $$ \left\{ \begin{array}[c]{ll} - Δu +ϕu = f(u) &\ \mbox{in } Ω, -Δϕ- \varepsilon^{4}Δ_4 ϕ= u^{2} & \ \mbox{in } Ω, u=ϕ=0 & \ \mbox{on } \partialΩ\end{array} \right. $$ depending on the parameter $\varepsilon>0$. The nonlinearity $f$ is assumed to have critical exponencial growth. We first prove existence of nontrivial solutions $(u_{\varepsilon}, ϕ_{\varepsilon})$ and then we show that as $\varepsilon\to0^{+}$ these solutions converges to a nontrivial solution of the associated Schrödinger-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 Schrödinger-Poisson system under an exponential critical nonlinearity: existence and asymptotic of solutions. https://arxiv.org/abs/1802.07289

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