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November/December 2017 Multiple solutions of a Kirchhoff type elliptic problem with the Trudinger-Moser growth
D. Naimen, C. Tarsi
Adv. Differential Equations 22(11/12): 983-1012 (November/December 2017).

Abstract

We consider a Kirchhoff type elliptic problem; \begin{equation*} \begin{cases} -\left(1+\alpha \int_{\Omega}|\nabla u|^2dx\right)\Delta u =f(x,u),\ u\ge0\text{ in }\Omega,\\ u=0\text{ on }\partial \Omega, \end{cases} \end{equation*} where $\Omega\subset \mathbb{R}^2$ is a bounded domain with a smooth boundary $\partial \Omega$, $\alpha > 0$ and $f$ is a continuous function in $\overline{\Omega}\times \mathbb{R}$. Moreover, we assume $f$ has the Trudinger-Moser growth. We prove the existence of solutions of (P), so extending a former result by de Figueiredo-Miyagaki-Ruf [11] for the case $\alpha =0$ to the case $\alpha>0$. We emphasize that we also show a new multiplicity result induced by the nonlocal dependence. In order to prove this, we carefully discuss the geometry of the associated energy functional and the concentration compactness analysis for the critical case.

Citation

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D. Naimen. C. Tarsi. "Multiple solutions of a Kirchhoff type elliptic problem with the Trudinger-Moser growth." Adv. Differential Equations 22 (11/12) 983 - 1012, November/December 2017.

Information

Published: November/December 2017
First available in Project Euclid: 1 September 2017

zbMATH: 1379.35122
MathSciNet: MR3692916

Subjects:
Primary: 35J20 , 35J25 , 35J62

Rights: Copyright © 2017 Khayyam Publishing, Inc.

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Vol.22 • No. 11/12 • November/December 2017
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