Advances in Differential Equations

Multiple solutions of a Kirchhoff type elliptic problem with the Trudinger-Moser growth

D. Naimen and C. Tarsi

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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.

Article information

Adv. Differential Equations, Volume 22, Number 11/12 (2017), 983-1012.

First available in Project Euclid: 1 September 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J20: Variational methods for second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations 35J62: Quasilinear elliptic equations


Naimen, D.; Tarsi, C. Multiple solutions of a Kirchhoff type elliptic problem with the Trudinger-Moser growth. Adv. Differential Equations 22 (2017), no. 11/12, 983--1012.

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