Multiple solutions for an anisotropic elliptic equation of Kirchhoff type in bounded domain

Authors : Nguyen Thanh CHUNG

Pages : 116-127

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Publication Date : 2018-11-14

Article Type : Research Paper

Abstract :In this paper, we consider a class of anisotropic elliptic equations of Kirchhoff type  $$ \begin{cases} - M\leftinsert ignore into journalissuearticles values(\sum\limits_{i=1}^N\int_\Omega\frac{1}{p_iinsert ignore into journalissuearticles values(x);}|\partial_{x_i}u|^{p_iinsert ignore into journalissuearticles values(x);}\,dx\right);\sum\limits_{i=1}^N\partial_{x_i} \Biginsert ignore into journalissuearticles values(|\partial_{x_i}u|^{p_iinsert ignore into journalissuearticles values(x);-2}\partial_{x_i}u\Big); = finsert ignore into journalissuearticles values(x,u); + hinsert ignore into journalissuearticles values(x);, \quad x\in \Omega,\\ u  =  0, \quad x\in \partial\Omega, \end{cases} $$ where $\Omega \subset \R^N$ insert ignore into journalissuearticles values($N \geq 3$); is a bounded domain with smooth boundary $\partial\Omega$, $Minsert ignore into journalissuearticles values(t); = a+bt^\tau$, $\tau>0$ is a positive constant, and $p_i$, $i = 1, 2, ..., N$ are continuous functions on $\overline\Omega$ such that $2 \leq p_iinsert ignore into journalissuearticles values(x);0$, $b\geq 0$. Under appropriate assumptions on $f$ and $h$, we prove the existence of as least two weak solutions for the problem by using the Ekeland variational principle and the mountain pass theorem in critical point theory.
Keywords : Anisotropic elliptic equations, Kirchhoff type equations, Variable exponents, Variational methods

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