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

Abstract

In this paper, we consider a class of anisotropic elliptic equations of Kirchhoff type 

$$

\begin{cases}

- M\left(\sum\limits_{i=1}^N\int_\Omega\frac{1}{p_i(x)}|\partial_{x_i}u|^{p_i(x)}\,dx\right)\sum\limits_{i=1}^N\partial_{x_i} \Big(|\partial_{x_i}u|^{p_i(x)-2}\partial_{x_i}u\Big) = f(x,u) + h(x), \quad x\in \Omega,\\

u  =  0, \quad x\in \partial\Omega,

\end{cases}

$$

where $\Omega \subset \R^N$ ($N \geq 3$) is a bounded domain with smooth boundary $\partial\Omega$, $M(t) = a+bt^\tau$, $\tau>0$ is a positive constant, and $p_i$, $i = 1, 2, ..., N$ are continuous functions on $\overline\Omega$ such that $2 \leq p_i(x)<N$, $a>0$, $b\geq 0$. 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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