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

Year 2018, Volume: 1 Issue: 3, 116 - 127, 14.11.2018

Nguyen Thanh Chung

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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