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

• Main Author: Nguyen Thanh Chung
• Format: Article
• Language: English
• Published: Erdal KARAPINAR 2018-12-01
• Series: Results in Nonlinear Analysis
• Subjects: anisotropic elliptic equations; kirchhoff type equations; variable exponents; variational methods
• Online Access: http://static.dergipark.org.tr/article-download/23eb/d64d/1587/5c18122d2e27c.pdf?

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≥3) is a bounded domain with smooth boundary ∂Ω, M(t)=a+bt^τ, τ>0 is a positive constant, and p_i​ , i=1,2,...,N are continuous functions on ¯Ω​​ such that 2≤p_i(x)<N, a>0, b≥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.