On the existence of solutions for a class of fourth order elliptic equations of Kirchhoff type with variable exponent

Authors : Nguyen Thanh CHUNG

Pages : 35-45

Doi:10.31197/atnaa.495567

View : 9 | Download : 4

Publication Date : 2019-03-31

Article Type : Research Paper

Abstract :In this paper, we consider a class of fourth order elliptic equations of Kirchhoff type with variable exponent $$ \left\{\begin{array}{ll} \Delta^2_{pinsert ignore into journalissuearticles values(x);}u-M\leftinsert ignore into journalissuearticles values(\int_\Omega\frac{1}{pinsert ignore into journalissuearticles values(x);}|\nabla u|^{pinsert ignore into journalissuearticles values(x);}\,dx\right);\Delta_{pinsert ignore into journalissuearticles values(x);} u  = \lambda finsert ignore into journalissuearticles values(x,u); \quad \text{ in }\Omega,\\ u=\Delta u = 0 \quad \text{ on } \partial\Omega,  \end{array}\right. $$ where $\Omega \subset \R^N$, $N \geq 3$, is a smooth bounded domain, $Minsert ignore into journalissuearticles values(t);=a+bt^\kappa$, $a, \kappa>0$, $b \geq 0$, $\lambda$ is a positive parameter, $\Delta_{pinsert ignore into journalissuearticles values(x);}^2u=\Delta insert ignore into journalissuearticles values(|\Delta u|^{pinsert ignore into journalissuearticles values(x);-2} \Delta u);$ is the operator of fourth order called the $pinsert ignore into journalissuearticles values(x);$-biharmonic operator, $\Delta_{pinsert ignore into journalissuearticles values(x);}u = \operatorname{div} \leftinsert ignore into journalissuearticles values(|\nabla u|^{pinsert ignore into journalissuearticles values(x);-2}\nabla u\right);$ is the $pinsert ignore into journalissuearticles values(x);$-Laplacian, $p:\overline\Omega \to \R$ is a log-H\`{o}lder continuous function and $f: \overline\Omega\times \R\to \R$ is a continuous function satisfying some certain conditions. Using Ekeland`s variational principle combined with variational techniques, an existence result is established in an appropriate function space.
Keywords : Fourth order elliptic equations, Kirchhoff type problems, Variable exponents, Ekeland`s variational principle