On the existence of a weak solution for some singular p ⁢ ( x ) p(x) -biharmonic equation with Navier boundary conditions

Abstract

Abstract In the present paper, we investigate the existence of solutions for the following inhomogeneous singular equation involving the {p(x)} -biharmonic operator: \left\{\begin{aligned} &\displaystyle\Delta(\lvert\Delta u\rvert^{p(x)-2}% \Delta u)=g(x)u^{-\gamma(x)}\mp\lambda f(x,u)&&\displaystyle\phantom{}\text{in% }\Omega,\\ &\displaystyle\Delta u=u=0&&\displaystyle\phantom{}\text{on }\partial\Omega,% \end{aligned}\right. where {\Omega\subset\mathbb{R}^{N}} ( {N\geq 3} ) is a bounded domain with {C^{2}} boundary, λ is a positive parameter, {\gamma:\overline{\Omega}\rightarrow(0,1)} is a continuous function, {p\kern-1.0pt\in\kern-1.0ptC(\overline{\Omega})} with {1\kern-1.0pt<\kern-1.0ptp^{-}\kern-1.0pt:=\kern-1.0pt\inf_{x\in\Omega}p(x)% \kern-1.0pt\leq\kern-1.0ptp^{+}\kern-1.0pt:=\kern-1.0pt\sup_{x\in\Omega}p(x)% \kern-1.0pt<\kern-1.0pt\frac{N}{2}} , as usual, {p^{*}(x)\kern-1.0pt=\kern-1.0pt\frac{Np(x)}{N-2p(x)}} , g\in L^{\frac{p^{*}(x)}{p^{*}(x)+\gamma(x)-1}}(\Omega), and {f(x,u)} is assumed to satisfy assumptions (f1)–(f6) in Section 3. In the proofs of our results, we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces. In addition, an example to illustrate our result is given.