For a class of \(p(x)\)-biharmonic operators with weights (Q2314682)

The paper is concerned with the fourth order nonlinear problem with \(p(x)\)-biharmonic operators \[ \begin{cases} \Delta(|\Delta u|^{p_1(x)-2}\Delta u)+\Delta (|\Delta u|^{p_2(x)-2}\Delta u)\\ \qquad =\lambda V_1(x)|u|^{q(x)-2}u-\mu V_2(x)|u|^{\alpha(x)-2}u,\,\,\,x\in\Omega, \\ u=\Delta u=0,\,\,\,x\in\partial \Omega, \end{cases}\tag{1} \] where \(\Omega\) is a bounded domain of \(\mathbb{R}^N\) with smooth boundary (\(N\ge 2\)), \(\lambda\) and \(\mu\) are positive real numbers, \(p_1,p_2,q\) and \(\alpha\) are continuous functions defined on \(\overline{\Omega}\), the functions \(V_1\) and \(V_2\) are given weight functions in generalized Lebesgue spaces, \(V_1\) may change sign in \(\Omega\) and \(V_2\ge 0\) on \(\Omega\). By using variational approaches and the Ekeland variational principle, the authors investigate the existence of positive weak solutions of problem \((1)\) when \(\lambda\) belongs to some intervals.