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Existence results for degenerate \(p(x)\)-Laplace equations with Leray-Lions type operators. (English) Zbl 1373.35123
The paper deals with the Dirichlet problem \[ \begin{cases} -\text{div\,}a(x,\nabla u)=\lambda f(x,u) & \text{in}\;\Omega,\\ u=0 & \text{on}\;\partial\Omega, \end{cases} \] where \(\Omega\subset \mathbb{R}^N\) is a bounded domain with Lipschitz smooth boundary, \(a\) and \(f\) are Carathéodory functions and \(\lambda\) is a positive parameter. The differential operator is modeled on the \(p(x)\)-Laplacian \(\text{div\,}\big(\omega(x)|\nabla u|^{p(x)-2}\nabla u\big)\) with \(p\in C(\overline{\Omega},(1,\infty))\) and where \(\omega\) is a suitable measurable and positive function in \(\Omega.\)
By means of standard techniques based on direct methods and critical point theories in the Calculus of Variations, the authors prove existence and multiplicity of solutions to the above problem when the differential operator is monotone and the nonlinearity is nonincreasing.

35J60 Nonlinear elliptic equations
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