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Weak solutions for nonlinear Neumann boundary value problems with \(p(x)\)-Laplacian operators. (English) Zbl 1390.35098

Summary: We study the nonlinear Neumann boundary value problem with a \(p(x)\)-Laplacian operator \[ \begin{cases} \Delta_{p(x)}u + a(x)|u|^{p(x)-2}u = f(x,u)\qquad &\text{in}\; \Omega, \\ |\nabla u|^{p(x)-2} \frac{\partial u}{\partial\nu} = |u|^{q(x)-2}u + \lambda |u|^{w(x)-2}u &\text{on}\; \partial\Omega, \end{cases} \] where \(\Omega \subset \mathbb{R}^N\), with \(N \geq 2\), is a bounded domain with smooth boundary and \(q(x)\) is critical in the context of variable exponent \(p_*(x) = (N-1)p(x)/(N-p(x))\). Using the variational method and a version of the concentration-compactness principle for the Sobolev trace immersion with variable exponents, we establish the existence and multiplicity of weak solutions for the above problem.

MSC:

35J62 Quasilinear elliptic equations
35J92 Quasilinear elliptic equations with \(p\)-Laplacian
35J25 Boundary value problems for second-order elliptic equations
35J35 Variational methods for higher-order elliptic equations
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