## 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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### References:

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