## Solutions for Neumann boundary value problems involving $$p(x)$$-Laplace operators.(English)Zbl 1158.35046

If $$N\in\mathbb{N}$$ and $$\Omega\subseteq\mathbb{R}^N$$ is a bounded domain with smooth boundary, consider the problem
$\begin{cases} -\text{div}(| \nabla u| ^{p(x)-2}\nabla u) +| u| ^{p(x)-2}u=\lambda f(x,u),&\text{in }\Omega,\\ | \nabla u| ^{p(x)-2}\frac{\partial u}{\partial\nu}= \mu g(x,u),&\text{on }\partial\Omega, \end{cases}\tag{1}$
where $$p\in C(\overline{\Omega})$$, $$p>1$$ in $$\overline{\Omega}$$, $$\lambda,\mu\in\mathbb{R}$$, and $$\lambda^2+\mu^2>0$$. Moreover, throughout the paper $$f$$ and $$g$$ denote Caratheodory functions with subcritical growth in a suitable sense, with respect to the variable exponent $$p(x)$$. The author proves existence results that are analogues of classical results for the case $$p\equiv2$$, using direct and minimax methods in a variational setting. The analogues for the following settings are covered: Sublinear and superlinear nonlinearities, superlinear odd nonlinearities, and odd concave-convex nonlinearities. Also the existence of a positive solution is considered in some of these cases.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J60 Nonlinear elliptic equations 35J70 Degenerate elliptic equations 35J20 Variational methods for second-order elliptic equations
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### References:

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