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.


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
Full Text: DOI


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