| Summary: | In this paper we study the following nonlinear boundary-value problem \[-\Delta_{p(x)} u=\lambda f(x,u) \quad \text{ in } \Omega,\] \[|\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu}+\beta(x)|u|^{p(x)-2}u=\mu g(x,u) \quad \text{ on } \partial\Omega,\] where \(\Omega\subset\mathbb{R}^N\) is a bounded domain with smooth boundary \(\partial\Omega\), \(\frac{\partial u}{\partial\nu}\) is the outer unit normal derivative on \(\partial\Omega\), \(\lambda, \mu\) are two real numbers such that \(\lambda^{2}+\mu^{2}\neq0\), \(p\) is a continuous function on \(\overline{\Omega}\) with \(\inf_{x\in \overline{\Omega}} p(x)\gt 1\), \(\beta\in L^{\infty}(\partial\Omega)\) with \(\beta^{-}:=\inf_{x\in \partial\Omega}\beta(x)\gt 0\) and \(f : \Omega\times\mathbb{R}\rightarrow \mathbb{R}\), \(g : \partial\Omega\times\mathbb{R}\rightarrow \mathbb{R}\) are continuous functions. Under appropriate assumptions on \(f\) and \(g\), we obtain the existence and multiplicity of solutions using the variational method. The positive solution of the problem is also considered.
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