Some Remarks on a Class of p(x)-Laplacian Robin Eigenvalue Problems

Abstract

We consider the p(x)-Laplacian Robin eigenvalue problem

$$\begin{aligned} \left\{ \begin{array}{ll} - \Delta _{p(x)}u = \lambda V(x) |u|^{q(x)-2}u, \quad x\in \Omega ,\\ |\nabla u|^{p(x)-2}\frac{\partial u}{\partial \nu }+\beta (x)|u|^{p(x)-2}u=0,\quad x\in \partial \Omega , \end{array}\right. \end{aligned}$$

where $\Omega$ is a bounded domain in $\mathbb {R}^N$ with smooth boundary $\partial \Omega$, $N\ge 2$, $\frac{\partial u}{\partial \nu }$ is the outer normal derivative of u with respect to $\partial \Omega$, $p,q\in C_+(\overline{\Omega })$, $1<p^-:= \inf _{x\in \overline{\Omega }}p(x) \le p^+:=\sup _{x\in \overline{\Omega }}p(x)<N$, $\beta \in L^\infty (\partial \Omega )$, $\beta ^-:=\inf _{x\in \partial \Omega }\beta (x)>0$, and $\lambda >0$ is a parameter. Under some suitable conditions on the functions q and V, we establish the existence of a continuous family of eigenvalues in a neighborhood of the origin using variational methods. The main results of this paper improve and generalize the previous ones introduced in Deng (J Math Anal Appl 360:548–560, 2009), Kefi (Zeitschrift für Analysis und ihre Anwendungen (ZAA) 37:25–38, 2018).