Eigenvalues of the \(p(x)\)-Laplacian Steklov problem (Q2465854)

The Steklov eigenvalue problem involving the \(p(x)\)-Laplacian in a bounded open set \(\Omega\subset \mathbb{R}^N,\;N\geq 2\) is considered. The existence of infinitely many eigenvalue sequences is proved. Unlike the \(p\)-Laplacian case, for a variable exponent \(p(x)\) there does not exist a principle eigenvalue and the set of all eigenvalues is not closed. Sufficient conditions are given in order the infimum of all eigenvalues to be zero or positive.