Eigenvalues of the \(p(x)\)-Laplacian Neumann problems (Q2381479)

The eigenvalues of the \(p(x)\)-Laplacian operator with zero Neumann boundary condition on a bounded domain are studied. Here \(p(x)>1\) is a continuous function defined on the domain \(\Omega\). The study is new. It is based on the properties of the set \(V_{p(x)}=\{u\in W^{1,p(x)}(\Omega ):\int_{\Omega }| v(x)| ^{p(x)-2}v(x)\,dx=0\}\) presented in Section 2. Main results on eigenvalues of the problem are given in Section 3 using variational method and Lusternik-Shnirelman theorem.