Existence of infinitely many solutions for a Neumann problem involving the \(p(x)\)-Laplacian
The authors consider the Neumann problem involving the \(p(x)\)-Laplacian of the type \[ \begin{gathered} -\text{div}(|\nabla u|^{p(x)-2}\nabla u)+ \lambda(x)|u(x)|^{p(x)- 2} u= f(x,u)+ g(x,u)\text{ in }\Omega,\\ {\partial u\over\partial n}= 0\quad\text{on }\partial\Omega,\end{gathered}\tag{1} \] where \(p\in L^\infty(\Omega)\), \(\Omega\subset \mathbb{R}^d\) is a bounded open set with boundary of class \(C^1\), \(n\) is the outward unit normal to the boundary \(\partial\Omega\), and \(p> d\). Using a variational method (under some suitably assumptions on \(p\), \(f\), \(g\)), the authors prove the existence of infinitely many solutions for (1).
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