Existence of solutions for nonlocal elliptic systems involving \(p(x)\)-Laplace operator
The authors of the paper study the existence and the multiplicity of solutions of the following nonlocal elliptic systems \[-M\left( \int_{\Omega}\frac{1}{p(x)}\left\vert \Delta u\right\vert ^{p(x)}\,dx\right) \Delta_{p(x)}u=a(x)\left\vert u\right\vert ^{p(x)-2}\text{in }\Omega, \] \[\left\vert \nabla u\right\vert ^{p(x)-2}\frac{\partial u}{\partial v}=\lambda f(x,u)\text{ in }\partial \Omega ,\] where \(\Omega \)\ is a bounded smooth domain of \(\mathbb{R}^{n}\), \(v\)\ is the outer unit normal to \(\partial \Omega ,\) \(\lambda \) is a positive scalar, \(\Delta_{p(x)}\)\ denotes the \(p(x)\)-Laplace operator with \(p\in C(\overline{\Omega}),\) \(f:\partial \Omega \times \mathbb{R}\rightarrow \mathbb{R}\)\ is a Carathéodory function, \(a\in L^{\infty}(\Omega)\)\ with \(\operatorname{essinf}_{\Omega}a>0\) and \(M\in C(\mathbb{R})\). The results proposed by the authors are obtained in Sobolev spaces with variable exponent under suitable hypothesis on the given functions \(p\), \(f\), \(a\) and \(M\).
scientific article; zbMATH DE number 7479506
scientific article