Existence results for a class of nonlocal problems involving \(p(x)\)-Laplacian
The author finds sufficient conditions so that the boundary value problems below admit nontrivial weak solutions. The equations considered are \[ -M \Delta_{p(x)} u = \lambda \left( a(x)|u|^{q(x)-2}u+b(x) |u|^{r(x)-2}u\right) \;\text{ in } \Omega, \;\;\;u = 0 \text{ on } \partial \Omega. \] Here \(\Delta_{p(x)} u \) is the variable exponent \(p(x)\)-Laplacian, and \[ M = M\left( \int_\Omega \frac{1}{p(x)}|\nabla u|^{p(x)} \;dx\right) \] with \(M : \mathbb{R}^+\to \mathbb{R}\) a function such as for example \(M(t) = t\arctan(t)\). The existence of weak solutions is established via the Mountain Pass Theorem.
scientific article; zbMATH DE number 6419560