Mountain pass solution for the weighted Dirichlet ( p ( z ) , q ( z ) ) $(p(z),q(z))$ -problem

Abstract

Abstract We consider the Dirichlet boundary value problem for equations involving the ( p ( z ) , q ( z ) ) $(p(z),q(z))$ -Laplacian operator in the principal part on an open bounded domain Ω ⊂ R n $\Omega \subset \mathbb{R}^{n}$ . Here, the p ( z ) $p(z)$ -Laplacian is weighted by a function a ∈ L ∞ ( Ω ) + $a \in L^{\infty}(\Omega )_{+}$ , and the nonlinearity in the reaction term is allowed to depend on the solution without imposing the Ambrosetti–Rabinowitz condition. The proof of the existence of solution to our problem is based on a mountain pass critical point approach with the Cerami condition at level c.

Keywords

• ( p ( z ) , q ( z ) ) $(p(z),q(z))$ -Laplacian operator
• ( C c ) $(C_{c})$ -condition
• Nonlinear regularity
• Weak solution