Anisotropic \((p, q)\)-equations with superlinear reaction (Q6601218)

This paper discusses the anisotropic \((p,q)\) equation \(-\Delta_{p(z)} u-\Delta_{q(z)} u=f(z, u)\) in \(\Omega\), subject to an homogeneous Dirichlet boundary condition. In this context, \(\Omega\) is a smooth and bounded domain, \(\Delta_p\), \(\Delta_q\) stand for the variable exponents \(p\) and \(q\) Laplace operator respectively, \(f(z,u)\) is a Caratheodory function which is superlinear in some respect but does not satisfy the Ambrosetti-Rabinowitz condition. The main result of the article establishes the existence of at least five weak solutions with a precise sign information. The approach relies on a number of methods ranging from critical point theory, truncation and comparison techniques as well as critical group theory.