The behavior of solutions of a parametric weighted (p, q)-laplacian equation

We study the behavior of solutions for the parametric equation (Formula presented), under Dirichlet condition, where (Formula presented) is a bounded domain with a C2-boundary (Formula presented) are weighted versions of p-Laplacian and q-Laplacian. We prove existence and nonexistence of nontrivial solutions, when f (z, x) asymptotically as x → ±∞ can be resonant. In the studied cases, we adopt a variational approach and use truncation and comparison techniques. When λ is large, we establish the existence of at least three nontrivial smooth solutions with sign information and ordered. Moreover, the critical parameter value is determined in terms of the spectrum of one of the differential operators.