\((p, q)\)-Laplacian problems with parameters in bounded and unbounded domains (Q6589529)

This article is concerned with the existence and nonexistence of positive solutions for differential equations of the type \(-\Delta_p u-\Delta_q u=\lambda f(x,u)+\mu g(x, u)\) in bounded and unbounded domains. The nonlinearities \(f\) and \(g\) satisfy some growth conditions such as \(|f(x,u)|\leq C|u|^{p-1}\), \(|g(x,u)|\leq Cu^{q-1}\). Under some extra conditions involving \(\lambda\) and \(\mu\), the authors derive estiatence and nonexistence of positiove solutions. The approach is variational and relies on critical point theorems with Pale-Smale condition.