On a Class of Elliptic (p, q)-Kirchhoff Type Systems with Multiple Parameters

Abstract

This article concerns the existence of positive solutions for elliptic (p, q)-Kirchhoff type systems with multiple parameters. Our approach is based on the method of sub and super-solutions. The concepts of sub- and super-solution were introduced by Nagumo (Proc Phys-Math Soc Jpn19:861–866, 1937) in 1937 who proved, using also the shooting method, the existence of at least one solution for a class of nonlinear Sturm-Liouville problems. In fact, the premises of the sub- and super-solution method can be traced back to Picard. He applied, in the early 1880s, the method of successive approximations to argue the existence of solutions for nonlinear elliptic equations that are suitable perturbations of uniquely solvable linear problems. This is the starting point of the use of sub- and super-solutions in connection with monotone methods. Picard’s techniques were applied later by Poincare (J Math Pures Appl 4:137–230, 1898) in connection with problems arising in astrophysics. We refer to Rădulescu (Qualitative analysis ofnonlinear elliptic partial differential equations: monotonicity, analytic, and variational methods, contemporary mathematics and its applications, Hindawi Publishing Corporation, New York, 2008).