Abstract
Abstract We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction which is dependent on the gradient. We look for positive solutions and we do not assume that the reaction is nonnegative. Using a mixture of variational and topological methods (the "frozen variable" technique), we prove the existence of a positive smooth solution.
Highlights
Let Ω ⊂ RN be a bounded domain with a C -boundary ∂Ω
We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction which is dependent on the gradient
We look for positive solutions and we do not assume that the reaction is nonnegative
Summary
Let Ω ⊂ RN be a bounded domain with a C -boundary ∂Ω. In this paper, we study the following (p, q)equation with convection. Papageorgiou, Positive solutions for (p, q)-equations negative and two nodal (sign changing)) and their method is based on variational methods together with truncation and comparison techniques and Morse theory (critical groups); and Marano-Winkert [24] dealt with the existence of solutions to a class of quasilinear elliptic problems involving a general nonhomogeneous di erential operator, and two Carathéodory functions f and g, in which the function f was considered to depend on the gradient term of the unknown function. We solve this variational problem and we show that we can choose in a canonical way such a solution This way we have a well-de ned map which we show that it satis es the requirements of the Leray-Schauder Alternative Principle. We obtain a xed point for the solution map and this xed point is a solution of (1.1)
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