Positive solutions for \((p, q)\)-equations with convection and a sign-changing reaction
The paper is concerned with positive solutions to a gradient dependent Dirichlet problem with \((p,q)\)-Laplacian, of type \[ \left\{ \begin{array}{ll} -\Delta_p u - \Delta_q u= f(z,u, Du) \quad \hbox{ in \(\Omega\)},\\ u=0 \quad \hbox{ on } \partial \Omega, \end{array} \right. \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\). Here, among other conditions, the Carathéodory function \(f(z,x,y):\Omega \times \mathbb{R} \times \mathbb{R}^N \to \mathbb{R}\) satisfies a Nagumo-Hartman condition and a nonuniform nonresonance condition at zero; it is not assumed that \(f\) is nonnegative. In the main result, the authors obtain the existence of a smooth positive solution. In the approach, the \(y\)-variable is firstly fixed and a problem with variational structure is treated, then the conclusion is inferred by the a priori estimate method applied to the minimal solution map.
scientific article; zbMATH DE number 7394297
scientific article