Positive solutions for $(p,2)$-equations with superlinear reaction and a concave boundary term

Abstract

We consider a nonlinear boundary value problem driven by the $(p,2)$-Laplacian, with a $(p-1)$-superlinear reaction and a parametric concave boundary term (a "concave-convex" problem). Using variational tools (critical point theory) together with truncation and comparison techniques, we prove a bifurcation type theorem describing the changes in the set of positive solutions as the parameter $\lambda>0$ varies. We also show that for every admissible parameter $\lambda>0$, the problem has a minimal positive solution $\overline{u}_\lambda$ and determine the monotonicity and continuity properties of the map $\lambda \mapsto \overline{u}_\lambda$.

Keywords

• concave boundary term
• superlinear reaction
• $(p
• 2)$-laplacian
• nonlinear regularity
• nonlinear maximum principle
• positive solutions