Singular (p,q)-equations with competing perturbations

Abstract

We consider a nonlinear Dirichlet problem driven by the $(p,q)$-Laplacian and with a reaction involving the combined effects of a singular term, of a concave term and of a parametric convex term. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter $\lambda >0$ varies. Also, we show that for every admissible parameter $\lambda >0$, the problem has a minimal positive solution $u_{\lambda }^{\ast }$ and we establish the monotonicity and continuity properties of the map $\lambda \to u_{\lambda }^{\ast }$.

Keywords:

• $(p,q)$-Laplacian
• concave and convex nonlinearities
• trunctions
• comparisons
• nonlinear regularity
• minimal positive solutions
• Dirichlet problem

2010 Mathematics Subject Classifications:

• Primary 35J20
• 35J75
• Secondary 35J92

Acknowledgments

The authors wish to thank a knowledgeable referee for his/her remarks. This work was supported by the National Natural Science Foundation of China (No. 12071098).

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by National Natural Science Foundation of China [12071098].