Existence and multiplicity of solutions for resonant-superlinear problems

ABSTRACT

We consider a semilinear Dirichlet problem with a reaction which exhibits an asymmetric behaviour as $x\to \pm \mathrm{\infty }$ (it is superlinear as $x\to +\mathrm{\infty }$ and resonant as $x\to -\mathrm{\infty }$). Using variational tools from the critical point theory, together with truncation and comparison techniques and critical groups, we prove two multiplicity theorems producing two and three nontrivial solutions. Also we show that the problem can not have negative solutions.

KEYWORDS:

• Asymmetric reaction
• regularity theory
• Hopf maximum principle
• critical groups
• solutions of constant sign

AMS SUBJECT CLASSIFICATIONS:

• 35J20
• 35J60
• 58E05

Acknowledgments

The authors thank the anonymous reviewers for their careful reading of the manuscript and their comments and suggestions. All authors contribute equally to this paper. All authors read and approved the final manuscript.

Disclosure statement

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