Superlinear (p(z), q(z))-equations

ABSTRACT

We consider Dirichlet boundary value problems for equations involving the (p(z), q(z))-Laplacian operator in the principal part and prove the existence of one and three nontrivial weak solutions, respectively. Here, the nonlinearity in the reaction term is allowed to depend on the solution, but does not satisfy the Ambrosetti–Rabinowitz condition. The hypotheses on the reaction term ensure that the Euler–Lagrange functional, associated to the problem, satisfies both the -condition and a mountain pass geometry.

KEYWORDS:

• (p(z)
• q(z))-Laplacian operator
• -condition
• critical groups
• nonlinear regularity
• weak solution

AMS SUBJECT CLASSIFICATIONS:

• 35J20
• 35J92
• 58E05

Acknowledgements

The authors wish to thank the two knowledgeable referees for their corrections and helpful remarks.

Notes

No potential conflict of interest was reported by the authors.