Positive solutions of indefinite equations with p-Laplacian and supercritical nonlinearity

Abstract

This article is devoted to the study of the Dirichlet boundary problem for equations with p-Laplacian

where n ≥ 1, 1 < p < q < +∞, and a(·) may change the sign in Ω. In the main result, we establish necessary and sufficient conditions when the problem possesses a weak positive solution with λ ∈ [λ₁, λ*) where λ₁ > 0 is the first eigenvalue of the Dirichlet p-Laplacian and λ* is an extremal point.

Keywords:

• supercritical exponent
• indefinite sign nonlinearity
• extremal point
• super- and sub-solutions

AMS Subject Classifications::

• 35B50
• 35J25
• 35J65
• 47A75
• 49R50

Acknowledgement

This research was supported by grants nos. DFG 436RUS17/59/05, RFBR 08-01-00441-a, 08-01-97020-.