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Existence of positive solutions for a class of p-Laplacian superlinear semipositone problems

Part of: Elliptic equations and systems

Published online by Cambridge University Press:  21 July 2015

M. Chhetri ,
P. Drábek  and
R. Shivaji
M. Chhetri
Affiliation:
Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC 27402, USA, (maya@uncg.edu)
P. Drábek
Affiliation:
KMA-FAV, Západočeská Univerzita v Plzni, Univerzitní 22, 30614 Plzeň, Czech Republic, (pdrabek@kma.zcu.cz)
R. Shivaji
Affiliation:
Department of Mathematics and Statistics, The University of North Carolina at Greensboro, Greensboro, NC 27402, USA, (shivaji@uncg.edu)
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Abstract

We consider a quasilinear elliptic problem of the form

where λ > 0 is a parameter, 1 < p < 2 and Ω is a strictly convex bounded domain in ℝN, N > p, with C2 boundary ∂Ω. The nonlinearity f : [0, ∞) → ℝ is a continuous function that is semipositone (f(0) < 0) and p-superlinear at infinity. Using degree theory, combined with a rescaling argument and uniform L∞a priori bound, we establish the existence of a positive solution for λ small. Moreover, we show that there exists a connected component of positive solutions bifurcating from infinity at λ = 0. We also extend our study to systems.

Keywords

p-Laplaciansystemssemipositonesuperlinearpositive solutions

MSC classification

Primary: 35J60: Nonlinear elliptic equations
Secondary: 35J25: Boundary value problems for second-order elliptic equations 35J65: Nonlinear boundary value problems for linear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 145 , Issue 5 , October 2015 , pp. 925 - 936
Copyright
Copyright © Royal Society of Edinburgh 2015 

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