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A variational approach to superlinear semipositone elliptic problems


Authors: David G. Costa, Humberto Ramos Quoirin and Hossein Tehrani
Journal: Proc. Amer. Math. Soc. 145 (2017), 2661-2675
MSC (2010): Primary 35J15, 35J20, 35J61, 35J91
DOI: https://doi.org/10.1090/proc/13426
Published electronically: December 15, 2016
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Abstract: In this paper we present a variational approach to a class of elliptic problems with superlinear semipositone nonlinearities. We consider the parametrized family of problems

$\displaystyle \left \{ \begin {array}{lll} -\Delta u =\lambda a(x)(f(u)-l)& {\rm in } & \Omega ,\\ u = 0 & {\rm on } & \partial \Omega , \end{array}\right . $

with $ l>0$, $ a$ continuous, and $ f$ subcritical and superlinear at infinity. We obtain positive solutions of such problems for $ 0< \lambda < \lambda _0$ by combining a suitable rescaling with a continuity argument. In doing so, we require $ f$ to be of regular variation at infinity, so that $ f$ does not need to be asymptotic to a power. Furthermore, $ a$ may vanish in open parts of $ \Omega $ or change sign.

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Additional Information

David G. Costa
Affiliation: Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, Nevada 89154-4020
Email: costa@unlv.nevada.edu

Humberto Ramos Quoirin
Affiliation: Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
Email: humberto.ramos@usach.cl

Hossein Tehrani
Affiliation: Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, Nevada 89154-4020
Email: tehranih@unlv.nevada.edu

DOI: https://doi.org/10.1090/proc/13426
Keywords: Semipositone problem, superlinear problem, variational methods, indefinite problem, regular variation
Received by editor(s): June 22, 2016
Received by editor(s) in revised form: August 1, 2016
Published electronically: December 15, 2016
Additional Notes: The second author was supported by the FONDECYT project 1161635
Communicated by: Joachim Krieger
Article copyright: © Copyright 2016 American Mathematical Society