A variational approach to superlinear semipositone elliptic problems
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- by David G. Costa, Humberto Ramos Quoirin and Hossein Tehrani PDF
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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 \[ \left \{ \begin {array}{lll} -\Delta u =\lambda a(x)(f(u)-l)& \textrm {in } & \Omega ,\\ u = 0 & \textrm {on } & \partial \Omega , \end {array}\right . \] 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.References
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Additional Information
- David G. Costa
- Affiliation: Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, Nevada 89154-4020
- MR Author ID: 51945
- Email: costa@unlv.nevada.edu
- Humberto Ramos Quoirin
- Affiliation: Universidad de Santiago de Chile, Casilla 307, Correo 2, Santiago, Chile
- MR Author ID: 876954
- Email: humberto.ramos@usach.cl
- Hossein Tehrani
- Affiliation: Department of Mathematical Sciences, University of Nevada, Las Vegas, Las Vegas, Nevada 89154-4020
- MR Author ID: 604345
- Email: tehranih@unlv.nevada.edu
- 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
- © Copyright 2016 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 145 (2017), 2661-2675
- MSC (2010): Primary 35J15, 35J20, 35J61, 35J91
- DOI: https://doi.org/10.1090/proc/13426
- MathSciNet review: 3626519