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Fall 2015 Existence result for a class of quasilinear elliptic equations with ($p$-$q$)-Laplacian and vanishing potentials
M. J. Alves, R. B. Assunção, O. H. Miyagaki
Illinois J. Math. 59(3): 545-575 (Fall 2015). DOI: 10.1215/ijm/1475266397


The main purpose of this paper is to establish the existence of positive solutions to a class of quasilinear elliptic equations involving the ($p$-$q$)-Laplacian operator. We consider a nonlinearity that can be subcritical at infinity and supercritical at the origin; we also consider potential functions that can vanish at infinity. The approach is based on variational arguments dealing with the mountain-pass lemma and an adaptation of the penalization method. In order to overcome the lack of compactness, we modify the original problem and the associated energy functional. Finally, to show that the solution of the modified problem is also a solution of the original problem we use an estimate obtained by the Moser iteration scheme.


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M. J. Alves. R. B. Assunção. O. H. Miyagaki. "Existence result for a class of quasilinear elliptic equations with ($p$-$q$)-Laplacian and vanishing potentials." Illinois J. Math. 59 (3) 545 - 575, Fall 2015.


Received: 28 August 2015; Revised: 17 March 2016; Published: Fall 2015
First available in Project Euclid: 30 September 2016

zbMATH: 1352.35061
MathSciNet: MR3554222
Digital Object Identifier: 10.1215/ijm/1475266397

Primary: 35J20, 35J92
Secondary: 35B09, 35B38, 35B45, 35J10

Rights: Copyright © 2015 University of Illinois at Urbana-Champaign


Vol.59 • No. 3 • Fall 2015
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