Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Explicit solutions and multiplicity results for some equations with the $ p$-Laplacian

Author: Philip Korman
Journal: Quart. Appl. Math. 75 (2017), 635-647
MSC (2010): Primary 35J25, 35J61
DOI: https://doi.org/10.1090/qam/1471
Published electronically: April 19, 2017
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Abstract: We derive explicit ground state solutions for several equations with the $ p$-Laplacian in $ R^n$, including (here $ \varphi (z)=z\vert z\vert^{p-2}$, with $ p>1$)

$\displaystyle \varphi \left (u'(r)\right )' +\frac {n-1}{r} \varphi \left (u'(r)\right )+u^M+u^Q=0 \,. $

The constant $ M>0$ is assumed to be below the critical power, while $ Q=\frac {M p-p+1}{p-1}$ is above the critical power. This explicit solution is used to give a multiplicity result, similarly to C. S. Lin and W.-M. Ni (1998). We also give the $ p$-Laplace version of G. Bratu's solution, connected to combustion theory.

In another direction, we present a change of variables which removes the non-autonomous term $ r^{\alpha }$ in

$\displaystyle \varphi \left (u'(r)\right )' +\frac {n-1}{r} \varphi \left (u'(r)\right )+r^{\alpha } f(u)=0 \,, $

while preserving the form of this equation. In particular, we study singular equations, when $ \alpha <0$, that occur often in applications. The Coulomb case $ \alpha =-1$ turned out to give the critical power.

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Additional Information

Philip Korman
Affiliation: Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio 45221-0025
Email: kormanp@ucmail.uc.edu

DOI: https://doi.org/10.1090/qam/1471
Keywords: Explicit solutions, multiplicity results
Received by editor(s): August 21, 2016
Received by editor(s) in revised form: March 20, 2017
Published electronically: April 19, 2017
Article copyright: © Copyright 2017 Brown University

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