On stable entire solutions of semi-linear elliptic equations with weights
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- by Craig Cowan and Mostafa Fazly PDF
- Proc. Amer. Math. Soc. 140 (2012), 2003-2012 Request permission
Abstract:
We are interested in the existence versus non-existence of non-trivial stable sub- and super-solutions of \begin{equation} -\operatorname {div}( \omega _1 \nabla u) = \omega _2 f(u) \qquad \text {in}\ \ \mathbb {R}^N, \end{equation} with positive smooth weights $\omega _1(x),\omega _2(x)$. We consider the cases $f(u) = e^u, u^p$ where $p>1$ and $-u^{-p}$ where $p>0$. We obtain various non-existence results which depend on the dimension $N$ and also on $p$ and the behaviour of $\omega _1,\omega _2$ near infinity. Also the monotonicity of $\omega _1$ is involved in some results. Our methods here are the methods developed by Farina. We examine a specific class of weights $\omega _1(x) = ( |x|^2 +1)^\frac {\alpha }{2}$ and $\omega _2(x) = ( |x|^2+1)^\frac { \beta }{2} g(x)$, where $g(x)$ is a positive function with a finite limit at $\infty$. For this class of weights, non-existence results are optimal. To show the optimality we use various generalized Hardy inequalities.References
- Daniele Castorina, Pierpaolo Esposito, and Berardino Sciunzi, Low dimensional instability for semilinear and quasilinear problems in $\Bbb R^N$, Commun. Pure Appl. Anal. 8 (2009), no.Β 6, 1779β1793. MR 2552149, DOI 10.3934/cpaa.2009.8.1779
- Craig Cowan, Optimal Hardy inequalities for general elliptic operators with improvements, Commun. Pure Appl. Anal. 9 (2010), no.Β 1, 109β140. MR 2556749, DOI 10.3934/cpaa.2010.9.109
- E. N. Dancer, Stable and finite Morse index solutions on $\mathbf R^n$ or on bounded domains with small diffusion, Trans. Amer. Math. Soc. 357 (2005), no.Β 3, 1225β1243. MR 2110438, DOI 10.1090/S0002-9947-04-03543-3
- L. Dupaigne and A. Farina, Stable solutions of $-\Delta u=f(u)$ in $\Bbb R^N$, J. Eur. Math. Soc. (JEMS) 12 (2010), no.Β 4, 855β882. MR 2654082, DOI 10.4171/JEMS/217
- Pierpaolo Esposito, Linear instability of entire solutions for a class of non-autonomous elliptic equations, Proc. Roy. Soc. Edinburgh Sect. A 138 (2008), no.Β 5, 1005β1018. MR 2477449, DOI 10.1017/S0308210507000212
- Pierpaolo Esposito, Compactness of a nonlinear eigenvalue problem with a singular nonlinearity, Commun. Contemp. Math. 10 (2008), no.Β 1, 17β45. MR 2387858, DOI 10.1142/S0219199708002697
- Pierpaolo Esposito, Nassif Ghoussoub, and Yujin Guo, Mathematical analysis of partial differential equations modeling electrostatic MEMS, Courant Lecture Notes in Mathematics, vol. 20, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 2010. MR 2604963, DOI 10.1090/cln/020
- Pierpaolo Esposito, Nassif Ghoussoub, and Yujin Guo, Compactness along the branch of semistable and unstable solutions for an elliptic problem with a singular nonlinearity, Comm. Pure Appl. Math. 60 (2007), no.Β 12, 1731β1768. MR 2358647, DOI 10.1002/cpa.20189
- Alberto Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\Bbb R^N$, J. Math. Pures Appl. (9) 87 (2007), no.Β 5, 537β561 (English, with English and French summaries). MR 2322150, DOI 10.1016/j.matpur.2007.03.001
- Alberto Farina, Stable solutions of $-\Delta u=e^u$ on $\Bbb R^N$, C. R. Math. Acad. Sci. Paris 345 (2007), no.Β 2, 63β66 (English, with English and French summaries). MR 2343553, DOI 10.1016/j.crma.2007.05.021
- Li Ma and J. C. Wei, Properties of positive solutions to an elliptic equation with negative exponent, J. Funct. Anal. 254 (2008), no.Β 4, 1058β1087. MR 2381203, DOI 10.1016/j.jfa.2007.09.017
- Wei Ming Ni, A nonlinear Dirichlet problem on the unit ball and its applications, Indiana Univ. Math. J. 31 (1982), no.Β 6, 801β807. MR 674869, DOI 10.1512/iumj.1982.31.31056
Additional Information
- Craig Cowan
- Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
- MR Author ID: 815665
- Email: ctcowan@stanford.edu
- Mostafa Fazly
- Affiliation: Department of Mathematics, University of British Columbia, Vancouver, B.C., Canada V6T 1Z2
- MR Author ID: 822619
- Email: fazly@math.ubc.ca
- Received by editor(s): February 3, 2011
- Published electronically: September 30, 2011
- Additional Notes: This work is supported by a University Graduate Fellowship and is part of the second authorβs Ph.D. dissertation in preparation under the supervision of N. Ghoussoub.
- Communicated by: James E. Colliander
- © Copyright 2011
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 2003-2012
- MSC (2010): Primary 35B08; Secondary 35J61, 35A01
- DOI: https://doi.org/10.1090/S0002-9939-2011-11351-0
- MathSciNet review: 2888188