Stable weak solutions of weighted nonlinear elliptic equations

Xia Huang

doi:10.3934/cpaa.2014.13.293

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2014, Volume 13, Issue 1: 293-305. Doi: 10.3934/cpaa.2014.13.293

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Stable weak solutions of weighted nonlinear elliptic equations

Xia Huang^1,

1.
Department of Mathematics and Center for Partial Differential Equations, East China Normal University, Shanghai, 200241

Received: November 30, 2012

Revised: March 31, 2013

Published: December 2013

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Abstract

This paper deals with the weighted nonlinear elliptic equation \begin{eqnarray} -\mathrm{div}(|x|^\alpha \nabla u )=|x|^\gamma e^u \ in\ \Omega ,\\ u = 0 \ on \ \partial \Omega, \end{eqnarray} where $\alpha, \gamma \in R$ satisfy $N + \alpha > 2$ and $\gamma - \alpha > -2$, and the domain $\Omega \subset R^N (N \geq 2)$ is bounded or not. Moreover, when $\alpha\neq 0$, we prove that, for $N + \alpha > 2$, $\gamma - \alpha \leq -2$, the above equation admits no weak solution. We also study Liouville type results for the equation in $R^N$.

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Mathematics Subject Classification: Primary: 35J91; Secondary: 35B35, 35B53.

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References

[1]	L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259-275.
[2]	P. Caldiroli and R. Musina, On a variational degenerate elliptic problem, Nonlinear differ. Equ. Appl., 7 (2000), 187-199.doi: 10.1007/s000300050004.
[3]	F. Catrina and Z. Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229-258.doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I.
[4]	W. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations, Duke Math. J., 63 (1991), 615-622.doi: 10.1215/S0012-7094-91-06325-8.
[5]	C. Cowan and M. Fazly, On stable entire solutions of semilinear elliptic equations with weights, Proc. Amer. Math. Soc., 140 (2012), 2003-2012.doi: 10.1090/S0002-9939-2011-11351-0.
[6]	M. G. Crandall and P. Rabinowitz, Some continuation and variational methods for positive solutions of nonlinear ellptic eigenvalue problems, Arch. Ration. Mech. Anal., 58 (1975), 207-218.
[7]	E. N. Dancer, Y. Du and Z. M. Guo, Finite Morse index solutions of an elliptic equation with supercritical exponent, J. Differential Equations, 250 (2011), 3281-3310.doi: 10.1016/j.jde.2011.02.005.
[8]	E. N. Dancer and A. Farina, On the classification of solutions of $-\Delta u = e^u$ on $\mathbbR^N$: stability outside a compact set and applications, Proc. Amer. Math. Soc., 137 (2009), 1333-1338.doi: 10.1090/S0002-9939-08-09772-4.
[9]	Y. Du and Z. M. Guo, Finite Morse index solutions and asymptotics of weighted nonlinear elliptic equations, Adv. Differential Equations, 7 (2013), 737-768.
[10]	R. Dautray and J. L. Lions, "Mathematical Analysis and Numerical Methods for Science and Technology," Vol. 1: physical origins and classical methods, Springer-Verlag, Berlin, 1990.
[11]	D. E. Edmunds and L. A. Peletier, A Harnack inequality for weak solutions of degenerate quasilinear elliptic equations, J. London Math. Soc., 5 (1972), 21-31.
[12]	A. Farina, Stable solutions of $-\Delta u = e^u$ on $\mathbbR^N$, C. R. Acad. Sci. Paris I, 345 (2007), 63-66.doi: 10.1016/j.crma.2007.05.021.
[13]	A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of $\mathbbR^N$, J. Math. Pures Appl., 87 (2007), 537-561.doi: 10.1016/j.matpur.2007.03.001.
[14]	D. Gilbarg and N. S. Trudinger, "Elliptic Partial Differential Equations Of Second Order," Berlin, New York: Springer, 1998.
[15]	W. Jeong and Y. Lee, Stable solutions and finite Morse index solutions of nonlinear ellptic equations with Hardy potential, Nonlinear Anal., 87 (2013), 126-145.doi: 10.1016/j.na.2013.04.007.
[16]	F. Mignot and J. P. Puel, Solution radiale singulière de $-\Delta u=\lambda e^u$, C. R. Acad. Sci., Paris, Sér. I., 307 (1988), 379-382.
[17]	P. Pucci and J. Serrin, "The Maximum Principle," Progr. Nolinear Differential Equations Appl, Vol. 73, Birkhäuser, Basel, 2007.
[18]	J. Serrin, Local behavior of solutions of quasi-linear equations, Acta Math., 111 (1964), 247-302.
[19]	C. Wang and D. Ye, Some Liouville theorems for Hénon type elliptic equations, J. Funct Anal., 262 (2012), 1705-1727.doi: 10.1016/j.jfa.2011.11.017.