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January  2014, 13(1): 293-305. doi: 10.3934/cpaa.2014.13.293

## Stable weak solutions of weighted nonlinear elliptic equations

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

Received  December 2012 Revised  April 2013 Published  July 2013

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 $R^N$.
Citation: Xia Huang. Stable weak solutions of weighted nonlinear elliptic equations. Communications on Pure and Applied Analysis, 2014, 13 (1) : 293-305. doi: 10.3934/cpaa.2014.13.293
##### 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 $\mathbb{R}^N2$: 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 $\mathbb{R}^N2$, 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 $\mathbb{R}^N2$, 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.

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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 $\mathbb{R}^N2$: 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 $\mathbb{R}^N2$, 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 $\mathbb{R}^N2$, 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.
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