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May  2013, 12(3): 1243-1257. doi: 10.3934/cpaa.2013.12.1243

Infinite multiplicity for an inhomogeneous supercritical problem in entire space

1. 

Department of mathematics, East China Normal University, 500 Dong Chuan Road, Shanghai 200241

2. 

Department of Mathematics, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

Received  December 2011 Revised  July 2012 Published  September 2012

Let $K(x)$ be a positive function in $R^N, N \geq 3$ and satisfy $\lim\limits_{|x|\rightarrow \infty} K(x) = K_\infty$ where $K_\infty$ is a positive constant. When $p > \frac{N + 1}{N - 3}, N \geq 4$, we prove the existence of infinitely many positive solutions to the following supercritical problem: \begin{eqnarray*} \Delta u(x) + K(x)u^p = 0, u>0 \quad in \quad R^N, \lim_{|x|\rightarrow \infty} u(x) = 0. \end{eqnarray*} If in addition we have, for instance, $\lim\limits_{|x| \rightarrow \infty}|x|^\mu (K(x) - K_\infty ) = C_0 \neq 0, 0 < \mu \leq N - \frac{2p+2}{p-1}$, then this result still holds provided that $p > \frac{N + 2}{N - 2}$.
Citation: Liping Wang, Juncheng Wei. Infinite multiplicity for an inhomogeneous supercritical problem in entire space. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1243-1257. doi: 10.3934/cpaa.2013.12.1243
References:
[1]

S. Bae, Infinite multiplicity and separation structure of positive solutions for a semilinear elliptic equation in $\mathbb{R}^{n}$, J. Diff. Eqns., 200 (2004), 274-311. doi: 10.1016/j.jde.2003.11.006.

[2]

G. Bernard, An inhomogeneous semilinear equation in entire space, J. Differential Equations, 125 (1996), 184-214. doi: 10.1006/jdeq.1996.0029.

[3]

S. Bae and W.-M. Ni, Existence and infinite multiplicity for an inhomogeneous semilinear elliptic equationon $R^n$, Math. Ann., 320 (2001), 191-210. doi: 10.1007/PL00004468.

[4]

J. Dávila, M. del Pino and M. Musso, The supercritical Lane-Emden-Fowler equation in exterior domains, Commun. Part. Diff. Equations, 32 (2007), 1225-1243. doi: 10.1080/03605300600854209.

[5]

J. Dávila, M. Del Pino, M. Musso and J. Wei, Standing waves for supercritical nonlinear Schrödinger equations, J. Differential Equations, 236 (2007), 164-198. doi: 10.1016/j.jde.2007.01.016.

[6]

W.-Y. Ding and W.-M. Ni, On the elliptic equation $\Delta u + Ku^{\frac{N + 2}{N - 2}} = 0$ and related topics, Duke Math. J., 52 (1985), 485-506. doi: 10.1215/S0012-7094-85-05224-X.

[7]

C.-F. Gui, Positive entire solutions of equation $\Delta u + f(x, u) = 0$, J. Diff. Eqns., 99 (1992), 245-280. doi: 10.1016/0022-0396(92)90023-G.

[8]

C.-F. Gui, On positive entire solutions of the elliptic equation $\Delta u+K(x) u^p=0$ and its applications to Riemannian geometry, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 225-237. doi: 10.1017/S0308210500022708.

[9]

C.-F. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat euqation in $R^N$, Comm. Pure Appl. Math., 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906.

[10]

Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u+ K(x) u^p=0$ in $R^n$, J. Differential Equations, 95 (1992), 304-330. doi: 10.1016/0022-0396(92)90034-K.

[11]

X.-F. Wang and J.-C. Wei, On the equation $\Delta u +Ku^{\frac{N+ 2}{N - 2} \pm \epsilon^2} = 0$ in $R^n$, Rend. Circ. Mat. Palermo, 44 (1995), 365-400. doi: 10.1007/BF02844676.

[12]

E. Yanagida and S. Yotsutani, Classification of the structure of positive radial solutions to $\Delta u + K(|x|) u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 124 (1993), 239-259. doi: 10.1007/BF00953068.

show all references

References:
[1]

S. Bae, Infinite multiplicity and separation structure of positive solutions for a semilinear elliptic equation in $\mathbb{R}^{n}$, J. Diff. Eqns., 200 (2004), 274-311. doi: 10.1016/j.jde.2003.11.006.

[2]

G. Bernard, An inhomogeneous semilinear equation in entire space, J. Differential Equations, 125 (1996), 184-214. doi: 10.1006/jdeq.1996.0029.

[3]

S. Bae and W.-M. Ni, Existence and infinite multiplicity for an inhomogeneous semilinear elliptic equationon $R^n$, Math. Ann., 320 (2001), 191-210. doi: 10.1007/PL00004468.

[4]

J. Dávila, M. del Pino and M. Musso, The supercritical Lane-Emden-Fowler equation in exterior domains, Commun. Part. Diff. Equations, 32 (2007), 1225-1243. doi: 10.1080/03605300600854209.

[5]

J. Dávila, M. Del Pino, M. Musso and J. Wei, Standing waves for supercritical nonlinear Schrödinger equations, J. Differential Equations, 236 (2007), 164-198. doi: 10.1016/j.jde.2007.01.016.

[6]

W.-Y. Ding and W.-M. Ni, On the elliptic equation $\Delta u + Ku^{\frac{N + 2}{N - 2}} = 0$ and related topics, Duke Math. J., 52 (1985), 485-506. doi: 10.1215/S0012-7094-85-05224-X.

[7]

C.-F. Gui, Positive entire solutions of equation $\Delta u + f(x, u) = 0$, J. Diff. Eqns., 99 (1992), 245-280. doi: 10.1016/0022-0396(92)90023-G.

[8]

C.-F. Gui, On positive entire solutions of the elliptic equation $\Delta u+K(x) u^p=0$ and its applications to Riemannian geometry, Proc. Roy. Soc. Edinburgh Sect. A, 126 (1996), 225-237. doi: 10.1017/S0308210500022708.

[9]

C.-F. Gui, W.-M. Ni and X. Wang, On the stability and instability of positive steady states of a semilinear heat euqation in $R^N$, Comm. Pure Appl. Math., 45 (1992), 1153-1181. doi: 10.1002/cpa.3160450906.

[10]

Y. Li, Asymptotic behavior of positive solutions of equation $\Delta u+ K(x) u^p=0$ in $R^n$, J. Differential Equations, 95 (1992), 304-330. doi: 10.1016/0022-0396(92)90034-K.

[11]

X.-F. Wang and J.-C. Wei, On the equation $\Delta u +Ku^{\frac{N+ 2}{N - 2} \pm \epsilon^2} = 0$ in $R^n$, Rend. Circ. Mat. Palermo, 44 (1995), 365-400. doi: 10.1007/BF02844676.

[12]

E. Yanagida and S. Yotsutani, Classification of the structure of positive radial solutions to $\Delta u + K(|x|) u^p=0$ in $R^n$, Arch. Rational Mech. Anal., 124 (1993), 239-259. doi: 10.1007/BF00953068.

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