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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

Liping Wang 1, and  Juncheng Wei 2,

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 $p > \frac{N + 2}{N - 2}$.
Keywords: entire space., Supercritical, infinite multiplicity.
Mathematics Subject Classification: Primary 35B35, 92C15; Secondary 35B40, 92D2.
Citation: Liping Wang, Juncheng Wei. Infinite multiplicity for an inhomogeneous supercritical problem in entire space. Communications on Pure & 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 $\R^n$,, J. Diff. Eqns., 200 (2004), 274.  doi: 10.1016/j.jde.2003.11.006.  Google Scholar

[2]

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

[3]

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

[4]

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

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

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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.  doi: 10.1215/S0012-7094-85-05224-X.  Google Scholar

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

[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.  doi: 10.1017/S0308210500022708.  Google Scholar

[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.  doi: 10.1002/cpa.3160450906.  Google Scholar

[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.  doi: 10.1016/0022-0396(92)90034-K.  Google Scholar

[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.  doi: 10.1007/BF02844676.  Google Scholar

[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.  doi: 10.1007/BF00953068.  Google Scholar

show all references

References:
[1]

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

[2]

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

[3]

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

[4]

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

[5]

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

[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.  doi: 10.1215/S0012-7094-85-05224-X.  Google Scholar

[7]

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

[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.  doi: 10.1017/S0308210500022708.  Google Scholar

[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.  doi: 10.1002/cpa.3160450906.  Google Scholar

[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.  doi: 10.1016/0022-0396(92)90034-K.  Google Scholar

[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.  doi: 10.1007/BF02844676.  Google Scholar

[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.  doi: 10.1007/BF00953068.  Google Scholar

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