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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 & 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), 274-311. 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), 184-214. 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), 191-210. 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, 32 (2007), 1225-1243. 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), 164-198. 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), 485-506. 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), 245-280. 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), 225-237. 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), 1153-1181. 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), 304-330. 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), 365-400. 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), 239-259. doi: 10.1007/BF00953068.  Google Scholar

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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), 274-311. 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), 184-214. 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), 191-210. 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, 32 (2007), 1225-1243. 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), 164-198. 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), 485-506. 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), 245-280. 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), 225-237. 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), 1153-1181. 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), 304-330. 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), 365-400. 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), 239-259. doi: 10.1007/BF00953068.  Google Scholar
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