# American Institute of Mathematical Sciences

• Previous Article
On vector solutions for coupled nonlinear Schrödinger equations with critical exponents
• CPAA Home
• This Issue
• Next Article
On the number of maximum points of least energy solution to a two-dimensional Hénon equation with large exponent
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. 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

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
 [1] Vianney Perchet, Marc Quincampoix. A differential game on Wasserstein space. Application to weak approachability with partial monitoring. Journal of Dynamics & Games, 2019, 6 (1) : 65-85. doi: 10.3934/jdg.2019005 [2] Soohyun Bae. Positive entire solutions of inhomogeneous semilinear elliptic equations with supercritical exponent. Conference Publications, 2005, 2005 (Special) : 50-59. doi: 10.3934/proc.2005.2005.50 [3] Angela Alberico, Andrea Cianchi, Luboš Pick, Lenka Slavíková. Sharp Sobolev type embeddings on the entire Euclidean space. Communications on Pure & Applied Analysis, 2018, 17 (5) : 2011-2037. doi: 10.3934/cpaa.2018096 [4] Yajing Zhang, Jianghao Hao. Existence of positive entire solutions for semilinear elliptic systems in the whole space. Communications on Pure & Applied Analysis, 2009, 8 (2) : 719-724. doi: 10.3934/cpaa.2009.8.719 [5] Peter Poláčik, Darío A. Valdebenito. Existence of quasiperiodic solutions of elliptic equations on the entire space with a quadratic nonlinearity. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 1-25. doi: 10.3934/dcdss.2020077 [6] Mrinal Kanti Roychowdhury. Quantization coefficients for ergodic measures on infinite symbolic space. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 2829-2846. doi: 10.3934/dcds.2014.34.2829 [7] Koh Katagata. Transcendental entire functions whose Julia sets contain any infinite collection of quasiconformal copies of quadratic Julia sets. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 5319-5337. doi: 10.3934/dcds.2019217 [8] Andrew E.B. Lim, John B. Moore. A path following algorithm for infinite quadratic programming on a Hilbert space. Discrete & Continuous Dynamical Systems - A, 1998, 4 (4) : 653-670. doi: 10.3934/dcds.1998.4.653 [9] Zhihua Liu, Pierre Magal. Functional differential equation with infinite delay in a space of exponentially bounded and uniformly continuous functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 0-0. doi: 10.3934/dcdsb.2019227 [10] Núria Fagella, Christian Henriksen. Deformation of entire functions with Baker domains. Discrete & Continuous Dynamical Systems - A, 2006, 15 (2) : 379-394. doi: 10.3934/dcds.2006.15.379 [11] Patricia Domínguez, Peter Makienko, Guillermo Sienra. Ruelle operator and transcendental entire maps. Discrete & Continuous Dynamical Systems - A, 2005, 12 (4) : 773-789. doi: 10.3934/dcds.2005.12.773 [12] Boris Kruglikov, Martin Rypdal. Entropy via multiplicity. Discrete & Continuous Dynamical Systems - A, 2006, 16 (2) : 395-410. doi: 10.3934/dcds.2006.16.395 [13] Giuseppina Autuori, Patrizia Pucci. Entire solutions of nonlocal elasticity models for composite materials. Discrete & Continuous Dynamical Systems - S, 2018, 11 (3) : 357-377. doi: 10.3934/dcdss.2018020 [14] Agnieszka Badeńska. No entire function with real multipliers in class $\mathcal{S}$. Discrete & Continuous Dynamical Systems - A, 2013, 33 (8) : 3321-3327. doi: 10.3934/dcds.2013.33.3321 [15] Antonio Vitolo. On the growth of positive entire solutions of elliptic PDEs and their gradients. Discrete & Continuous Dynamical Systems - S, 2014, 7 (6) : 1335-1346. doi: 10.3934/dcdss.2014.7.1335 [16] Wan-Tong Li, Li Zhang, Guo-Bao Zhang. Invasion entire solutions in a competition system with nonlocal dispersal. Discrete & Continuous Dynamical Systems - A, 2015, 35 (4) : 1531-1560. doi: 10.3934/dcds.2015.35.1531 [17] Patrizia Pucci, Marco Rigoli. Entire solutions of singular elliptic inequalities on complete manifolds. Discrete & Continuous Dynamical Systems - A, 2008, 20 (1) : 115-137. doi: 10.3934/dcds.2008.20.115 [18] Yu-Juan Sun, Li Zhang, Wan-Tong Li, Zhi-Cheng Wang. Entire solutions in nonlocal monostable equations: Asymmetric case. Communications on Pure & Applied Analysis, 2019, 18 (3) : 1049-1072. doi: 10.3934/cpaa.2019051 [19] Manuel del Pino, Jean Dolbeault, Monica Musso. Multiple bubbling for the exponential nonlinearity in the slightly supercritical case. Communications on Pure & Applied Analysis, 2006, 5 (3) : 463-482. doi: 10.3934/cpaa.2006.5.463 [20] Yansheng Zhong, Yongqing Li. On a p-Laplacian eigenvalue problem with supercritical exponent. Communications on Pure & Applied Analysis, 2019, 18 (1) : 227-236. doi: 10.3934/cpaa.2019012

2018 Impact Factor: 0.925