Infinite multiplicity for an inhomogeneous supercritical problem in entire space

Authors:
Baishun Lai and Zhihao Ge

Journal:
Proc. Amer. Math. Soc. **139** (2011), 4409-4418

MSC (2000):
Primary 35J25, 35J20

DOI:
https://doi.org/10.1090/S0002-9939-2011-10902-X

Published electronically:
April 27, 2011

MathSciNet review:
2823086

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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we will prove the existence of infinitely many positive solutions to the following supercritical problem by using the Liapunov-Schmidt reduction method and asymptotic analysis:

**1.**S. Bae and W-M.Ni, Existence and infinite multiplicity for an inhomogeneous semilinear elliptic equation on , Math. Ann. 320 (2001), 191-210. MR**1835068 (2002i:35052)****2.**S. Bae, T.-K. Chang and D.-H. Pank, Infinite multiplicity of positive entire solutions for a semilinear elliptic equation, J. Differential Equations 181 (2002), 367-387. MR**1907146 (2003i:35076)****3.**G. Bernard, An inhomogeneous semilinear equation in entire space, J. Differential Equations 125 (1996), 184-214. MR**1376065 (97d:35050)****4.**J. Davila, M. del Pino, M. Musso and J. Wei, Standing waves for supercritical nonlinear Schrödinger equations, J. Differential Equations 236 (2007), 164-198. MR**2319924 (2009b:35389)****5.**J. Davila, M. del Pino, M. Musso and J. Wei, Fast and slow decay solutions of supercritical problems in exterior domains, Calculus of Variations and PDE 32 (2008), 453-480. MR**2402919 (2009b:35140)****6.**M. del Pino, Supercritical elliptic problems from a perturbation viewpoint, Discrete and Continous Dynamical Systems 21 (2008), 69-89. MR**2379457 (2009b:35130)****7.**M. del Pino and J. Wei, Supercritical elliptic problems in domains with small holes, Ann. Inst. H. Poincaré Anal. Non Lineaire 24 (2007), 507-520. MR**2334989 (2008j:35077)****8.**Y.-B. Deng, Y. Li and F. Yang, On the stability of the positive steady states for a nonhomogeneous semilinear Cauchy problem, J. Differential Equations 228 (2006), 507-529. MR**2289543 (2008m:35172)****9.**R. H. Fowler, Further studies on Emden's and similar differential equations, Quart. J. Math. 2 (1931), 259-288.**10.**H. Egnell and I. Kaj, Positive global solutions of a nonhomogeneous semilinear elliptic equation, J. Math. Pures Appl. (9) 70, No. 3 (1991), 345-367. MR**1113816 (92m:35078)****11.**T.-Y. Lee, Some limit theorems for super-Brownian motion and semilinear differential equations, Ann. Probab. 21, No. 2 (1993), 979-995. MR**1217576 (94b:60038)****12.**S. I. Pokhozhaev, On the solvability of an elliptic problem in Rn with a supercritical index of nonlinearity, Soviet Math. Dokl. 42, No. 1 (1991), 215-219. MR**1080040 (91j:35100)****13.**C.-F. Gui, Positive entire solutions of equation , J. Differential Equations 99 (1992), 245-280. MR**1184056 (93h:35065)**

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Additional Information

**Baishun Lai**

Affiliation:
Institute of Contemporary Mathematics, Henan University, Kaifeng 475004, People’s Republic of China

Address at time of publication:
School of Mathematics and Information Science, Henan University, Kaifeng 475004, People’s Republic of China

Email:
laibaishun@henu.edu.cn

**Zhihao Ge**

Affiliation:
School of Mathematics and Information Science, Henan University, Kaifeng 475004, People’s Republic of China

Email:
zhihaoge@gmail.com

DOI:
https://doi.org/10.1090/S0002-9939-2011-10902-X

Keywords:
Critical exponents,
linearized operators,
supercritical problem.

Received by editor(s):
October 21, 2010

Published electronically:
April 27, 2011

Additional Notes:
The first author was supported in part by National Natural Science Foundation of China Grant 10971061 and Natural Science Foundation of Henan Province Grant 112300410054.

The second author was supported in part by National Natural Science Foundation of China Grant 10901047

Communicated by:
Walter Craig

Article copyright:
© Copyright 2011
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.