Infinite multiplicity for an inhomogeneous supercritical problem in entire space

Lai, Baishun; Ge, Zhihao

doi:10.1090/S0002-9939-2011-10902-X

Infinite multiplicity for an inhomogeneous supercritical problem in entire space
HTML articles powered by AMS MathViewer

by Baishun Lai and Zhihao Ge

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

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

Published electronically: April 27, 2011

PDF | Request permission

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: \begin{eqnarray*} \left \{ \begin {array}{ll} \Delta u + u^{p}+f(x)=0,\ \ u>0\ \ \mbox {in}\ \mathbb {R}^{n},\\ \lim _{|x|\to \infty }u(x)\to 0. \end{array} \right . \end{eqnarray*}

References

Soohyun Bae and Wei-Ming Ni, Existence and infinite multiplicity for an inhomogeneous semilinear elliptic equation on $\mathbf R^n$, Math. Ann. 320 (2001), no. 1, 191–210. MR 1835068, DOI 10.1007/PL00004468
Soohyun Bae, Tong Keun Chang, and Dae Hyeon Pahk, Infinite multiplicity of positive entire solutions for a semilinear elliptic equation, J. Differential Equations 181 (2002), no. 2, 367–387. MR 1907146, DOI 10.1006/jdeq.2001.4079
Guy Bernard, An inhomogeneous semilinear equation in entire space, J. Differential Equations 125 (1996), no. 1, 184–214. MR 1376065, DOI 10.1006/jdeq.1996.0029
Juan Dávila, Manuel del Pino, Monica Musso, and Juncheng Wei, Standing waves for supercritical nonlinear Schrödinger equations, J. Differential Equations 236 (2007), no. 1, 164–198. MR 2319924, DOI 10.1016/j.jde.2007.01.016
Juan Dávila, Manuel del Pino, Monica Musso, and Juncheng Wei, Fast and slow decay solutions for supercritical elliptic problems in exterior domains, Calc. Var. Partial Differential Equations 32 (2008), no. 4, 453–480. MR 2402919, DOI 10.1007/s00526-007-0154-1
Manuel del Pino, Supercritical elliptic problems from a perturbation viewpoint, Discrete Contin. Dyn. Syst. 21 (2008), no. 1, 69–89. MR 2379457, DOI 10.3934/dcds.2008.21.69
Manuel del Pino and Juncheng Wei, Problèmes elliptiques supercritiques dans des domaines avec de petits trous, Ann. Inst. H. Poincaré C Anal. Non Linéaire 24 (2007), no. 4, 507–520 (English, with English and French summaries). MR 2334989, DOI 10.1016/j.anihpc.2006.03.001
Yinbin Deng, Yi Li, and Fen Yang, On the stability of the positive steady states for a nonhomogeneous semilinear Cauchy problem, J. Differential Equations 228 (2006), no. 2, 507–529. MR 2289543, DOI 10.1016/j.jde.2006.02.010
R. H. Fowler, Further studies on Emden’s and similar differential equations, Quart. J. Math. 2 (1931), 259-288.
Henrik Egnell and Ingemar Kaj, Positive global solutions of a nonhomogeneous semilinear elliptic equation, J. Math. Pures Appl. (9) 70 (1991), no. 3, 345–367. MR 1113816
Tzong-Yow Lee, Some limit theorems for super-Brownian motion and semilinear differential equations, Ann. Probab. 21 (1993), no. 2, 979–995. MR 1217576
S. I. Pokhozhaev, Solvability of an elliptic problem in $\textbf {R}^n$ with a supercritical index of nonlinearity, Dokl. Akad. Nauk SSSR 313 (1990), no. 6, 1356–1360 (Russian); English transl., Soviet Math. Dokl. 42 (1991), no. 1, 215–219. MR 1080040
Changfeng Gui, Positive entire solutions of the equation $\Delta u+f(x,u)=0$, J. Differential Equations 99 (1992), no. 2, 245–280. MR 1184056, DOI 10.1016/0022-0396(92)90023-G