Nodal Bubble-Tower Solutions for a semilinear elliptic problem with competing powers

Zhongyuan Liu

doi:10.3934/dcds.2017230

Article Contents

2017, Volume 37, Issue 10: 5299-5317. Doi: 10.3934/dcds.2017230

This issue Previous Article A general law of large permanent Next Article Relations between Bohl and general exponents

Nodal Bubble-Tower Solutions for a semilinear elliptic problem with competing powers

Zhongyuan Liu^,

School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475004, China

* Corresponding author: Zhongyuan Liu
* Corresponding author: Zhongyuan Liu

Received: January 31, 2017

Revised: April 30, 2017

Published: October 2017

The author is supported by NSFC under grant No.11501166

Abstract Full Text(HTML) Related Papers Cited by

Abstract

In this paper, we consider the following semilinear elliptic problem

$\begin{equation*}\begin{cases}-Δ u=|u|^{p-1}u-|u|^{q-1}u~\text{in}~\mathbb{R}^N, \\ u(x)\to 0, ~~\text{as}~|x|\to ∞, \end{cases}\end{equation*}$

where $\frac{N}{N-2} < q < p < p^*$ or $q>p>p^*$, $p^*=\frac{N+2}{N-2}$, $N≥3$. We show that if $q$ is fixed and $p$ is close enough to $\frac{N+2}{N-2}$, the above problem has radial nodal bubble tower solutions, which behave like a superposition of bubbles with different orders and blow up at the origin.

Keywords:

Mathematics Subject Classification: Primary: 35J60; Secondary: 58C15, 35J61.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

	W. Ao , M. Musso , F. Pacard and J. Wei , Solutions without any symmetry for semilinear elliptic problems, J. Funct. Anal., 270 (2016) , 884-956. doi: 10.1016/j.jfa.2015.10.015.
	T. Bartsch and M. Willem , Infinitely many radial solutions of a semilinear elliptic problem on ${{\mathbb{R}}^{N}}$, Arch. Rational Mech. Anal., 124 (1993) , 261-276. doi: 10.1007/BF00953069.
	T. Bartsch and M. Willem , Infinitely many nonradial solutions of a Euclidean scalar field equation, J. Funct. Anal., 117 (1993) , 447-460. doi: 10.1006/jfan.1993.1133.
	H. Berestycki and P. L. Lions , Nonlinear scalar field equations. Ⅰ. Existence of a ground state, Arch. Ration. Mech. Anal., 82 (1983) , 313-345. doi: 10.1007/BF00250555.
	H. Berestycki and P. L. Lions , Nonlinear scalar field equations, Ⅱ, Arch. Ration. Mech. Anal., 82 (1983) , 347-375. doi: 10.1007/BF00250556.
	H. Brezis and L. Nirenberg , Positive solutions of nonlinear elliptic equations involving critical Sobolev exponents, Comm. Pure Appl. Math., 36 (1983) , 437-477. doi: 10.1002/cpa.3160360405.
	L. A. Caffarelli , B. Gidas and J. Spruck , Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math., 42 (1989) , 271-297. doi: 10.1002/cpa.3160420304.
	J. Campos , "Bubble-Tower" phenomena in a semilinear elliptic equation with mixed Sobolev growth, Nonlinear Analysis, 68 (2008) , 1382-1397. doi: 10.1016/j.na.2006.12.032.
	D. Cao and X. Zhu , On the existence and nodal character of solutions of semilinear elliptic equations, Acta Math. Sci., 8 (1988) , 345-359.
	A. Contreras and M. Del Pino , Nodal bubble-tower solutions to radial elliptic problems near criticality, Discrete Contin. Dyn. Syst., 16 (2006) , 525-539. doi: 10.3934/dcds.2006.16.525.
	E. N. Dancer , New solutions of equations on $\mathbb{R}^n$, Ann. Sc. Norm. Super. Pisa Cl. Sci., 30 (2001) , 535-563.
	E. N. Dancer and S. Santra , Singular perturbed problems in the zero mass case: Asymptotic behavior of spikes, Ann. Mat. Pura Appl., 189 (2010) , 185-225. doi: 10.1007/s10231-009-0105-x.
	E. N. Dancer , S. Santra and J. Wei , Asymptotic behavior of the least energy solution of a problem with competing powers, J. Funct. Anal., 261 (2011) , 2094-2134. doi: 10.1016/j.jfa.2011.06.005.
	J. Davila and I. Guerra , Slowly decaying radial solutions of an elliptic equations with subcritical and supercritical exponents, J.D'Analyse Math., 129 (2016) , 367-391. doi: 10.1007/s11854-016-0025-9.
	M. Del Pino , J. Dolbeault and M. Musso , "Bubble-Tower" radial solutions in the slightly supercritical Brezis-Nirenberg problem, J. Differential Equations, 193 (2003) , 280-306. doi: 10.1016/S0022-0396(03)00151-7.
	C. Jones , Radial solutions of a semilinear elliptic equation at a critical exponent, Arch. Rational Mech. Anal., 104 (1988) , 251-270. doi: 10.1007/BF00281356.
	C. Jones and T. Kupper , On the infinitely many solutions of a semilinear elliptic equation, SIAM J. Math. Anal., 17 (1986) , 803-835. doi: 10.1137/0517059.
	M. K. Kwong , Uniqueness of positive solutions of $\Delta u+{{u}^{p}}-u=0\ \ \text{in}\ {{\mathbb{R}}^{N}}$, Arch. Rational Mech. Anal., 105 (1989) , 243-266. doi: 10.1007/BF00251502.
	M. K. Kwong , J. B. McLeod , L. A. Peletier and W. C. Troy , On groundstate solutions of $-\Delta u={{u}^{p}}-{{u}^{q}}$, J. Differential Equations, 95 (1992) , 218-239. doi: 10.1016/0022-0396(92)90030-Q.
	M. K. Kwong and L. Zhang , Uniqueness of the positive solution of $Δ u +f(u)=0$ in an annulus, Differential Integral Equations, 4 (1991) , 588-599.
	Y. Li and W. M. Ni , Radial symmetry of positive solutions of a nonlinear elliptic equations in ${{\mathbb{R}}^{N}}$, Comm. Partial Differential Equations, 18 (1993) , 1043-1054. doi: 10.1080/03605309308820960.
	S. A. Lorca and P. Ubilla , Symmetric and nonsymmetric solutions for an elliptic equation on ${{\mathbb{R}}^{N}}$, Nonlinear Anal., 58 (2004) , 961-968. doi: 10.1016/j.na.2004.03.034.
	A. Malchiodi , Some new entire solutions of semilinear elliptic equations on ${{\mathbb{R}}^{n}}$, Adv. Math., 221 (2009) , 1843-1909. doi: 10.1016/j.aim.2009.03.012.
	K. McLeod , W. C. Troy and F. B. Weissler , Radial solutions of $\Delta u+f\left( u \right)=0$ with prescribed numbers of zeros, J. Differential Equations, 83 (1990) , 368-378. doi: 10.1016/0022-0396(90)90063-U.
	F. Merle and L. A. Peletier , Asymptotic behavior of positive solutions of elliptic equations with critical and supercritical growth. Ⅰ. The radial case, Arch. Ration. Mech. Anal., 112 (1990) , 1-19. doi: 10.1007/BF00431720.
	F. Merle and L. A. Peletier , Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth. Ⅱ. The nonradial case, J. Funct. Anal., 105 (1992) , 1-41. doi: 10.1016/0022-1236(92)90070-Y.
	M. Musso , F. Pacard and J. Wei , Finite-energy sign-changing solutions with dihedral symmetry for the stationary nonlinear Schr¨odinger equation, J. Eur. Math. Soc., 14 (2012) , 1923-1953. doi: 10.4171/JEMS/351.
	M. Musso and A. Pistoia , Tower of bubbles for almost critical problems in general domains, J. Math. Pures Appl., 93 (2010) , 1-40. doi: 10.1016/j.matpur.2009.08.001.
	A. Pistoia and T. Weth , Sign changing bubble tower solutions in a slightly subcritical semilinear Dirichlet problem, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007) , 325-340. doi: 10.1016/j.anihpc.2006.03.002.
	O. Rey , The role of the Green function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct.Anal., 89 (1990) , 1-52. doi: 10.1016/0022-1236(90)90002-3.
	G. Talenti , Best constant in Sobolev inequality, Ann. Math. Pura Appl., 110 (1976) , 353-372. doi: 10.1007/BF02418013.
	W. C. Troy , Bounded solutions of $\Delta u+\|u{{\|}^{p-1}}u-\|u{{\|}^{q-1}}u=0$ in the supercritical case, SIAM J. Math. Anal., 21 (1990) , 1326-1334. doi: 10.1137/0521073.