# American Institute of Mathematical Sciences

October  2017, 37(10): 5299-5317. doi: 10.3934/dcds.2017230

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

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

* Corresponding author: Zhongyuan Liu

Received  February 2017 Revised  May 2017 Published  June 2017

Fund Project: The author is supported by NSFC under grant No.11501166.

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.
Citation: Zhongyuan Liu. Nodal Bubble-Tower Solutions for a semilinear elliptic problem with competing powers. Discrete and Continuous Dynamical Systems, 2017, 37 (10) : 5299-5317. doi: 10.3934/dcds.2017230
##### References:
 [1] 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. [2] 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. [3] 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. [4] 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. [5] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, Ⅱ, Arch. Ration. Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556. [6] 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. [7] 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. [8] 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. [9] D. Cao and X. Zhu, On the existence and nodal character of solutions of semilinear elliptic equations, Acta Math. Sci., 8 (1988), 345-359. [10] 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. [11] E. N. Dancer, New solutions of equations on $\mathbb{R}^n$, Ann. Sc. Norm. Super. Pisa Cl. Sci.(4), 30 (2001), 535-563. [12] 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. [13] 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. [14] 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. [15] 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. [16] 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. [17] 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. [18] 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. [19] 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. [20] 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. [21] 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. [22] 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. [23] 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. [24] 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. [25] 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. [26] 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. [27] 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. [28] 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. [29] 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. [30] 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. [31] G. Talenti, Best constant in Sobolev inequality, Ann. Math. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013. [32] 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.

show all references

##### References:
 [1] 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. [2] 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. [3] 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. [4] 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. [5] H. Berestycki and P. L. Lions, Nonlinear scalar field equations, Ⅱ, Arch. Ration. Mech. Anal., 82 (1983), 347-375.  doi: 10.1007/BF00250556. [6] 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. [7] 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. [8] 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. [9] D. Cao and X. Zhu, On the existence and nodal character of solutions of semilinear elliptic equations, Acta Math. Sci., 8 (1988), 345-359. [10] 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. [11] E. N. Dancer, New solutions of equations on $\mathbb{R}^n$, Ann. Sc. Norm. Super. Pisa Cl. Sci.(4), 30 (2001), 535-563. [12] 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. [13] 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. [14] 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. [15] 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. [16] 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. [17] 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. [18] 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. [19] 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. [20] 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. [21] 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. [22] 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. [23] 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. [24] 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. [25] 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. [26] 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. [27] 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. [28] 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. [29] 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. [30] 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. [31] G. Talenti, Best constant in Sobolev inequality, Ann. Math. Pura Appl., 110 (1976), 353-372.  doi: 10.1007/BF02418013. [32] 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.
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