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A general law of large permanent
Nodal Bubble-Tower Solutions for a semilinear elliptic problem with competing powers
School of Mathematics and Statistics, Henan University, Kaifeng, Henan 475004, China |
$\begin{equation*}\begin{cases}-Δ u=|u|^{p-1}u-|u|^{q-1}u~\text{in}~\mathbb{R}^N, \\ u(x)\to 0, ~~\text{as}~|x|\to ∞, \end{cases}\end{equation*}$ |
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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