Multiplicity of ground states for the scalar curvature equation. (English) Zbl 1440.35134
This interesting paper concerns the existence and multiplicity of radial ground states of the scalar curvature equation
\[
\Delta u+K(|x|)u^{\frac{n+2}{n-2}}=0,\quad x\in\mathbb{R}^n,
\tag{1} \]
where \(n\geq 3\), and the function \(K(r)\) is bounded between two positive constants \(\underline{K}\) and \(\overline{K}\). In the past several decades, the effects of the behavior of \(K(r)\) on the existence or non-existence of ground states have been studied by many authors. For instance, W. Ding and W. Ni [Duke Math. J. 52, 485–506 (1985; Zbl 0592.35048)] found that, for decreasing non-constant \(K(r)\), (1) admits infinitely many ground states with slow decay, and for increasing non-constant \(K(r)\) (1) has no ground states. R. A. Johnson et al. [Indiana Univ. Math. J. 43, No. 3, 1045–1077 (1994; Zbl 0818.35025)] showed that multiplicity of ground states can be produced if the derivative \(K'(r)\) changes sign many times. C.-C. Chen and C.-S. Lin [Commun. Partial Differ. Equations 24, No. 5–6, 785–799 (1999; Zbl 0953.58023)] found that if \(K(r)\) has a minimum and is sufficiently close to a constant, then (1) has many ground states with fast decay. J. Wei and S. Yan [J. Funct. Anal. 258, No. 9, 3048–3081 (2010; Zbl 1209.53028)] found infinitely many non-radial ground states when \(K(r)\) has a positive maximum.
In the present paper, the authors improve the results of Chen and Lin [loc. cit.] by considering a non-perturbative situation, and prove existence of multiple ground states assuming that the ratio \(\overline{K}/\underline{K}\) is smaller than some computable values. To obtain these accurate results, the authors perform a very careful phase plane analysis, and combine the geometric structure of the flows with an asymptotic result of Chen and Lin [loc. cit].
In the present paper, the authors improve the results of Chen and Lin [loc. cit.] by considering a non-perturbative situation, and prove existence of multiple ground states assuming that the ratio \(\overline{K}/\underline{K}\) is smaller than some computable values. To obtain these accurate results, the authors perform a very careful phase plane analysis, and combine the geometric structure of the flows with an asymptotic result of Chen and Lin [loc. cit].
Reviewer: Xingbin Pan (Shanghai)
MSC:
| 35J61 | Semilinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
References:
| [1] | Alarcón, S.; Quaas, A., Large number of fast decay ground states to Matukuma-type equations, J. Differ. Equ., 248, 866-892 (2010) · Zbl 1187.35077 · doi:10.1016/j.jde.2009.10.006 |
| [2] | Battelli, F.; Johnson, R., On positive solutions of the scalar curvature equation when the curvature has variable sign, Nonlinear Anal., 47, 1029-1037 (2001) · Zbl 1042.34550 · doi:10.1016/S0362-546X(01)00243-7 |
| [3] | Bianchi, G., Non-existence and symmetry of solutions to the scalar curvature equation, Commun. Partial Differ. Equ., 21, 229-234 (1996) · Zbl 0844.35025 · doi:10.1080/03605309608821182 |
| [4] | Bianchi, G., The scalar curvature equation on \({\mathbb{R}}^n\) and \({\mathbb{S}}^n\), Adv. Differ. Equ., 1, 857-880 (1996) · Zbl 0865.35044 |
| [5] | Bianchi, G., Non-existence of positive solutions to semilinear elliptic equations on \({\mathbb{R}}^n\) or \({\mathbb{R}}^n_+\) through the method of moving planes, Commun. Partial Differ. Equ., 22, 1671-1690 (1997) · Zbl 0910.35048 · doi:10.1080/03605309708821315 |
| [6] | Bianchi, G.; Egnell, H., An ODE approach to the equation \(\Delta u+K u^{\frac{n+2}{n-2}} =0\), in \({\mathbb{R}}^n\), Math. Z., 210, 137-166 (1992) · Zbl 0759.35019 · doi:10.1007/BF02571788 |
| [7] | Bianchi, G.; Egnell, H., A variational approach to the equation \(\Delta u+K u^{\frac{n+2}{n-2}} =0\) in \({\mathbb{R}}^n\), Arch. Ration. Mech. Anal., 122, 159-182 (1993) · Zbl 0803.35033 · doi:10.1007/BF00378166 |
| [8] | Cao, D.; Noussair, E.; Yan, S., On the scalar curvature equation \(-\Delta u = (1 + \varepsilon K)u^{(N+2)/(N-2)}\) in \({\mathbb{R}}^N\), Calc. Var. Partial Differ. Equ., 15, 403-419 (2002) · Zbl 1148.35319 · doi:10.1007/s00526-002-0137-1 |
| [9] | Chen, Cc; Lin, Cs, Blowing up with infinite energy of conformal metrics on \(S^n\), Commun. Partial Differ. Equ., 24, 785-799 (1999) · Zbl 0953.58023 · doi:10.1080/03605309908821446 |
| [10] | Chen, Cc; Lin, Cs, On the asymptotic symmetry of singular solutions of the scalar curvature equations, Math. Ann., 313, 229-245 (1999) · Zbl 0927.35034 · doi:10.1007/s002080050259 |
| [11] | Cheng, Ks; Chern, Jl, Existence of positive entire solutions of some semilinear elliptic equations, J. Differ. Equ., 98, 169-180 (1992) · Zbl 0781.35022 · doi:10.1016/0022-0396(92)90110-9 |
| [12] | Coddington, E.; Levinson, N., Theory of Ordinary Differential Equations (1955), New York: Mc Graw Hill, New York · Zbl 0064.33002 |
| [13] | Dalbono, F.; Franca, M., Nodal solutions for supercritical Laplace equations, Commun. Math. Phys., 347, 875-901 (2016) · Zbl 1433.35132 · doi:10.1007/s00220-015-2546-y |
| [14] | Ding, Wy; Ni, Wm, On the elliptic equation \(\Delta u+K u^{\frac{n+2}{n-2}} =0\) and related topics, Duke Math. J., 52, 485-506 (1985) · Zbl 0592.35048 · doi:10.1215/S0012-7094-85-05224-X |
| [15] | Flores, I.; Franca, M., Multiplicity results for the scalar curvature equation, J. Differ. Equ., 259, 4327-4355 (2015) · Zbl 1325.35050 · doi:10.1016/j.jde.2015.05.020 |
| [16] | Franca, M., Non-autonomous quasilinear elliptic equations and Ważewski’s principle, Topol. Methods Nonlinear Anal., 23, 213-238 (2004) · Zbl 1075.35013 · doi:10.12775/TMNA.2004.010 |
| [17] | Franca, M., Structure theorems for positive radial solutions of the generalized scalar curvature equation, Funkc. Ekvacioj, 52, 343-369 (2009) · Zbl 1188.35094 · doi:10.1619/fesi.52.343 |
| [18] | Franca, M., Fowler transformation and radial solutions for quasilinear elliptic equations. Part 2: nonlinearities of mixed type, Ann. Mat. Pura Appl., 189, 67-94 (2010) · Zbl 1183.35148 · doi:10.1007/s10231-009-0101-1 |
| [19] | Franca, M., Radial ground states and singular ground states for a spatial-dependent \(p\)-Laplace equation, J. Differ. Equ., 248, 2629-2656 (2010) · Zbl 1195.35156 · doi:10.1016/j.jde.2010.02.012 |
| [20] | Franca, M., Positive solutions for semilinear elliptic equations: two simple models with several bifurcations, J. Dyn. Differ. Equ., 23, 573-611 (2011) · Zbl 1254.35087 · doi:10.1007/s10884-010-9198-6 |
| [21] | Franca, M., Positive solutions of semilinear elliptic equations: a dynamical approach, Differ. Integr. Equ., 26, 505-554 (2013) · Zbl 1299.35127 |
| [22] | Franca, M.; Johnson, R., Ground states and singular ground states for quasilinear partial differential equations with critical exponent in the perturbative case, Adv. Nonlinear Stud., 4, 93-120 (2004) · Zbl 1045.35027 · doi:10.1515/ans-2004-0106 |
| [23] | Franca, M.; Sfecci, A., Entire solutions of superlinear problems with indefinite weights and Hardy potentials, J. Dyn. Differ. Equ., 30, 1081-1118 (2018) · Zbl 1405.35060 · doi:10.1007/s10884-017-9589-z |
| [24] | García-Huidobro, M.; Manasevich, R.; Yarur, C., On the structure of positive radial solutions to an equation containing \(p\)-Laplacian with weights, J. Differ. Equ., 223, 51-95 (2006) · Zbl 1170.35404 · doi:10.1016/j.jde.2005.04.012 |
| [25] | Johnson, R.; Pan, Xb; Yi, Yf, The Melnikov method and elliptic equation with critical exponent, Indiana Math. J., 43, 1045-1077 (1994) · Zbl 0818.35025 · doi:10.1512/iumj.1994.43.43046 |
| [26] | Kawano, N.; Ni, Wm; Yotsutani, S., A generalized Pohozaev identity and its applications, J. Math. Soc. Jpn., 42, 541-564 (1990) · Zbl 0714.35028 · doi:10.2969/jmsj/04230541 |
| [27] | Kawano, N.; Yanagida, E.; Yotsutani, S., Structure theorems for positive radial solutions to \(\Delta u+K(|x|) u^p =0\) in \({\mathbb{R}}^n\), Funkcial. Ekvac., 36, 557-579 (1993) · Zbl 0793.34024 |
| [28] | Li, Y.; Ni, Wm, On conformal scalar curvature equations in \({\mathbb{R}}^n\), Duke Math. J., 57, 895-924 (1988) · Zbl 0674.53048 · doi:10.1215/S0012-7094-88-05740-7 |
| [29] | Lin, Cs; Lin, Ss, Positive radial solutions for \(\Delta u+K u^{\frac{n+2}{n-2}} =0\) in \({\mathbb{R}}^n\) and related topics, Appl. Anal., 38, 121-159 (1990) · Zbl 0687.35034 · doi:10.1080/00036819008839959 |
| [30] | Lin, Ls; Liu, Zl, Multi-bump solutions and multi-tower solutions for equations on \({\mathbb{R}}^n\), J. Funct. Anal., 257, 485-505 (2009) · Zbl 1171.35114 · doi:10.1016/j.jfa.2009.02.001 |
| [31] | Naito, Y., Bounded solutions with prescribed numbers of zeros for the Emden-Fowler differential equation, Hiroshima Math. J., 24, 177-220 (1994) · Zbl 0805.34028 · doi:10.32917/hmj/1206128140 |
| [32] | Ni, Wm, On the elliptic equation \(\Delta u+K u^{\frac{n+2}{n-2}} =0\), its generalizations, and applications in geometry, Indiana Univ. Math. J., 31, 493-529 (1982) · Zbl 0496.35036 · doi:10.1512/iumj.1982.31.31040 |
| [33] | Noussair, Es; Yan, S., The scalar curvature equation on \({\mathbb{R}}^N\), Nonlinear Anal., 45, 483-514 (2001) · Zbl 1116.35050 · doi:10.1016/S0362-546X(99)00428-9 |
| [34] | Papini, D.; Zanolin, F., On the periodic boundary value problem and chaotic-like dynamics for nonlinear Hill’s equations, Adv. Nonlinear Stud., 4, 71-91 (2004) · Zbl 1069.34024 · doi:10.1515/ans-2004-0105 |
| [35] | Wei, J.; Yan, S., Infinitely many solutions for the prescribed scalar curvature problem on \({\mathbb{S}}^N\), J. Funct. Anal., 258, 3048-3081 (2010) · Zbl 1209.53028 · doi:10.1016/j.jfa.2009.12.008 |
| [36] | Yan, S., Concentration of solutions for the scalar curvature equation on \({\mathbb{R}}^N\), J. Differ. Equ., 163, 239-264 (2000) · Zbl 0954.35058 · doi:10.1006/jdeq.1999.3718 |
| [37] | Yanagida, E.; Yotsutani, S., Classification of the structure of positive radial solutions to \(\Delta u +K(|x|) u^p=0\) in \({\mathbb{R}}^n\), Arch. Ration. Mech. Anal., 124, 239-259 (1993) · Zbl 0819.35010 · doi:10.1007/BF00953068 |
| [38] | Yanagida, E.; Yotsutani, S., Existence of nodal fast-decay solutions to \(\Delta u +K(|x|) |u|^{p-1}u=0\) in \({\mathbb{R}}^n\), Nonlinear Anal., 22, 1005-1015 (1994) · Zbl 0823.35053 · doi:10.1016/0362-546X(94)90063-9 |
| [39] | Yanagida, E.; Yotsutani, S., Existence of positive radial solutions to \(\Delta u +K(|x|) u^p=0\) in \({\mathbb{R}}^n\), J. Differ. Equ., 115, 477-502 (1995) · Zbl 0813.34036 · doi:10.1006/jdeq.1995.1024 |
| [40] | Yanagida, E.; Yotsutani, S., Global structure of positive solutions to equations of Matukuma type, Arch. Ration. Mech. Anal., 134, 199-226 (1996) · Zbl 0862.35007 · doi:10.1007/BF00379534 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
