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Asymptotic behavior of least energy solutions to the Lane-Emden system near the critical hyperbola. (English. French summary) Zbl 1437.35065

The following Lane-Emden system \[ -\Delta u(x) = v(x)^p,\;\; -\Delta v(x) = u(x)^q, \ u(x)>0,\; v(x) >0, \; x \in \Omega, \;\; u|_{\partial \Omega} = v|_{\partial \Omega} = 0, \] where \(\Omega \subset \mathbb{R}^n\) is a smooth bounded domain (\(n \geq 3\)) and \(0 < p < q < \infty\) is considered. A general result on the asymptotic behavior of least energy solutions for the above system near the critical hyperbola is proved in this paper. For \(p\geq 1\) and the domain \(\Omega\) is convex, this kind of result was obtained by I. A. Guerra [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 25, No. 1, 181–200 (2008; Zbl 1136.35025)]. In this paper, the author allows \(p<1\) and \(\Omega\) being arbitrary smooth bounded domains, which also answers a conjecture proposed by I. A. Guerra.

MSC:

35B40 Asymptotic behavior of solutions to PDEs
35B33 Critical exponents in context of PDEs
35J47 Second-order elliptic systems
35J61 Semilinear elliptic equations

Citations:

Zbl 1136.35025
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References:

[1] Ackermann, N.; Clapp, M.; Pistoia, A., Boundary clustered layers near the higher critical exponents, J. Differ. Equ., 254, 4168-4193 (2013) · Zbl 1288.35240
[2] Bahri, A.; Li, Y. Y.; Rey, O., On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calc. Var. Partial Differ. Equ., 3, 67-93 (1995) · Zbl 0814.35032
[3] Ben Ayed, M.; El Mehdi, K., On a biharmonic equation involving nearly critical exponent, Nonlinear Differ. Equ. Appl., 13, 485-509 (2006) · Zbl 1162.35375
[4] Bonheure, D.; Moreira dos Santos, E.; Ramos, M., Ground state and non-ground state solutions of some strongly coupled elliptic systems, Trans. Am. Math. Soc., 364, 447-491 (2012) · Zbl 1250.35093
[5] Bonheure, D.; Moreira dos Santos, E.; Ramos, M., Symmetry and symmetry breaking for ground state solutions of some strongly coupled elliptic systems, J. Funct. Anal., 264, 62-96 (2013) · Zbl 1278.35069
[6] Bonheure, D.; Moreira dos Santos, E.; Ramos, M.; Tavares, H., Existence and symmetry of least energy nodal solutions for Hamiltonian elliptic systems, J. Math. Pures Appl., 104, 1075-1107 (2015) · Zbl 1328.35039
[7] Bonheure, D.; Moreira dos Santos, E.; Tavares, H., Hamiltonian elliptic systems: a guide to variational frameworks, Port. Math., 71, 301-395 (2014) · Zbl 1326.35126
[8] Calanchi, M.; Ruf, B., Radial and non radial solutions for Hardy-Hénon type elliptic systems, Calc. Var. Partial Differ. Equ., 38, 111-133 (2010) · Zbl 1193.35035
[9] Chen, X.; Li, C.; Ou, B., Classification of solutions for a system of integral equations, Commun. Partial Differ. Equ., 30, 59-65 (2005) · Zbl 1073.45005
[10] Choi, W.; Kim, S., Minimal energy solutions to the fractional Lane-Emden system: existence and singularity formation, Rev. Mat. Iberoam. (2019), in press · Zbl 1418.35358
[11] Chou, K.; Geng, D., Asymptotics of positive solutions for a biharmonic equation involving critical exponent, Differ. Integral Equ., 13, 921-940 (2000) · Zbl 0977.35043
[12] Clément, P.; de Figueiredo, D. G.; Mitidieri, E., Positive solutions of semilinear elliptic systems, Commun. Partial Differ. Equ., 17, 923-940 (1992) · Zbl 0818.35027
[13] El Mehdi, K., Single blow-up solutions for a slightly subcritical biharmonic equation, Abstr. Appl. Anal., 2006, Article 18387 pp. (2006) · Zbl 1155.35331
[14] de Figueiredo, D. G.; Felmer, P., On superquadratic elliptic systems, Trans. Am. Math. Soc., 343, 99-116 (1994) · Zbl 0799.35063
[15] de Figueiredo, D. G.; Lions, P. L.; Nussbaum, R. D., A priori estimates and existence of positive solutions of semilinear elliptic equations, J. Math. Pures Appl., 61, 41-63 (1982) · Zbl 0452.35030
[16] Geng, D., Location of the blow up point for positive solutions of a biharmonic equation involving nearly critical exponent, Acta Math. Sci. Ser. B Engl. Ed., 25, 283-295 (2005) · Zbl 1141.35367
[17] Guerra, I. A., Solutions of an elliptic system with a nearly critical exponent, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 25, 181-200 (2008) · Zbl 1136.35025
[18] Han, Z. C., Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 8, 159-174 (1991) · Zbl 0729.35014
[19] He, H.; Yang, J., Asymptotic behavior of solutions for Hénon systems with nearly critical exponent, J. Math. Anal. Appl., 347, 459-471 (2008) · Zbl 1156.35023
[20] Hulshof, J.; Mitidieri, E.; Van der Vorst, R. C.A. M., Strongly indefinite systems with critical Sobolev exponents, Trans. Am. Math. Soc., 350, 2349-2365 (1998) · Zbl 0908.35034
[21] Hulshof, J.; Van der Vorst, R. C.A. M., Differential systems with strongly indefinite structure, J. Funct. Anal., 114, 32-58 (1993) · Zbl 0793.35038
[22] Hulshof, J.; Van der Vorst, R. C.A. M., Asymptotic behaviour of ground states, Proc. Am. Math. Soc., 124, 2423-2431 (1996) · Zbl 0860.35029
[23] Mitidieri, E., A Rellich type identity and applications: identity and applications, Commun. Partial Differ. Equ., 18, 125-151 (1993) · Zbl 0816.35027
[24] Musso, M.; Pistoia, A., Multispike solutions for a nonlinear elliptic problem involving the critical Sobolev exponent, Indiana Univ. Math. J., 51, 541-579 (2002) · Zbl 1074.35037
[25] Quittner, P.; Souplet, P., Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts: Basler Lehrbücher (2007), Birkhäuser Verlag: Birkhäuser Verlag Basel · Zbl 1128.35003
[26] Rey, O., The role of the Green’s function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal., 89, 1-52 (1990) · Zbl 0786.35059
[27] Rey, O., The topological impact of critical points at infinity in a variational problem with lack of compactness: the dimension 3, Adv. Differ. Equ., 4, 581-616 (1999) · Zbl 0952.35051
[28] Stein, E. M.; Weiss, G., Fractional integrals in n-dimensional Euclidean space, J. Math. Mech., 7, 503-514 (1958) · Zbl 0082.27201
[29] Van der Vorst, R. C.A. M., Variational identities and applications to differential systems, Arch. Ration. Mech. Anal., 116, 375-398 (1992) · Zbl 0796.35059
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