Multi-bubble nodal solutions for slightly subcritical elliptic problems in domains with symmetries

Bartsch, Thomas ; DʼAprile, Teresa ; Pistoia, Angela

Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013), p. 1027-1047 / Harvested from Numdam

Résumé

We study the existence of sign-changing solutions with multiple bubbles to the slightly subcritical problem -Δu=|u| 2 ⁎ -2-ϵ uinΩ,u=0on∂Ω, where Ω is a smooth bounded domain in $ {\mathbb{R}}^{N}$, $ N⩾3$, $ {2}^{⁎}=\frac{2N}{N-2}$ and $ \epsilon >0$ is a small parameter. In particular we prove that if Ω is convex and satisfies a certain symmetry, then a nodal four-bubble solution exists with two positive and two negative bubbles.

Détail

Publié le : 2013-01-01

MR 3132415 | Zbl 1288.35212

DOI : https://doi.org/10.1016/j.anihpc.2013.01.001
Classification:  35B40,  35J20,  35J65

Bibliographie

[1] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 (1976), 573-598 | MR 448404 | Zbl 0371.46011

[2] A. Bahri, J.M. Coron, On a nonlinear elliptic equation involving the critical Sobolev exponent: The effect of the topology of the domain, Comm. Pure Appl. Math. 41 (1988), 253-294 | MR 929280 | Zbl 0649.35033

[3] A. Bahri, Y. Li, O. Rey, On a variational problem with lack of compactness: The topological effect of the critical points at infinity, Calc. Var. Partial Differential Equations 3 (1995), 67-93 | MR 1384837 | Zbl 0814.35032

[4] T. Bartsch, Critical point theory on partially ordered Hilbert spaces, J. Funct. Anal. 186 (2001), 117-152 | MR 1863294 | Zbl 1211.58003

[5] T. Bartsch, T. DʼAprile, A. Pistoia, On the profile of sign-changing solutions of an almost critical problem in the ball, arXiv:1208.5903 | MR 3138492

[6] T. Bartsch, A. Micheletti, A. Pistoia, On the existence and the profile of nodal solutions of elliptic equations involving critical growth, Calc. Var. Partial Differential Equations 26 (2006), 265-282 | MR 2232205 | Zbl 1104.35009

[7] T. Bartsch, T. Weth, A note on additional properties of sign changing solutions to superlinear elliptic equations, Topol. Methods Nonlinear Anal. 22 (2003), 1-14 | MR 2037264 | Zbl 1094.35041

[8] M. Ben Ayed, K. El Mehdi, F. Pacella, Classification of low energy sign-changing solutions of an almost critical problem, J. Funct. Anal. 50 (2007), 347-373 | MR 2352484 | Zbl 1155.35371

[9] H. Brézis, L.A. Peletier, Asymptotics for elliptic equations involving critical growth, Partial Differential Equations and the Calculus of Variations, vol. I, Progr. Nonlinear Differential Equations Appl. vol. 1, Birkhäuser, Boston, MA (1989), 149-192 | MR 1034005

[10] L. Caffarelli, B. Gidas, J. Spruck, Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth, Comm. Pure Appl. Math. 42 (1989), 271-297 | MR 982351 | Zbl 0702.35085

[11] P. Cardaliaguet, R. Tahraoui, On the strict concavity of the harmonic radius in dimension $ N⩾3$, J. Math. Pures Appl. 81 (2002), 223-240 | MR 1894062 | Zbl 1027.31003

[12] M. Del Pino, P. Felmer, M. Musso, Multi-bubble solutions for slightly super-critical elliptic problems in domains with symmetries, Bull. Lond. Math. Soc. 35 (2003), 513-521 | MR 1979006 | Zbl 1109.35334

[13] M. Del Pino, P. Felmer, M. Musso, Multi-peak solutions for super-critical elliptic problems in domains with small holes, J. Differential Equations 182 (2002), 511-540 | MR 1900333 | Zbl 1014.35028

[14] M. Del Pino, P. Felmer, M. Musso, Two-bubble solutions in the super-critical Bahri–Coronʼs problem, Calc. Var. Partial Differential Equations 16 (2003), 113-145 | MR 1956850 | Zbl 1142.35421

[15] M. Grossi, F. Takahashi, Nonexistence of multi-bubble solutions to some elliptic equations on convex domains, J. Funct. Anal. 259 (2010), 904-917 | MR 2652176 | Zbl 1195.35147

[16] M. Flucher, J. Wei, Semilinear Dirichlet problem with nearly critical exponent, asymptotic location of hot spots, Manuscripta Math. 94 (1997), 337-346 | MR 1485441 | Zbl 0892.35061

[17] Z.C. Han, Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. Inst. H. Poincaré, Anal. Non Linéaire 8 (1991), 159-174 | Numdam | MR 1096602 | Zbl 0729.35014

[18] J. Kazdan, F. Warner, Remarks on some quasilinear elliptic equations, Comm. Pure Appl. Math. 28 (1975), 567-597 | MR 477445 | Zbl 0325.35038

[19] M. Musso, A. Pistoia, Tower of bubbles for almost critical problems in general domains, J. Math. Pures Appl. 93 (2010), 1-40 | MR 2579374 | Zbl 1183.35143

[20] A. Pistoia, O. Rey, Multiplicity of solutions to the supercritical Bahri–Coronʼs problem in pierced domains, Adv. Differential Equations 11 (2006), 647-666 | MR 2238023 | Zbl 1166.35333

[21] S.I. Pohoz̆Aev, On the eigenfunctions of the equation $ \mathrm{\Delta }u+\lambda f(u)=0$, Soviet Math. Dokl. 6 (1965), 1408-1411 | MR 192184 | Zbl 0141.30202

[22] O. Rey, Blow-up points of solutions to elliptic equations with limiting nonlinearity, Differential Integral Equations 4 (1991), 1155-1167 | MR 1133750 | Zbl 0830.35043

[23] O. Rey, The role of the Greenʼs function in a nonlinear elliptic equation involving the critical Sobolev exponent, J. Funct. Anal. 89 (1990), 1-52 | MR 1040954 | Zbl 0786.35059

[24] O. Rey, Proof of two conjectures of H. Brezis and L.A. Peletier, Manuscripta Math. 65 (1989), 19-37 | MR 1006624 | Zbl 0708.35032

[25] A. Pistoia, 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 | Numdam | MR 2310698 | Zbl 1166.35018

[26] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976), 353-372 | MR 463908 | Zbl 0353.46018