×

Multi-peak solutions for super-critical elliptic problems in domains with small holes. (English) Zbl 1014.35028

This paper is devoted to the construction of solutions of the problem \[ \begin{cases} -\Delta u=u^{{N+2 \over N-2}+\varepsilon} \quad &\text{in } \Omega,\\ u>0\quad &\text{in }\Omega,\\ u=0 \quad &\text{on }\partial\Omega, \end{cases}\tag{1} \] where \(\Omega\) is a bounded domain with smooth boundary in \(\mathbb{R}^N\), \(N\geq 3\), and \(\varepsilon >0\) is a small parameter. The goal of this paper is to raise the issue of solvability (1) in a domain exhibiting multiple holes. The main result of the authors states that in such a situation, multi-peak solutions exist, consisting of the glueing of double-spikes associated with each of the holes.

MSC:

35J60 Nonlinear elliptic equations
PDF BibTeX XML Cite
Full Text: DOI Link

References:

[1] Bahri, A., Critical points at infinity in some variational problems, (1989), Longman New York · Zbl 0676.58021
[2] Bahri, A.; Coron, J.M., On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain, Comm. pure appl. math., 41, 255-294, (1988) · Zbl 0649.35033
[3] Bahri, A.; Li, Y.; Rey, O., On a variational problem with lack of compactness: the topological effect of the critical points at infinity, Calc. var. partial differential equations, 3, 67-93, (1995) · Zbl 0814.35032
[4] H. Brezis, Elliptic equations with limiting Sobolev exponent—the impact of topology, in, Proceedings 50th Anniv. Courant Inst, Comm. Pure Appl. Math. 39, 1986. · Zbl 0601.35043
[5] Brezis, H.; Peletier, L.A., Asymptotics for elliptic equations involving critical growth, () · Zbl 1151.35035
[6] Coron, J.M., Topologie et cas limite des injections de Sobolev, C. R. acad. sci. Paris, ser. I, 299, 209-212, (1984) · Zbl 0569.35032
[7] M. del Pino, P. Felmer, and, M. Musso, Two-bubble solutions in the super-critical Bahri-Coron’s problem, preprint. · Zbl 1142.35421
[8] Fitzpatrick, P.; Massabó, I.; Pejsachowicz, J., Global several-parameter bifurcation and continuation theorem: a unified approach via complementing maps, Math. ann., 263, 61-73, (1983) · Zbl 0519.58024
[9] Han, Z.C., Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent, Ann. inst. Poincaré, anal. non linéaire, 8, 159-174, (1991) · Zbl 0729.35014
[10] Kazdan, J.; Warner, F., Remarks on some quasilinear elliptic equations, Comm. pure appl. math., 28, 349-374, (1983)
[11] Passaseo, D., New nonexistence results for elliptic equations with supercritical nonlinearity, Differential integral equations, 8, 577-586, (1995) · Zbl 0821.35056
[12] Passaseo, D., Nontrivial solutions of elliptic equations with supercritical exponent in contractible domains, Duke math. J., 92, 429-457, (1998) · Zbl 0943.35034
[13] Pohozaev, S., Eigenfunctions of the equation δu +λf (u)=0, Soviet. math. dokl., 6, 1408-1411, (1965) · Zbl 0141.30202
[14] 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
[15] Rey, O., A multiplicity result for a variational problem with lack of compactness, J. nonlinear anal. TMA, 13, 1241-1249, (1989) · Zbl 0702.35101
[16] Rey, O., The topological impact of critical points in a variational problem with lack of compactness: the dimension 3, Adv. differential equations, 4, 581-616, (1999) · Zbl 0952.35051
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.