## Multiple solutions to the pure critical exponent problem in domains with a hole of arbitrary size.(English)Zbl 1143.35052

Summary: We consider the problem $-\Delta u = \left| u\right|^{2^\ast-2} u\,{\text{in}}\,\Omega, \quad u = 0\,{\text{on}}\,\partial\Omega,$ where $$\Omega$$ is a bounded smooth domain in $$\mathbb{R}^{N}$$, $$N \geq 3$$, and $$2^{\ast}=\frac{2N}{N-2}$$ is the critical Sobolev exponent. We assume that $$\Omega$$ is annular shaped, i.e. there are constants $$R_{2} > R_{1} > 0$$ such that $$\{x \in \mathbb{R}^{N} : R_{1} < |x| < R_{2}\} \subset \Omega$$ and $$0 \not\in \Omega$$. We also assume that $$\Omega$$ is invariant under a group $$\Gamma$$ of orthogonal transformations of $$\mathbb{R}^{N}$$ without fixed points. We establish the existence of multiple sign changing solutions if, either $$\Gamma$$ is arbitrary and $$R_{1}/ R_{2}$$ is small enough, or $$R_{1}/ R_{2}$$ is arbitrary and the minimal $$\Gamma$$-orbit of $$\Omega$$ is large enough. We believe this is the first existence result for sign changing solutions in domains with holes of arbitrary size. The proof takes advantage of the invariance of this problem under the group of Möbius transformations.

### MSC:

 35J65 Nonlinear boundary value problems for linear elliptic equations 35J20 Variational methods for second-order elliptic equations 35B33 Critical exponents in context of PDEs
Full Text:

### References:

 [1] Bahri A. and Coron J.M. (1988). On a nonlinear elliptic equation involving the critical Sobolev exponent: the effect of the topology of the domain. Commun. Pure Appl. Math. 41: 253–294 · Zbl 0649.35033 [2] Beardon A.F. (1983). The Geometry of Discrete Groups. Springer, New York · Zbl 0528.30001 [3] Castro A., Cossio J. and Neuberger J.M. (1997). A sign changing solution for a superlinear Dirichlet problem. Rocky Mountain J. Math. 27: 1041–1053 · Zbl 0907.35050 [4] Clapp, M.: A global compactness result for elliptic problems with critical nonlinearity on symmetric domains, in Nonlinear Equations: Methods, Models and Applications, 117–126, PNLDE 54, Birkhauser, Boston (2003) · Zbl 1108.35332 [5] Clapp M. and Weth T. (2004). Minimal nodal solutions of the pure critical exponent problem on a symmetric domain. Calc. Var. 21: 1–14 · Zbl 1097.35048 [6] Clapp M. and Weth T. (2005). Multiple solutions for the Brezis–Nirenberg problem. Adv. Diff. Eq. 10: 463–480 · Zbl 1284.35151 [7] Clapp, M., Weth, T.: Two solutions of the Bahri–Coron problem in punctured domains via the fixed point transfer, Commun. Contemp. Math., (in press) · Zbl 1157.35048 [8] Coron, J.M.: Topologie et cas limite des injections de Sobolev, C.R. Acadamy of Science, Paris 299, Ser. I, 209–212 (1984) · Zbl 0569.35032 [9] Dancer E.N. (1988). A note on an equation with critical exponent. Bull. Lond. Math. Soc. 20: 600–602 · Zbl 0646.35027 [10] Ding W. (1989). Positive solutions of $$\Delta u + u^{2^{\ast} -1} = 0$$ on contractible domains. J. Partial Diff. Eq. 2: 83–88 · Zbl 0694.35067 [11] Kazdan J. and Warner F. (1975). Remarks on some quasilinear elliptic equations. Commun. Pure Appl. Math. 38: 557–569 · Zbl 0325.35038 [12] Lions, P.L.: Symmetries and the concentration compactness method, in Nonlinear Variational Problems, Pitman, London 47–56 (1985) · Zbl 0601.49005 [13] Marchi M.V. and Pacella F. (1993). On the existence of nodal solutions of the equation $$-\Delta u = \left| u\right| ^{2^\ast-2} u$$ with Dirichlet boundary conditions Diff. Int. Eq. 6: 849–862 · Zbl 0782.35022 [14] Musso M. and Pistoia A. (2006). Sign changing solutions to a nonlinear elliptic problem involving the critical Sobolev exponent in pierced domains. J. Math. Pure Appl. 86: 510–528 · Zbl 1215.35070 [15] Passaseo D. (1989). Multiplicity of positive solutions of nonlinear elliptic equations with critical Sobolev exponent in some contractible domains. Manuscripta Math 65: 147–165 · Zbl 0701.35068 [16] Passaseo D. (1994). The effect of the domain shape on the existence of positive solutions of the equation $$\Delta u + u^{2^{\ast}-1} = 0$$ Top. Meth. Nonl. Anal. 3: 27–54 [17] Pohožaev S. (1965). Eigenfunctions of the equation $$\Delta u + \lambda f (u) = 0$$ . Soviet Math. Dokl. 6: 1408–1411 · Zbl 0141.30202 [18] Struwe M. (1984). A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187: 511–517 · Zbl 0545.35034 [19] Struwe M. (1990). Variational Methods. Applications to nonlinear partial differential equations and Hamiltonian systems. Springer, Berlin · Zbl 0746.49010 [20] Willem M. (1996). Minimax theorems, PNLDE 24. Birkhäuser, Boston
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.