×

zbMATH — the first resource for mathematics

A Palais-Smale approach to problems in Esteban-Lions domains with holes. (English) Zbl 0951.35043
In this paper, the author considers the problem of finding positive solutions \(u>0\) to the equation (1) \(-\delta u+u= u^{p-1}\) in a domain \(\Omega \subset {\mathbb R}^N\) where \(2<p<2N/(N-1)\). It is known that (1) admits positive solutions when \(\Omega\) is bounded or when \(\Omega={\mathbb R}^N\). When \(\Omega\) is an unbounded proper domain, a non-existence result in (what the author calls) “Esteban-Lions domains” was proved by Esteban and Lions in [Proc. R. Soc. Edinb., Sect. A 93, 1-14 (1982; Zbl 0506.35035)]. Similarly to the famous non-existence result of Pohozaev in starshaped domains for a class of elliptic problems, both the topology and the geometry of the domain seem to influence strongly the existence of solutions of (1). For example, in [Differ. Integral Equ. 6, No. 6, 1281-1298 (1993; Zbl 0837.35051)], it is proved that when adding a ball to the Esteban-Lions domain, the existence of a ground state solution may be obtained. In this paper, the author asserts that, although (1) admits no ground state solution, a higher energy solution indeed exists in various Esteban-Lions domains with a hole, i.e. with a ball taken out from. This is done by a close study of the asymptotic behaviour of possible positive solutions to (1).
Dynamic systems of solutions of (1) are also studied and multiple solutions of (1) in the presence of a forcing term are examined.

MSC:
35J60 Nonlinear elliptic equations
35B40 Asymptotic behavior of solutions to PDEs
35A15 Variational methods applied to PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35J20 Variational methods for second-order elliptic equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Robert A. Adams, Sobolev spaces, Academic Press [A subsidiary of Harcourt Brace Jovanovich, Publishers], New York-London, 1975. Pure and Applied Mathematics, Vol. 65. · Zbl 0314.46030
[2] Vieri Benci and Giovanna Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal. 99 (1987), no. 4, 283 – 300. · Zbl 0635.35036
[3] H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Rational Mech. Anal. 82 (1983), no. 4, 313 – 345. , https://doi.org/10.1007/BF00250555 H. Berestycki and P.-L. Lions, Nonlinear scalar field equations. II. Existence of infinitely many solutions, Arch. Rational Mech. Anal. 82 (1983), no. 4, 347 – 375. · Zbl 0533.35029
[4] Haïm Brézis and Tosio Kato, Remarks on the Schrödinger operator with singular complex potentials, J. Math. Pures Appl. (9) 58 (1979), no. 2, 137 – 151. · Zbl 0408.35025
[5] Haïm Brezis and Louis Nirenberg, Remarks on finding critical points, Comm. Pure Appl. Math. 44 (1991), no. 8-9, 939 – 963. · Zbl 0751.58006
[6] K. J. Chen, K. C. Chen, and H. C. Wang, Symmetry of positive solutions of semilinear elliptic equations in infinite strip domains, J. Differential Equations, 148 (1998), 1-8. CMP 98:16
[7] K. J. Chen, C. S. Lee, and H. C. Wang, Semilinear elliptic problems in interior and exterior flask domains, commun. Appl. Nonlinear Anal., 6 (1999). · Zbl 1110.35303
[8] Jean-Michel Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Sér. I Math. 299 (1984), no. 7, 209 – 212 (French, with English summary). · Zbl 0569.35032
[9] E. N. Dancer, On the influence of domain shape on the existence of large solutions of some superlinear problems, Math. Ann. 285 (1989), no. 4, 647 – 669. · Zbl 0699.35103
[10] Ivar Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc. (N.S.) 1 (1979), no. 3, 443 – 474. · Zbl 0441.49011
[11] Maria J. Esteban and P.-L. Lions, Existence and nonexistence results for semilinear elliptic problems in unbounded domains, Proc. Roy. Soc. Edinburgh Sect. A 93 (1982/83), no. 1-2, 1 – 14. · Zbl 0506.35035
[12] B. Gidas, Wei Ming Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in \?\(^{n}\), Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York-London, 1981, pp. 369 – 402. · Zbl 0469.35052
[13] David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[14] Tsing-san Hsu and Hwai-chiuan Wang, A perturbation result of semilinear elliptic equations in exterior strip domains, Proc. Roy. Soc. Edinburgh Sect. A 127 (1997), no. 5, 983 – 1004. · Zbl 0884.35038
[15] Man Kam Kwong, Uniqueness of positive solutions of \Delta \?-\?+\?^{\?}=0 in \?\(^{n}\), Arch. Rational Mech. Anal. 105 (1989), no. 3, 243 – 266. · Zbl 0676.35032
[16] Wen Ching Lien, Shyuh Yaur Tzeng, and Hwai Chiuan Wang, Existence of solutions of semilinear elliptic problems on unbounded domains, Differential Integral Equations 6 (1993), no. 6, 1281 – 1298. · Zbl 0837.35051
[17] Jacques-Louis Lions and Enrique Zuazua, Approximate controllability of a hydro-elastic coupled system, ESAIM Contrôle Optim. Calc. Var. 1 (1995/96), 1 – 15. · Zbl 0878.93034
[18] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana 1 (1985), no. 2, 45 – 121. , https://doi.org/10.4171/RMI/12 P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana 1 (1985), no. 1, 145 – 201. · Zbl 0704.49005
[19] S. I. Pohozaev, Eigenfunctions of the equation \(\Delta u+\lambda f(u)=0\), Soviet Math. Dokl., 6 (1965), 1408-1411. · Zbl 0141.30202
[20] C. A. Stuart, Bifurcation in \(L^{p}({\mathbb{R} }^{N})\) for a semilinear elliptic equations, Proc. London Math. Soc., 45 (1982), 169-192.
[21] Xi Ping Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equation, J. Differential Equations 92 (1991), no. 2, 163 – 178. · Zbl 0739.35027
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.