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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
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##### 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
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