Transactions of the American Mathematical Society

Author(s): Hwai-Chiuan F. Wang
Journal: Trans. Amer. Math. Soc. 352 (2000), 4237-4256.
MSC (1991): Primary 35J20, 35J25
Posted: March 16, 2000
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Let $\Omega \subset {\mathbb{R} }^{N}$ be the upper half strip with a hole. In this paper, we show there exists a positive higher energy solution of semilinear elliptic equations in $\Omega$ and describe the dynamic systems of solutions of equation

in various $\Omega$ . We also show there exist at least two positive solutions of perturbed semilinear elliptic equations in $\Omega$ .

1.: R.A. Adams, Sobolev Spaces, Academic Press, New York, 1975. MR 56:9247
2.: V. Benci and G. Cerami, Positive solutions of some nonlinear elliptic problems in exterior domains, Arch. Rational Mech. Anal., 99 (1987), 282-300. MR 88f:35014
3.: H. Berestycki and P. L. Lions, Nonlinear scalar field equations. I. Existence of ground state, Arch. Rat. Mech. Anal., 82 (1983), 313-345. MR 84h:35054a
4.: H. Brezis and T. Kato, Remarks on the Schrodinger operator with singular complex potentials, J. Math. Pures Appl., 58 (1979), 137-151. MR 80i:35135
5.: H. Brezis and L. Nirenberg, Remarks on finding critical points, Comm. on Pure and Appl. Math., 44 (1991), 939-963. MR 92i:58032
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).
8.: J. M. Coron, Topologie et cas limite des injections de Sobolev, C. R. Acad. Sci. Paris Ser. I Math., 299 (1984), 209-212. MR 86b:35059
9.: E. N. Dancer, On the influence of domain shape on the existence of large solutions of some superlinear problems, Math. Ann., 285 (1989), 647-669. MR 91h:35122
10.: I. Ekeland, Nonconvex minimization problems, Bull. Amer. Math. Soc., 1 (1979), 443-474. MR 80h:49007
11.: M. J. Esteban and P. L. Lions, Existence and non-existence results for semilinear elliptic problems in unbounded domains, Proc. Royal Soc. Edinburgh, 93A (1982), 1-14. MR 85e:35047
12.: B. Gidas, W. M. Ni and L. Nirenburg, Symmetry of positive solutions of nonlinear elliptic equations in ${\mathbb{R} }^{N}$ , Adv. in Math. Suppl. Stud., 7A (1981), 369-402. MR 84a:35083
13.: D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of Second order, Springer-Verlag, New York, 1983. MR 86c:35035
14.: T. S. Hsu and H. C. Wang, A perturbation result of semilinear elliptic equations in exterior strip domains, Proc. Royal Soc. Edinburgh, 127A (1997), 983-1004. MR 98h:35063
15.: M. K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^{p}=0$ in ${\mathbb{R} }^{N}$ , Arch. Rat. Mech. Anal., 105 (1989), 243-266. MR 90d:35015
16.: W.C. Lien, S.Y. Tzeng and H.C. Wang, Existence of solutions of semilinear elliptic problems on unbounded domains, Differential Integral Equations 6 (1993), 1281-1298. MR 94h:35061
17.: J. L. Lions and E. Zuazua, Approximate controllability of a hydro-elastic coupled system, ESMAIM Contrôle Optim. Calc. Var. 1 (1995/96), 1-15. MR 97a:93012
18.: P. L. Lions, The concentration-compactness principle in the calculus of variations, the limit case, Rev. Mat. Iberoamericana, 1 (1985), Part 1, 45-121, Part 2, 145-201. MR 87j:49012; MR 87c:49007
19.: S. I. Pohozaev, Eigenfunctions of the equation $\Delta u+\lambda f(u)=0$ , Soviet Math. Dokl., 6 (1965), 1408-1411.
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.: X.P. Zhu, A perturbation result on positive entire solutions of a semilinear elliptic equations, J. Differential Equations, 92 (1991), 163-178. MR 92g:35019

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