×

Positive solutions of some nonlinear elliptic problems in exterior domains. (English) Zbl 0635.35036

Consider the question of the existence of positive solutions u in \(H^ 1_ 0(\Omega)\) of (*): \(-\Delta u+\lambda u=| u|^{p-2} u\) in \(\Omega\), where \(\Omega \subset R^ N\) is an unbounded domain, \(\partial \Omega \neq \phi\) is bounded, \(\lambda \in R_+\), \(N\geq 3\), and \(2\leq p<2N/(N-2).\) After a nice discussion of recent results and the difficulty encountered when \(\Omega\) is unbounded - a lack of compactness of the embedding \(J: H^ 1_ 0(\Omega)\to L^ p(\Omega)\)- the authors examine the obstruction to the compactness and obtain some estimates of the energy levels where the Palais-Smale condition can fail. This enables them to prove the following two theorems.
Theorem A: If \(p<(2N-2)/(N-2)\) for \(N=3,4\) and \(p=1+8/N\) for \(4<N<8\), then there exists a \(\lambda_ c(\Omega)\) such that for \(\lambda \in (0,\lambda_ c)\), problem (*) has at least one positive solution. Theorem B: Let N and p be as in Theorem A and \(x_ 0\in R^ n-\Omega\). Then for all \(\lambda\) there is a \(\rho\) (\(\lambda)\) such that if \(R^ N-\Omega \subset B_{\rho}(x_ 0)=\{x\in R^ N:| x-x_ 0| \leq \rho \}\), problem (*) has at least one positive solution. The conditions on N and p are a consequence of K. McLeod and J. Serrin [Proc. Natl. Acad. Sci. USA 78, 6592-6595 (1981; Zbl 0474.35047)] and Theorem B shows that the geometry of \(\Omega\) plays a role in the existence question.
Reviewer: P.W.Schaefer

MSC:

35J65 Nonlinear boundary value problems for linear elliptic equations
35J20 Variational methods for second-order elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs

Citations:

Zbl 0474.35047
PDF BibTeX XML Cite
Full Text: DOI

References:

[1] Ambrosetti, A., & Rabinowitz, P. H., Dual variational methods in critical point theory and applications. J. Funct. Anal. 14 (1973), 349-381. · Zbl 0273.49063
[2] Bahri, A., & Coron, J. M., Sur une équation elliptique avec l’exposant critique de Sobolev. C. R. Acad. Sc. Paris 301 (1985), 345-348, and detailed paper to appear. · Zbl 0601.35040
[3] Benci, V., & Cerami, G. In preparation.
[4] Benci, V., & Fortunato, D., Some compact embedding theorems for weighted Sobolev spaces. Boll. U. M. I. (5), 13 B (1976), 832-843. · Zbl 0382.46018
[5] Berestycki, H., & Lions, P. L., Nonlinear scalar field equations. I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983), 313-346. · Zbl 0533.35029
[6] Berestycki, H., & Lions, P. L., Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch. Rational Mech. Anal. 82 (1983), 347-376. · Zbl 0556.35046
[7] Brezis, H., & Lieb, E., A relation between pointwise convergence of functions and convergence of functionals. Proc. Amer. Math. Soc. 88 (1983), 486-490. · Zbl 0526.46037
[8] Coffman, C. V., Uniqueness of the ground state solution for ?u ? u + u 3 = 0 and a variational characterization of the other solutions. Arch. Rational Mech. Anal. 46 (1972), 81-85. · Zbl 0249.35029
[9] Coffman, C. V., & Marcus, M. M., Existence theorems for superlinear elliptic Dirichlet problems in exterior domains. Preprint. · Zbl 0596.35048
[10] Coron, J. M., Topologie et cas limite des injections de Sobolev. C. R. Acad. Sci. Paris 299 (1984), 209-212. · Zbl 0569.35032
[11] Esteban, M. J., & Lions, P. L., Existence and non-existence results for semilinear elliptic problems in unbounded domains. Proc. Royal Edinbourgh Soc. 93 A (1982), 1-14. · Zbl 0506.35035
[12] Gidas, B., Ni, W. M., & Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in ?N. Mathematical Analysis and Applications, Part A, Advances in Mathematics Supplementary Studies, vol. 7 A, Academic Press (1981). · Zbl 0469.35052
[13] Hempel, J. A., Multiple solutions for a class of nonlinear boundary value problems. Indiana Univ. Math. J. 20 (1971), 983-996. · Zbl 0225.35045
[14] Hofer, H., Variational and topological methods in partially ordered Hilbert spaces. Math. Ann. 261 (1982), 493-514. · Zbl 0488.47034
[15] Lions, P. L., La méthode de concentration-compacité en calcul de variations. Séminaire Goulaouic-Meyer-Schwartz 1982-83, Exposé XIV, (Février 1983). École Polytechnique Palaiseau.
[16] Lions, P. L., The concentration-compactness principle in the Calculus of Variations ?The locally compact case?Part I. Ann. Inst. H. Poincaré?Analyse Nonlinéaire 1 (1984), 109-145. · Zbl 0541.49009
[17] Lions, P. L., The concentration-compactness principle in the Calculus of Variations. The locally compact case. Part II. Ann. Inst. H. Poincaré. Analyse Nonlinéaire 1 (1984), 223-283. · Zbl 0704.49004
[18] Lions, P. L., Solutions of Hartree-Fock equations for Coulomb systems. Preprint n. 8607 CEREMADE. · Zbl 0618.35111
[19] McLeod, K., & Serrin, J., Uniqueness of solutions of semilinear Poisson equations. Proc. Natl. Acad. Sci. U. S. A. 78, n. 11 (1981), 6592-6595. · Zbl 0474.35047
[20] Poho?aev, S., Eigenfunctions of the equation ?u + ?f(u). Soviet Math. Doklady 6 (1965), 1408-1411. · Zbl 0141.30202
[21] Rabinowitz, P. H., Variational methods for nonlinear eigenvalue problems, Eigenvalues of Nonlinear Problems. (G. Prodi, Ed.), C. I. M. E., Edizioni Cremonese, Roma, 1975, 141-195.
[22] Rabinowitz, P. H., Théorie du degré topologique et applications à des problèmes aux limites nonlinéaires. Notes. Lab. Analyse Numerique, Un. Paris VI, n. 75010 (1975).
[23] Strauss, W. A., Existence of solitary waves in higher dimensions. Comm. Math. Phys. 55 (1977), 149-162. · Zbl 0356.35028
[24] Struwe, M., A global compactness result for elliptic boundary value problems involving limiting nonlinearities. Math. Z. 187 (1984), 511-517. · Zbl 0545.35034
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.