Multiple entire solutions of a semilinear elliptic equation. (English) Zbl 0671.35023

By the concentration-compactness principle the author proves the existence of a second entire solution to the semilinear elliptic equation \[ -\Delta u+u=q(x)| u|^{\gamma -1}u\quad on\quad {\mathbb{R}}^ n, \] \(1<\gamma <(n+2)/(n-2)\), if \(n\geq 5\), \(q(x)\geq q_ 0\geq 0\), \(\lim_{x\to \infty} q(x)=q_ 0\) and \(q(x)-q_ 0\geq c| x|^{- m}\) for \(| x|\) large in addition to the known positive solution given e.g. by P. L. Lions [Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1, 109-145 (1984; Zbl 0541.49009) and 223-283 (1984)] or E. S. Noussair and C. A. Swanson [Hiroshima Math. J. 15, 127-140 (1985; Zbl 0575.35025)]. Compare also the totally different technique of E. S. Noussair [Bull. Lond. Math. Soc. 19, 443-448 (1987; Zbl 0633.35025)] where the existence of a second positive solution for the slightly different equation \(-\Delta u=q(x)u^{\gamma}\) is proved.
Reviewer: M.Wiegner


35J60 Nonlinear elliptic equations
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
35J20 Variational methods for second-order elliptic equations
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[1] Adams, R. A., Sobolev Spaces (1975), Academic Press: Academic Press New York · Zbl 0186.19101
[2] Berestycki, H.; Lions, P. L., Nonlinear scalar field equations I, II, Archs ration. Mech. Analysis, 82, 313-376 (1983)
[3] Cerami, G.; Solomini, S.; Struwe, M., Some existence results for superlinear elliptic boundary value problems involving critical exponents, J. Funct. Analysis, 69, 289-306 (1986) · Zbl 0614.35035
[4] Ding, W. Y.; Ni, W. M., On the existence of positive entire solutions of semilinear elliptic equations, Archs ration. Mech. Analysis, 91, 283-308 (1986) · Zbl 0616.35029
[5] Gidas, B.; Ni, W. M.; Nirenberg, L., Symmetry of positive solutions of nonlinear elliptic equations in \(R^N\), Adv. Math. Suppl. Studies, 7A, 369-402 (1981)
[6] Gilbarg, D.; Trudinger, N. S., Elliptic Partial Differential Equations of Second Order (1983), Springer: Springer New York · Zbl 0691.35001
[7] Hofer, H., Variational and topological methods in partially ordered Hilbert spaces, Math. Annln, 261, 493-514 (1982) · Zbl 0488.47034
[8] Kato, T., Growth properties of solutions of the reduced wave equation with a variable coefficient, Communs pure appl. Math., 12, 403-425 (1959) · Zbl 0091.09502
[9] Lions, P. L., The concentration-compactness principle in the calculus of variations, Ann. Inst. H. Poincaré Analyse nonlinéaire, 1, 109-145 (1984), The locally compact case, part 1 · Zbl 0541.49009
[10] Lions, P. L., The concentration-compactness principle in the calculus of variations, Ann. Inst. H. Poincaré Analyse nonlinéaire, 1, 223-283 (1984), The locally compact case, part 2 · Zbl 0704.49004
[11] Miranda, C., Un’ osservazione sul teorema di Brouwer, Boll. Un. mat. ital. Ser. II, Ann. II u. \(1\), XIX, 5-7 (1940) · JFM 66.0217.01
[12] Nehari, Z., On a nonlinear differential equation arising in nuclear physics, Proc. R. Irish. Acad., 62, 117-135 (1963) · Zbl 0124.30204
[13] Strauss, W., Existence of solitary waves in higher dimensions, Communs math. Phys., 55, 149-162 (1977) · Zbl 0356.35028
[14] Yang, J. F.; Zhu, X. P., On the existence of nontrivial solutions of a quasilinear elliptic boundary value problems for unbounded domains, Acta math. Sci., 7, 341-359 (1987) · Zbl 0674.35030
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