×

zbMATH — the first resource for mathematics

Nonlinear elliptic equations on compact Riemannian manifolds and asymptotics of Emden equations. (English) Zbl 0755.35036
Summary: We study the asymptotics and the global solutions of the following Emden equations: \(-\Delta u=\lambda e^ u\) in a 3-dim domain \((\lambda>0)\) or \(-\Delta u=u^ q+\ell| x|^{-2}u\) \((q>1)\) in an \(N\)-dim domain. Precise behaviour is obtained by the use of Simon’s results on analytic geometric functionals. In the case of the first equation, or the second equation with \(\ell=0\) and \(q=(N+1)/(N-3)\) \((N>3)\), we point out how the asymptotics are described via the Möbius group on \(S^{N-1}\). For a conformally invariant equation \(-\Delta u=\varepsilon| u|^{4/(N-2)}u+\ell| x|^{-2}u\) \((\varepsilon=\pm 1)\) we prove the existence of a new type of solution of the form \(u(x)=| x|^{(2-N)/2}\omega(\Gamma(Ln| x|)(x/| x|))\), where \(\omega\) is defined on \(S^{N-1}\) and \(\Gamma\in C^ \infty\) \((\mathbb R;O(N))\). Finally, we extend and simplify the results of B. Gidas and J. Spruck [Commun. Pure Appl. Math. 34, 525–598 (1981; Zbl 0465.35003)] on semilinear elliptic equations on compact Riemannian manifolds by a systematic use of the Bochner-Lichnerowicz-Weitzenböck formula.

MSC:
35J60 Nonlinear elliptic equations
58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
35B40 Asymptotic behavior of solutions to PDEs
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
PDF BibTeX XML Cite
Full Text: DOI EuDML
References:
[1] Ambrosetti, A., Rabinowitz, P.H.: Dual variational methods in critical point theory and applications. J. Funct. Anal.41, 349-381 (1973) · Zbl 0273.49063 · doi:10.1016/0022-1236(73)90051-7
[2] Aubin, Th.: Problèmes isopérimétriques et espaces de Sobolev. J. Differ. Geom.11, 573-598 (1976) · Zbl 0371.46011
[3] Aubin, Th.: Nonlinear analysis on manifolds. Monge-Ampere equations. (Grundlehren Math. Wiss., vol. 252) Berlin Heidelberg New York: Springer 1982
[4] Aviles, P.: Local behaviour of solutions of some elliptic equations. Commun. Math. Phys.108, 177-192 (1987) · Zbl 0617.35040 · doi:10.1007/BF01210610
[5] Berger, M., Gauduchon, P., Mazet, E.: Le spectre d’une variété riemannienne. (Lect. Notes Math., vol. 194) Berlin Heidelberg New York: Springer 1971 · Zbl 0223.53034
[6] Bidaut-Veron, M.F., Veron, L.: Groupe conforme deS 2 et propriétés limites des solutions de ??u=?e u .C. R. Acad. Sci., Paris Sér. I.308, 493-498 (1989)
[7] Brezis, H., Lions, P.L.: A note on isolated singularities for linear elliptic equations. Math. Anal. Appl. Adv. Math., Suppl. Stud.7A, 263-266 (1981)
[8] Brezis, H., Veron, L.: Removable singularities for some nonlinear elliptic equations. Arch. Ration. Mech. Anal.75, 1-6 (1980) · Zbl 0459.35032 · doi:10.1007/BF00284616
[9] Caffarelli, L.A., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math.42, 271-297 (1989) · Zbl 0702.35085 · doi:10.1002/cpa.3160420304
[10] Chandrasekhar, S.: An introduction to the study of stellar structure. Dover: Publ. Inc. 1967 · Zbl 0149.24301
[11] Chang, S.Y.A., Yang, P.C.: Prescribing Gaussian curvature onS 2. Acta Math.159, 215-259 (1987) · Zbl 0636.53053 · doi:10.1007/BF02392560
[12] Chen, X.Y., Matano, H., Veron, L.: Anisotropic singularities of solutions of nonlinear elliptic equations in ?2. J. Funct. Anal.83, 50-97 (1989) · Zbl 0687.35020 · doi:10.1016/0022-1236(89)90031-1
[13] Crandall, M.C., Rabinowitz, P.H.: Bifurcation, perturbation of simple eigenvalues and linearized stability. Arch. Ration. Mech. Anal.52, 161-180 (1973) · Zbl 0275.47044 · doi:10.1007/BF00282325
[14] Emden, V.R.: Gaskugeln. Leipzig: Teubner 1897
[15] Fowler, R.H.: Further studies on Emden’s and similar differential equations. Q. J. Math.2, 259-288 (1931) · Zbl 0003.23502 · doi:10.1093/qmath/os-2.1.259
[16] Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. (Grundlehren Math. Wiss., vol. 224) Berlin Heidelberg New York: Springer 1983 · Zbl 0562.35001
[17] Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and related properties via the maximum principle. Commun. Math. Phys.68, 209-243 (1979) · Zbl 0425.35020 · doi:10.1007/BF01221125
[18] Gidas, B., Ni, W.M., Nirenberg, L.: Symmetry and positive solutions of nonlinear equations in ? N . Math. Anal. Appl. Part A. Adv. Math., Suppl. Stud.7A, 369-402 (1981) · Zbl 0469.35052
[19] Gidas, B., Spruck, J.: Global and local behaviour of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math.34, 525-598 (1981) · Zbl 0465.35003 · doi:10.1002/cpa.3160340406
[20] Kazdan, J., Warner, F.: Curvature functions for compact 2-manifolds. Ann. Math.99, 14-47 (1974) · Zbl 0273.53034 · doi:10.2307/1971012
[21] Lee, J.M., Parker, T.H.: The Yamabe problem. Bull. Am. Math. Soc.17 (1), 37-91 (1987) · Zbl 0633.53062 · doi:10.1090/S0273-0979-1987-15514-5
[22] Lichenerowicz, A.: Géométrie des groupes de transformations. Paris: Dunod 1958
[23] Lojasiewicz, S.: Ensembles semi-analytiques. Inst. Hautes Étud. Sci. notes (1965)
[24] Mignot, F., Puel, J.P.: Solutions radiales singulières de ??u=?e u ,C.R. Acad. Sci., Paris, Sér. I.307, 379-382 (1988) · Zbl 0683.35032
[25] Moser, J.: On nonlinear problem in differential geometry. In: Peixoto (ed.) Dynamical Systems, pp. 273-280. New York: Academic Press 1973
[26] Obata, M.: The conjectures on conformal transformations of Riemannian manifolds. J. Differ. Geom.6, 247-258 (1971) · Zbl 0236.53042
[27] Onofri, E.: On the positivity of the effective action in a theory of random surfaces. Commun. Math. Phys.86, 321-326 (1982) · Zbl 0506.47031 · doi:10.1007/BF01212171
[28] Rabinowitz, P.H.: Minimax methods in critical point theory with applications to differential equations Expository Lectures from the CBMS Regional Conference, Miami 1984. (Math. Sci. Regional Conf. Ser. in Math., Providence, RI: American Mathematical Society 1986 vol. 65)
[29] Simon, L.: Asymptotics for a class of nonlinear evolution equations with applications to geometric problems. Ann. Math.118, 525-571 (1983) · Zbl 0549.35071 · doi:10.2307/2006981
[30] Simon, L.: Isolated singularities of extrema of geometric variational problems, In: Giusti, E. (ed.) Harmonic Mappings and Minimal Immersions. (Lect. Notes Math., vol. 1161, pp. 206-277) Berlin Heidelberg New York: Springer 1985 · Zbl 0583.49028
[31] Simon, L.: Entire solutions of the minimal surface equation. J. Differ. Geom.30, 643-688 (1989) · Zbl 0687.53009
[32] Stein, E.M.: Topics in harmonic analysis. (Ann. Math. Stud.) Princeton: Princeton University Press 1970 · Zbl 0193.10502
[33] Talenti, G.: Best constant in Sobolev inequality. Ann. Mat. Pura Appl.110, 353-372 (1976) · Zbl 0353.46018 · doi:10.1007/BF02418013
[34] Veron, L.: Comportement asymptotique des solutions d’équations elliptiques semi-linéaires dans ? N . Ann. Mat. Pura Appl.127, 25-50 (1981) · Zbl 0467.35013 · doi:10.1007/BF01811717
[35] Veron, L.: Singular solutions of some nonlinear elliptic equations. Nonlinear Anal.5, 225-242 (1981) · Zbl 0457.35031 · doi:10.1016/0362-546X(81)90028-6
[36] Veron, L.: Singularities of some quasilinear equations. In: Ni, W.-M., Peletier, L.A., Serrin, J. (eds.) Nonlinear Diffusion Equations and their Equilibrium States, vol. II, Proceedings, Berkeley, CA 1986, pp. 333-365. Berlin Heidelberg New York: Springer 1988
[37] Vilenkin, N.: Fonctions spéciales et théorie de la représentation des groupes. Paris: Dunod 1969
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.