Previous |  Up |  Next

Article

Keywords:
conformal Hessian inequality; entire positive solution
Summary:
In this paper, we study the nonexistence of entire positive solution for a conformal $k$-Hessian inequality in $\mathbb {R}^n$ via the method of proof by contradiction.
References:
[1] Aubin, T.: Équations différentielles non linéaires et problème de Yamabe concernant la courbure scalaire. J. Math. Pur. Appl., IX. Sér. 55 (1976), 269-296 French. MR 0431287 | Zbl 0336.53033
[2] Aubin, T.: Problèmes isopérimétriques et espaces de Sobolev. J. Diff. Geom. 11 (1976), 573-598 French. DOI 10.4310/jdg/1214433725 | MR 0448404 | Zbl 0371.46011
[3] Bao, J., Ji, X., Li, H.: Existence and nonexistence theorem for entire subsolutions of $k$-Yamabe type equations. J. Differ. Equations 253 (2012), 2140-2160. DOI 10.1016/j.jde.2012.06.018 | MR 2946967 | Zbl 1260.35044
[4] Brezis, H.: Semilinear equations in $\mathbb{R}^N$ without condition at infinity. Appl. Math. Optimization 12 (1984), 271-282. DOI 10.1007/BF01449045 | MR 0768633 | Zbl 0562.35035
[5] Caffarelli, L., Gidas, B., Spruck, J.: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth. Commun. Pure Appl. Math. 42 (1989), 271-297. DOI 10.1002/cpa.3160420304 | MR 0982351 | Zbl 0702.35085
[6] Caffarelli, L., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations III: Functions of the eigenvalues of the Hessian. Acta Math. 155 (1985), 261-301. DOI 10.1007/BF02392544 | MR 0806416 | Zbl 0654.35031
[7] Gidas, B., Spruck, J.: Global and local behavior of positive solutions of nonlinear elliptic equations. Commun. Pure Appl. Math. 34 (1981), 525-598. DOI 10.1002/cpa.3160340406 | MR 0615628 | Zbl 0465.35003
[8] Ji, X., Bao, J.: Necessary and sufficient conditions on solvability for Hessian inequalities. Proc. Am. Math. Soc. 138 (2010), 175-188. DOI 10.1090/S0002-9939-09-10032-1 | MR 2550182 | Zbl 1180.35234
[9] Jiang, F., Trudinger, N. S.: Oblique boundary value problems for augmented Hessian equations I. Bull. Math. Sci. 8 (2018), 353-411. DOI 10.1007/s13373-018-0124-2 | MR 3826768 | Zbl 1411.35116
[10] Jiang, F., Trudinger, N. S., Yang, X.-P.: On the Dirichlet problem for a class of augmented Hessian equations. J. Differ. Equations 258 (2015), 1548-1576. DOI 10.1016/j.jde.2014.11.005 | MR 3295592 | Zbl 1309.35027
[11] Jin, Q., Li, Y., Xu, H.: Nonexistence of positive solutions for some fully nonlinear elliptic equations. Methods Appl. Anal. 12 (2005), 441-449. DOI 10.4310/MAA.2005.v12.n4.a5 | MR 2258318 | Zbl 1143.35322
[12] Keller, J. B.: On solutions of $\Delta u=f(u)$. Commun. Pure Appl. Math. 10 (1957), 503-510. DOI 10.1002/cpa.3160100402 | MR 0091407 | Zbl 0090.31801
[13] Li, A., Li, Y. Y.: On some conformally invariant fully nonlinear equations II: Liouville, Harnack and Yamabe. Acta Math. 195 (2005), 117-154. DOI 10.1007/BF02588052 | MR 2233687 | Zbl 1216.35038
[14] Lieberman, G. M.: Second Order Parabolic Differential Equations. World Scientific Publishing, Singapore (1996). DOI 10.1142/3302 | MR 1465184 | Zbl 0884.35001
[15] Loewner, C., Nirenberg, L.: Partial differential equations invariant under conformal or projective transformations. Contributions to Analysis: A Collection of Papers Dedicated to Lipman Bers Academic Press, New York (1974), 245-272. MR 0358078 | Zbl 0298.35018
[16] Osserman, R.: On the inequality $\Delta u\ge f(u)$. Pac. J. Math. 7 (1957), 1641-1647. DOI 10.2140/pjm.1957.7.1641 | MR 0098239 | Zbl 0083.09402
[17] Ou, Q.: Nonexistence results for Hessian inequality. Methods Appl. Anal. 17 (2010), 213-224. DOI 10.4310/MAA.2010.v17.n2.a5 | MR 2763578 | Zbl 1211.35293
[18] Ou, Q.: Singularities and Liouville theorems for some special conformal Hessian equations. Pac. J. Math. 266 (2013), 117-128. DOI 10.2140/pjm.2013.266.117 | MR 3105779 | Zbl 1284.35179
[19] Ou, Q.: A note on nonexistence of conformal Hessian inequalities. Adv. Math., Beijing 46 (2017), 154-158. MR 3627002 | Zbl 1389.26062
[20] Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20 (1984), 479-495. DOI 10.4310/jdg/1214439291 | MR 0788292 | Zbl 0576.53028
[21] Sheng, W., Trudinger, N. S., Wang, X.-J.: The $k$-Yamabe problem. Surv. Differ. Geom. 17 (2012), 427-457. DOI 10.4310/SDG.2012.v17.n1.a10 | MR 3076067 | Zbl 1382.53013
[22] Trudinger, N. S.: Remarks concerning the conformal deformation of Riemannian structures on compact manifolds. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 22 (1968), 265-274. MR 0240748 | Zbl 0159.23801
[23] Trudinger, N. S.: Recent developments in elliptic partial differential equations of Monge-Ampère type. International Congress of Mathematicians. Vol. III European Mathematical Society, Zürich (2006), 291-302. DOI 10.4171/022-3/15 | MR 2275682 | Zbl 1130.35058
[24] Viaclovsky, J. A.: Conformal geometry, contact geometry, and the calculus of variations. Duke Math. J. 101 (2000), 283-316. DOI 10.1215/S0012-7094-00-10127-5 | MR 1738176 | Zbl 0990.53035
[25] Yamabe, H.: On a deformation of Riemannian structures on compact manifolds. Osaka Math. J. 12 (1960), 21-37. MR 0125546 | Zbl 0096.37201
Partner of
EuDML logo