Hostname: page-component-669899f699-7tmb6 Total loading time: 0 Render date: 2025-04-26T09:18:14.465Z Has data issue: false hasContentIssue false
  • English
  • Français

On critical exponents of a k-Hessian equation in the whole space

Part of: Partial differential equations Elliptic equations and systems

Published online by Cambridge University Press:  18 January 2019

Yun Wang  and
Yutian Lei
Yun Wang
Affiliation:
School of Mathematical Sciences, Institute of Mathematics, Nanjing Normal University, Nanjing 210023, China ([email protected])
Yutian Lei
Affiliation:
School of Mathematical Sciences, Institute of Mathematics, Nanjing Normal University, Nanjing 210023, China ([email protected])
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper, we study negative classical solutions and stable solutions of the following k-Hessian equation

$$F_k(D^2V) = (-V)^p\quad {\rm in}\;\; R^n$$
with radial structure, where n ⩾ 3, 1 < k < n/2 and p > 1. This equation is related to the extremal functions of the Hessian Sobolev inequality on the whole space. Several critical exponents including the Serrin type, the Sobolev type, and the Joseph-Lundgren type, play key roles in studying existence and decay rates. We believe that these critical exponents still come into play to research k-Hessian equations without radial structure.

Keywords

k-Hessian equationstable solutioncritical exponentLiouville theoremdecay rate

MSC classification

Primary: 35B33: Critical exponents
Secondary: 35J60: Nonlinear elliptic equations
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 149 , Issue 6 , December 2019 , pp. 1555 - 1575
Copyright
Copyright © Royal Society of Edinburgh 2019 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

1Bahri, A. and Lions, P. L.. Solutions of superlinear elliptic equations and their Morse indices. Comm. Pure Appl. Math. 45 (1992), 1205–1215.Google Scholar
2Caffarelli, L., Nirenberg, L. and Spruck, J.. The Dirichlet problem for nonlinear second elliptic equations, III. Functions of the eigenvalues of the Hessian. Acta Math. 155 (1985), 261–301.Google Scholar
3Chen, W. and Li, C.. Classification of solutions of some nonlinear elliptic equations. Duke Math. J. 63 (1991), 615–622.Google Scholar
4Chen, W. and Li, C.. Radial symmetry of solutions for some integral systems of Wolff type. Discrete Contin. Dyn. Syst. 30 (2011), 1083–1093.Google Scholar
5Chou, K. and Chu, C.. On the best constant for a weighted Sobolev-Hardy inequality. J. London Math. Soc. 2 (1993), 137–151.Google Scholar
6Clément, P., de Figueiredo, D. and Mitidieri, E.. Quasilinear elliptic equations with critical exponents. Topol. Methods Nonlinear Anal. 7 (1996), 133–170.Google Scholar
7Damascelli, L., Farina, A., Sciunzi, B. and Valdinoci, E.. Liouville results for m-Laplace equations of Lane-Emden-Fowler type. Ann. Inst. H. Poincare Anal. Non Lineaire 26 (2009), 1099–1119.Google Scholar
8Dancer, E. N.. Stable solutions on R n and the primary branch of some non-self-adjoint convex problems. Differ. Integral Equ. 17 (2004), 961–970.Google Scholar
9Dancer, E. N.. Stable and finite Morse index solutions on Rn or on bounded domains with small diffusion. Trans. Amer. Math. Soc. 357 (2005), 1225–1243.Google Scholar
10Dancer, E. N.. Stable and finite Morse index solutions on Rn or on bounded domains with small diffusion II. Indiana univ. Math. J. 53 (2004), 97–108.Google Scholar
11Dancer, E. N., Du, Y. and Guo, Z.. Finite Morse index solutions of an elliptic equation with supercritical exponent. J. Differ. Equ. 250 (2011), 3281–3310.Google Scholar
12Du, Y. and Guo, Z.. Positive solutions of an elliptic equation with negative exponent: stability and critical power. J. Differ. Equ. 246 (2009), 2387–2414.Google Scholar
13Farina, A.. On the classification of solutions of the Lane-Emden equation on unbounded domains of R n. J. Math. Pures Appl. 87 (2007), 537–561.Google Scholar
14Fazly, M. and Wei, J.. On finite Morse index solutions of higher order fractional Lane-Emden equations. Amer. J. Math. 139 (2017), 433–460.Google Scholar
15Gidas, B. and Spruck, J.. Global and local behavior of positive solutions of nonlinear elliptic equations. Comm. Pure Appl. Math. 34 (1981), 525–598.Google Scholar
16Gui, C., Ni, W.-M. and Wang, X.-F.. On the stability and instability of positive steady states of a semilinear heat equation in R n. Comm. Pure Appl. Math. 45 (1992), 1153–1181.Google Scholar
17Guo, Z. and Wei, J.. Global solution branch and Morse index estimates of a semilinear elliptic equation with super-critical exponent. Trans. Amer. Math. Soc. 363 (2011), 4777–4799.Google Scholar
18Ivochkina, N. M. and Ladyzhenskaya, O. A.. The first initial-boundary value problem for evolution equations generated by traces of order m of the Hessian of the unknown surface. Russian Acad. Sci. Dokl. Math. 50 (1995), 61–65.Google Scholar
19Joseph, D. and Lundgren, T.. Quasilinear Dirichlet problems driven by positive sources. Arch. Rational Meth. Anal. 49 (1972/73), 241–269.Google Scholar
20Labutin, D.. Potential estimates for a class of fully nonlinear elliptic equations. Duke Math. J. 111 (2002), 1–49.Google Scholar
21Lei, Y.. Critical conditions and finite energy solutions of several nonlinear elliptic PDEs in R n. J. Differ. Equ., 258 (2015), 4033–4061.Google Scholar
22Lei, Y. and Li, C.. Integrability and asymptotics of positive solutions of a γ-Laplace system. J. Differ. Equ. 252 (2012), 2739–2758.Google Scholar
23Lei, Y. and Li, C.. Sharp criteria of Liouville type for some nonlinear systems. Discrete Contin. Dyn. Syst. 36 (2016), 3277–3315.Google Scholar
24Li, Y. and Ni, W.-M.. On conformal scalar curvature equations in R n. Duke Math. J. 57 (1988), 895–924.Google Scholar
25Ma, C., Chen, W. and Li, C.. Regularity of solutions for an integral system of Wolff type. Adv. Math. 226 (2011), 2676–2699.Google Scholar
26Miyamoto, Y.. Intersection properties of radial solutions and global bifurcation diagrams for supercritical quasilinear elliptic equations. Nonlinear Diff. Equ. Appl. (NoDEA) 23 (2016), 1–24.Google Scholar
27Ni, W.-M.. On the elliptic equation Δu + K(x)u (n + 2)/(n − 2) = 0, its generalizations, and applications in geometry. Indiana Univ. Math. J. 31 (1982), 493–529.Google Scholar
28Ou, Q.. Nonexistence results for Hessian inequality. Methods Appl. Anal. 17 (2010), 213–224.Google Scholar
29Phuc, N. and Verbitsky, I.. Quasilinear and Hessian equations of Lane-Emden type. Ann. Math. 168 (2008), 859–914.Google Scholar
30Ren, C.. The first initial-boundary value problem for fully nonlinear parabolic equations generated by functions of the eigenvalues of the Hessian. J. Math. Anal. Appl. 339 (2009), 1362–1373.Google Scholar
31Serrin, J. and Zou, H.. Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta Math. 189 (2002), 79–142.Google Scholar
32Sun, S. and Lei, Y.. Fast decay estimates for integrable solutions of the Lane-Emden type integral systems involving the Wolff potentials. J. Funct. Anal. 263 (2012), 3857–3882.Google Scholar
33Tian, G.-J. and Wang, X.-J.. Moser-Trudinger type inequalities for the Hessian equation. J. Funct. Anal. 259 (2010), 1974–2002.Google Scholar
34Tso, K.. Remarks on critical exponents for Hessian operators. Ann. Inst. H. Poincare Anal. Non Lineaire 7 (1990), 113–122.Google Scholar
35Villavert, J.. A characterization of fast decaying solutions for quasilinear and Wolff type systems with singular coefficients. J. Math. Anal. Appl. 424 (2015), 1348–1373.Google Scholar
36Wang, X.-F.. On the Cauchy problem for reaction-diffusion equations. Trans. Amer. Math. Soc. 337 (1993), 549–590.Google Scholar
37Wang, X.-J.. A class of fully nonlinear elliptic equations and related functionals. Indiana Univ. Math. J. 43 (1994), 25–54.Google Scholar
38Wang, G. and Liu, H.. Some results on evolutionary equations involving functions of the eigenvalues of the Hessian. Northeast. Math. J. 13 (1997), 433–448.Google Scholar