Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-10T21:28:53.075Z Has data issue: false hasContentIssue false
  • English
  • Français

Existence Theorems for a Class of Edge-Degenerate Elliptic Equations on Singular Manifolds

Part of: Elliptic equations and systems Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  17 February 2015

Haining Fan
Haining Fan*
Affiliation:
School of Sciences, China University of Mining and Technology, Xuzhou 221116, People's Republic of China School of Mathematics and Statistics, Wuhan University, Wuhan 430072, People's Republic of China, (fanhaining888@163.com)
Get access Rights & Permissions [Opens in a new window]

Abstract

In this paper we establish the Nehari manifold on edge Sobolev spaces and study some of their properties. Furthermore, we use these results and the mountain pass theorem to get non-negative solutions of a class of edge-degenerate elliptic equations on singular manifolds under different conditions.

Keywords

Nehari manifoldsingular manifoldsedge-degenerate

MSC classification

Primary: 35J20: Variational methods for second-order elliptic equations
Secondary: 58J05: Elliptic equations on manifolds, general theory
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 58 , Issue 2 , June 2015 , pp. 355 - 377
Copyright
Copyright © Edinburgh Mathematical Society 2015 

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.)

References

1.Chen, H.Liu, X. and Wei, Y.Existence theorem for a class of semilinear elliptic equations with critical cone Sobolev exponent, Annals Global Analysis Geom. 39 (2011), 27–43.CrossRefGoogle Scholar
2.Chen, H.Liu, X. and Wei, Y.Cone Sobolev inequality and Dirichlet problem for nonlinear elliptic equations on manifold with conical singularities, Calc. Var. PDEs 7 (2011), 1–22.Google Scholar
3.Chen, H.Liu, X. and Wei, Y.Multiple solutions for semilinear totally characteristic equations with subcritical or critical cone Sobolev exponents, J. Diff. Eqns 252 (2012), 4200–4228.CrossRefGoogle Scholar
4.Chen, H.Liu, X. and Wei, Y.Dirichlet problem for semilinear edge-degenerate elliptic equations with singular potential term, J. Diff. Eqns 252 (2012), 4289–4314.CrossRefGoogle Scholar
5.Egorov, Ju. V. and Schulze, B.-W.Pseudo-differential operators, singularities, applications, Operator Theory: Advances and Applications, Volume 93 (Birkhäuser, 1997).CrossRefGoogle Scholar
6.Fan, H. and Liu, X.Multiple positive solutions for degenerate elliptic equations with critical cone Sobolev exponents on singular manifolds, Electron. J. Diff. Eqns 2013 (2013), 1–22.Google Scholar
7.Fan, H. and Liu, X.Existence results for degenerate elliptic equations with critical cone Sobolev exponents, Acta Math. Sci. B 34(6) (2014), 1907–1921.CrossRefGoogle Scholar
8.Kondratiev, V. A.Boundary value problems for elliptic equations in domains with conical points, Tr. Mosk. Mat. Obs. 16 (1967), 209–292.Google Scholar
9.Mazzeo, R.Elliptic theory of differential edge operators, I, Commun. PDEs 16 (1991), 1615–1664.Google Scholar
10.Melrose, R.B. and Mendoza, G.A.Elliptic operators of totally characteristic type, Math. Sci. Res. Inst. 96 (1983), 47–83.Google Scholar
11.Melrose, R.B. and Piazza, P.Analytic K-theory on manifolds with corners, Adv. Math. 92 (1992), 1–26.CrossRefGoogle Scholar
12.Nehari, Z.On a class of nonlinear second-order differential equations, Trans. Am. Math. Soc. 95 (1960), 101–123.CrossRefGoogle Scholar
13.Schrohe, E. and Seiler, J.Ellipticity and invertibility in the cone algebra on Lp-Sobolev spaces, Integ. Eqns Operat. Theory 41 (2001), 93–114.CrossRefGoogle Scholar
14.Schulze, B.-W.Boundary value problems and singular pseudo-differential operators (Wiley, 1998).Google Scholar
15.Struwe, M.Variational methods, 2nd edn (Springer, 1996).CrossRefGoogle Scholar
16.Willem, M.Minimax theorems (Birkhäuser, 1996).CrossRefGoogle Scholar
17.Wu, T.F.On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Analysis Applic. 318 (2006), 253–270.CrossRefGoogle Scholar