Lower semicontinuity of Kirchhoff-type energy functionals and spectral gaps on (sub)Riemannian manifolds
DOI:
https://doi.org/10.12775/TMNA.2022.034Keywords
Critical Kirchhoff-type energy functional, spectral gaps, Riemannian manifolds, Heisenberg groupsAbstract
In this paper we characterize the sequentially weakly lower semicontinuity of the parameter-depending energy functional associated with the critical Kirchhoff problem in context of (sub)Riemannian manifolds. We also present some spectral gap and convexity results.References
C.O. Alves, F.J. Corrêa and G.M. Figueiredo, On a class of nonlocal elliptic problems with critical growth, Differ. Equ. Appl. 2 (2010), 409–417.
C.O. Alves, F.J. Corrêa and T.F. Ma, Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. Math. Appl. 49 (2005), 85–93.
T. Aubin, Problémes isopérimétriques et espaces de Sobolev, J. Differential Geometry 11 (1976), no. 4, 573–598.
G. Autuori, A. Fiscella and P. Pucci, Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity, Nonlinear Anal. 125 (2015), 699–714.
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.
Z. Balogh and A. Kristály, Sharp geometric inequalities in spaces with nonnegative Ricci curvature and Euclidean volume growth (2021), arXiv: 2012.11862.
C.Y. Chen, Y.C. Kuo and T.F. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Differential Equations 250 (2011), 1876–1908.
M. Chipot and B. Lovat, Some remarks on nonlocal elliptic and parabolic problems, Nonlinear Anal. 30 (1997), no. 7, 4619–4627.
F.J. Corrêa and G.M. Figueiredo, On an elliptic equation of p-Kirchhoff type via variational methods, Bull. Austral. Math. Soc. 74 (2006), 263–277.
B. Dacorogna, Direct methods in the calculus of variations, second edition, Applied Mathematical Sciences, vol. 78, Springer, New York, 2008, xii+619 pp.
Y. Deng and W. Shuai, Sign-changing multi-bump solutions for Kirchhoff-type equations in R3 , Discrete Contin. Dyn. Syst. 38 (2018), no. 6, 3139–3168.
H. Fan, Multiple positive solutions for a class of Kirchhoff type problems involving critical Sobolev exponents, J. Math. Anal. Appl. 431 (2015), 150–168.
F. Faraci and K. Silva, On the Brezis–Nirenberg problem for a Kirchhoff type equation in high dimension, Calc. Var. Partial Differential Equations 60 (2021), no. 1, paper no. 22, 33 pp.
R.L. Frank and E.H. Lieb, Sharp constants in several inequalities on the Heisenberg group, Ann. Math. 176 (2012), 349–381.
G.M. Figueiredo, Existence of a positive solution for a Kirchhoff problem type with critical growth via truncation argument, J. Math. Anal. Appl. 401 (2013), 706–713.
G.M. Figueiredo and J.R. Santos, Multiplicity of solutions for a Kirchhoff equation with subcritical or critical growth, Differential Integral Equations 25 (2012), 853–868.
E. Hebey, Nonlinear analysis on manifolds: Sobolev spaces and inequalities, Courant Lecture Notes in Mathematics, vol. 5. New York University, Courant Institute of Mathematical Sciences, New York; American Mathematical Society, Providence, RI, 1999.
E. Hebey, Compactness and the Palais–Smale property for critical Kirchhoff equations in closed manifolds, Pacific J. Math. 280 (2016), 41–50.
E. Hebey, Multiplicity of solutions for critical Kirchhoff type equations, Comm. Partial Differential Equations 41 (2016), 913–924.
E. Hebey and P.-D. Thizy, Kirchhoff systems in closed high dimensional manifolds, Commun. Contemp. Math. 18 (2016), no. 2, 1550028, 53 pp.
D. Jerison, J. M. Lee, Extremals for the Sobolev inequality on the Heisenberg group and the CR Yamabe problem, J. Amer. Math. Soc. 1 (1988), 1–13.
G. Kirchhoff, Mechanik, Teubner, Leipzig, 1883.
C.Y. Lei, G.S. Liu and L.T. Guo, Multiple positive solutions for a Kirchhoff type problem with a critical nonlinearity, Nonlinear Anal. Real World Appl. 31 (2016), 343–355.
D. Naimen, The critical problem of Kirchhoff type elliptic equations in dimension four, J. Differential Equations 257 (2014), 1168–1193.
K. Perera and Z. Zhang, Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. Differential Equations 221 (2006), 246–255.
K. Perera and Z. Zhang, Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. Math. Anal. Appl. 317 (2006), 456–463.
G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353–372.
X. Yao and C. Mu, Multiplicity of solutions for Kirchhoff type equations involving critical Sobolev exponents in high dimension, Math. Methods Appl. Sci. 39 (2016), 3722–3734.
M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and their Applications, vol. 24, Birkhäuser, Boston, 1996.
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Csaba Farkas, Sándor Kajántó, Csaba Varga
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
Stats
Number of views and downloads: 0
Number of citations: 0
