×

zbMATH — the first resource for mathematics

Topological degree for mean field equations on \(S^2\). (English) Zbl 0964.35038
The very interesting paper under review is devoted to the topological degree theory for the mean field equation on the unit sphere \(S^2\) of \(\mathbb{R}^3\) equipped with the metric \(g_0\) induced from the flat metric of \(\mathbb{R}^3.\) Precisely, let \(f\) be a positive smooth function of \(S^2,\) \(\Delta\) the Laplace-Beltrami operator on \((S^2,g_0),\) \(d\mu\) the volume form with respect to \(g_0\) and \(\rho>0\) a constant. The mean field equation \[ \Delta \phi +\rho\left( {{f(y) e^\phi} \over {\int_{S^2} f(y) e^\phi d\mu}} - {1\over {4\pi}}\right)=0\quad \text{on} S^2 \tag{1} \] provides the Euler-Lagrange equation of the nonlinear functional \[ J_\rho(\phi)={{1}\over{2}} \int_{S^2} |\nabla \phi|^2 d\mu -\rho\log\left( \int_{S^2} f(y)e^\phi d\mu\right) \] for \(\phi\) satisfying the normalizing condition \[ \int_{S^2} \phi(y) d\mu(y)=0 \] and lying in the Sobolev space \(H^1(S^2)\) of functions with \(L^2\)-integrable first derivatives.
The author studies symmetric properties of the concentrated solutions to \((1)\) and apply them to the calculation of the Leray-Schauder degree in the cases \(f(y)\equiv 1\) and \(f(y)\equiv \exp (-\gamma\langle n, y\rangle)\) with positive constant \(\gamma\) and \(n\) being a unit vector in \(\mathbb{R}^3.\)

MSC:
35J60 Nonlinear elliptic equations
58J05 Elliptic equations on manifolds, general theory
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] C. Bandle, Isoperimetric Inequalities and Applications , Monogr. Stud. Math. 7 , Pitman, Boston, 1980. · Zbl 0436.35063
[2] E. Caglioti, P.-L. Lions, C. Marchioro, and M. Pulvirenti, A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description , Comm. Math. Phys. 143 (1992), 501–525. · Zbl 0745.76001
[3] –. –. –. –., A special class of stationary flows for two-dimensional Euler equations: A statistical mechanics description, II, Comm. Math. Phys. 174 (1995), 229–260. · Zbl 0840.76002
[4] S.-Y. A. Chang, M. J. Gursky, and P. C. Yang, The scalar curvature equation on 2- and 3-spheres , Calc. Var. Partial Differential Equations 1 (1993), 205–229. · Zbl 0822.35043
[5] S. Chanillo and M. Kiessling, Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry , Comm. Math. Phys. 160 (1994), 217–238. · Zbl 0821.35044
[6] W. X. Chen and C. Li, Classification of solutions of some nonlinear elliptic equations , Duke Math. J. 63 (1991), 615–622. · Zbl 0768.35025
[7] C.-C. Chen and C.-S. Lin, Estimate of the conformal scalar curvature equation via the method of moving planes, II , J. Differential Geom. 49 (1998), 115–178. · Zbl 0961.35047
[8] ——–, Singular limits of a nonlinear eigenvalue problem in two dimension ,
[9] K.-S. Cheng and C.-S. Lin, On the conformal Gaussian curvature equation in \(\R^2\) , J. Differential Equations 146 (1998), 226–250. · Zbl 0911.35044
[10] W. Ding, J. Jost, J. Li, and G. Wang, The differential equation \(\D u=8\pi-8\pi h e^u\) on a compact Riemann surface , Asian J. Math. 1 (1997), 230–248. · Zbl 0955.58010
[11] ——–, Existence results for mean field equations , · Zbl 0937.35055
[12] B. Gidas, W. M. Ni, and L. Nirenberg, “Symmetry of positive solutions of nonlinear elliptic equations in \(\R^n\)” in Mathematical Analysis and Applications, Part A , Adv. Math. Supp. Stud. 7a , Academic Press, New York, 1981, 369–402. · Zbl 0469.35052
[13] M. K.-H. Kiessling, Statistical mechanics of classical particles with logarithmic interactions , Comm. Pure Appl. Math. 46 (1993), 27–56. · Zbl 0811.76002
[14] Y. Y. Li, Harnack type inequality: The method of moving planes , Comm. Math. Phys. 200 (1999), 421–444. · Zbl 0928.35057
[15] Y. Y. Li and I. Shafrir, Blow-up analysis for solutions of \(-\D u=V e^u\) in dimension two , Indiana Univ. Math. J. 43 (1994), 1255–1270. · Zbl 0842.35011
[16] C. S. Lin, Uniqueness of conformal metrics with prescribed total curvature in \(\R^2\) , to appear in Calc. Var. Partial Differential Equations. · Zbl 0996.53007
[17] ——–, Uniqueness of solutions of the mean field equation on \(S^2\) , to appear in Arch. Rational Mech. Anal.
[18] M. Nolasco and G. Tarantello, On a sharp Sobolev-type inequality on two-dimensional compact manifolds , Arch. Rational Mech. Anal. 145 (1998), 161–195. · Zbl 0980.46022
[19] L. M. Polvani and D. G. Dritschel, Wave and vortex dynamics on the surface of a sphere , J. Fluid Mech. 255 (1993), 35–64. · Zbl 0793.76022
[20] J. Serrin, A symmetry problem in potential theory , Arch. Rational Mech. Anal. 43 (1971), 304–318. · Zbl 0222.31007
[21] M. Struwe and G. Tarantello, On multivortex solutions in Chern-Simons gauge theory , Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 1 (1998), 109–121. · Zbl 0912.58046
[22] G. Tarantello, Multiple condensate solutions for the Chern-Simons-Higgs theory , J. Math. Phys. 37 (1996), 3769–3796. · Zbl 0863.58081
[23] Z. Q. Wang, Symmetries and the calculations of degree , Chinese Ann. Math. Ser B. 10 (1989), 520–536. · Zbl 0705.58006
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.