The existence of topological solutions for the Chern-Simons equation with
two Higgs particles has been proved by Lin, Ponce and Yang . However, both the
uniqueness problem and the existence of non-topological solutions have been left open.
In this paper, we consider the case of one vortex at origin. Among others, we prove the
uniqueness of topological solutions and give a complete study of the radial solutions, in
particular, the existence of some non-topological solutions
Li Y, Lin C. A Nonlinear Elliptic PDE with Two Sobolev–Hardy Critical Exponents[J]. Archive for Rational Mechanics and Analysis, 2011, 203(3): 943-968.
Fall M M. On the Hardy-Poincar\\\u0027e inequality with boundary singularities[J]. Communications in Contemporary Mathematics, 2010, 14(03).
Ishiwata M, Nakamura M, Wadade H, et al. On the sharp constant for the weighted Trudinger–Moser type inequality of the scaling invariant form[J]. Annales De L Institut Henri Poincare-analyse Non Lineaire, 2014, 31(2): 297-314.
Shusen Yan · Jianfu Yang. Infinitely many solutions for an elliptic problem involving critical Sobolev and Hardy–Sobolev exponents. 2012.
Cerami G, Zhong X Y, Zou W, et al. On some nonlinear elliptic PDEs with Sobolev–Hardy critical exponents and a Li–Lin open problem[J]. Calculus of Variations and Partial Differential Equations, 2015, 54(2): 1793-1829.
Fall M M, Minlend I A, Thiam E H, et al. The role of the mean curvature in a Hardy-Sobolev trace inequality[J]. Nodea-nonlinear Differential Equations and Applications, 2014, 22(5): 1047-1066.
Yang J. Positive solutions for the Hardy–Sobolev–Maz\u0027ya equation with Neumann boundary condition[J]. Journal of Mathematical Analysis and Applications, 2015, 421(2): 1889-1916.
Jin T. Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy terms[J]. Annales De L Institut Henri Poincare-analyse Non Lineaire, 2011, 28(6): 965-981.
Chunhua Wang · Changlin Xiang. INFINITELY MANY SOLUTIONS FOR QUASILINEAR ELLIPTIC EQUATIONS INVOLVING DOUBLE CRITICAL TERMS AND BOUNDARY GEOMETRY. 2016.
Mousomi Bhakta. Infinitely many sign changing solutions of an elliptic problem involving critical Sobolev and Hardy-Sobolev exponent. 2014.
In this paper, we prove the uniqueness of topological multivortex solutions for the self-dual Maxwell–Chern–Simons U(1)U(1) model if the Chern–Simons coupling parameter is sufficiently large and the charge of electron is sufficiently small or large. On the other hand, we also establish the sharp region of the flux for non-topological solutions and provide the classification of radial solutions of all types in the case of one vortex point.