In this paper, we study entire solutions of the Allen-Chan equation in one-dimensional Euclidean space. This equation is a scalar reaction-diffusion equation with a bistable nonlinearity.
It is well-known that this equation admits three different types of traveling fronts connecting two of its three constant states.
Under certain conditions on the wave speeds, the existence of entire solutions with merging these three traveling fronts is shown by constructing a suitable pair of super-sub-solutions.
Chern J, Chen Z, Tang Y, et al. Structure of solutions to a singular Liouville system arising from modeling dissipative stationary plasmas[J]. Discrete and Continuous Dynamical Systems, 2012, 33(6): 2299-2318.
Zhiyou Chen · Jannlong Chern. Topological multivortex solutions for the Chern–Simons system with two Higgs particles. 2016.
An elliptic equation arising from the study o fstatic solutions with prescribed zeros and poles of the Einstein equations coupled with the classical sigma model and an Abelian gauge field, is considered. We classify the solutions and establish the uniqueness of radially symmetric solutions. We also complete a classification of symmetric solutions of an elliptic equation on the sphere.
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.