论文信息 - On a class of elliptic problems in R2: symmetry and uniqueness results

On a class of elliptic problems in R2: symmetry and uniqueness results

In the plane R2, we classify all solutions for an elliptic problem of Liouville type involving a (radial) weight function. As a consequence, we clarify the origin of the non-radially symmetric solutions for the given problem, as established by Chanillo and Kiessling. For a more general class of Liouville-type problems, we show that, rather than radial symmetry, the solutions always inherit the invariance of the problem under inversion with respect to suitable circles. This symmetry result is derived with the help of a 'shrinking-sphere' method.

J. Prajapat | G. Tarantello | Gabriella Tarantello

[1] Juncheng Wei,et al. On a conjecture of Wolansky , 2002 .

[2] M. Kiessling,et al. Surfaces with prescribed Gauss curvature , 2000 .

[3] Kuo‐shung Cheng,et al. Conformal metrics in R 2 with prescribed Gaussian curvature with positive total curvature , 1999 .

[4] G. Tarantello,et al. Double vortex condensates in the Chern-Simons-Higgs theory , 1999 .

[5] K. Chou,et al. Asymptotic radial symmetry for solutions of Δu + eu = 0 in a punctured disc , 1994 .

[6] M. Kiessling,et al. Rotational symmetry of solutions of some nonlinear problems in statistical mechanics and in geometry , 1994 .

[7] Congming Li,et al. QUALITATIVE PROPERTIES OF SOLUTIONS TO SOME NONLINEAR ELLIPTIC EQUATIONS IN R 2 , 1993 .

[8] Henri Berestycki,et al. On the method of moving planes and the sliding method , 1991 .

[9] Weinberg,et al. Self-dual Chern-Simons vortices. , 1990, Physical review letters.

[10] P. Aviles. Conformal complete metrics with prescribed non-negative Gaussian curvature in ℝ2 , 1986 .

[11] F. W. Warner,et al. Curvature Functions for Compact 2-Manifolds , 1974 .