Infinitely Many Solutions for the Prescribed Boundary Mean Curvature Problem in BN

Wang, Liping ; Zhao, Chunyi

Canadian journal of mathematics, Tome 65 (2013) no. 4, pp. 927-960

Résumé

We consider the prescribed boundary mean curvature problem in ${{\mathbb{B}}^{N}}$ with the Euclidean metric $$\{_{\frac{\partial u}{\partial v}+\frac{N-2}{2}u=\frac{N-2}{2}\tilde{K}\left( x \right){{u}^{{{2}^{\#-1}}}}\,\,\,\,\,\,\text{on}{{\mathbb{S}}^{N-1}},}^{-\Delta u=0,\,\,\,\,\,\,u>0\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\text{in}{{\mathbb{B}}^{N}},}$$ where $\tilde{K}\left( x \right)$ is positive and rotationally symmetric on ${{\mathbb{S}}^{N-1}},{{2}^{\#}}=\frac{2\left( N-1 \right)}{N-2}$ . We show that if $\tilde{K}\left( x \right)$ has a local maximum point, then this problem has infinitely many positive solutions that are not rotationally symmetric on ${{\mathbb{S}}^{N-1}}$ .

Détail

DOI : 10.4153/CJM-2012-054-2

Mots-clés : 35J25, 35J65, 35J67, infinitely many solutions, prescribed boundary mean curvature, variational reduction