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Search: MSC category 35J65 ( Nonlinear boundary value problems for linear elliptic equations )

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1. CJM 2012 (vol 65 pp. 927)

Wang, Liping; Zhao, Chunyi
 Infinitely Many Solutions for the Prescribed Boundary Mean Curvature Problem in $\mathbb B^N$ We consider the following prescribed boundary mean curvature problem in $\mathbb B^N$ with the Euclidean metric: $\begin{cases} \displaystyle -\Delta u =0,\quad u\gt 0 &\text{in }\mathbb B^N, \\[2ex] \displaystyle \frac{\partial u}{\partial\nu} + \frac{N-2}{2} u =\frac{N-2}{2} \widetilde K(x) u^{2^\#-1} \quad & \text{on }\mathbb S^{N-1}, \end{cases}$ where $\widetilde K(x)$ is positive and rotationally symmetric on $\mathbb S^{N-1}, 2^\#=\frac{2(N-1)}{N-2}$. We show that if $\widetilde K(x)$ has a local maximum point, then the above problem has infinitely many positive solutions that are not rotationally symmetric on $\mathbb S^{N-1}$. Keywords:infinitely many solutions, prescribed boundary mean curvature, variational reductionCategories:35J25, 35J65, 35J67

2. CJM 2012 (vol 65 pp. 702)

Taylor, Michael
 Regularity of Standing Waves on Lipschitz Domains We analyze the regularity of standing wave solutions to nonlinear SchrÃ¶dinger equations of power type on bounded domains, concentrating on Lipschitz domains. We establish optimal regularity results in this setting, in Besov spaces and in HÃ¶lder spaces. Keywords:standing waves, elliptic regularity, Lipschitz domainCategories:35J25, 35J65