Number of synchronized and segregated interior spike solutions for nonlinear coupled elliptic systems

作者：Tang, Zhongwei^*; Wang, Lushun

来源：Journal of Mathematical Analysis and Applications, 2017, 448(2): 1079-1119.

DOI：10.1016/j.jmaa.2016.11.044

• Lyapunov-Schmidt reduction methods
• Synchronized and segregated solution
• Interior spikes
• Elliptic systems

In this paper, we consider the following nonlinear coupled elliptic systems
{-epsilon(2)Delta u + u = mu(1)u(3) + beta uv(2) in Omega,
-epsilon(2)Delta v + v = mu(1)v(3) + beta u(2)v in Omega,
u > 0, v > 0 in Omega, (A(epsilon))
partial derivative u/partial derivative v = partial derivative u/partial derivative v = 0 on partial derivative Omega,
where epsilon > 0, mu(1) > 0, mu(2) > 0, beta is an element of R, and Omega is a bounded domain with smooth boundary in R-3. Due to Lyapunov Schmidt reduction method, we proved that (A(epsilon)) has at least O(1/epsilon(3)vertical bar In epsilon vertical bar) synchronized and segregated vector solutions for epsilon small enough and some beta is an element of R. Moreover, for each m is an element of (0, 3) there exist synchronized and segregated vector solutions for (A(epsilon)) with energies in the order of epsilon(3-m). Our result extends the result of Lin, Ni and Wei [20', from the Lin-Ni-Takagi problem to the nonlinear elliptic systems.

• 出版日期2017-4-15
• 单位北京师范大学