Asymptotic behavior of least energy solutions for a 2D nonlinear Neumann problem with large exponent

作者：Takahashi Futoshi^*

来源：Journal of Mathematical Analysis and Applications, 2014, 411(1): 95-106.

DOI：10.1016/j.jmaa.2013.09.044

• Least energy solution
• Nonlinear Neumann boundary condition
• Large exponent
• Concentration

In this paper, we consider the following elliptic problem with the nonlinear Neumann boundary condition: %26lt;br%26gt;(E-p) {-Delta u + u = 0 on Omega, %26lt;br%26gt;u %26gt; 0 on Omega, %26lt;br%26gt;partial derivative u/partial derivative v = u(p) on Omega, %26lt;br%26gt;where Omega is a smooth bounded domain in R-2, nu is the outer unit normal vector to partial derivative Omega, and p %26gt; 1 is any positive number. %26lt;br%26gt;We study the asymptotic behavior of least energy solutions to (E-p) when the nonlinear exponent p gets large. Following the arguments of X. Ren and J.C. Wei [13,14], we show that the least energy solutions remain bounded uniformly in p, and it develops one peak on the boundary, the location of which is controlled by the Green function associated to the linear problem.

• 出版日期2014-3-1