Asymptotic analysis for two-dimensional elliptic eigenvalue problems with exponentially dominated nonlinearities

Locating the peaks of solutions via the maximum principle II: A local version of the method of moving planes

“…In R 2 , a similar problem to (1.4) has been extensively studied for more than two decades. We refer the reader to [2,7,8,12] and the references therein. For solutions of (1.4), the method of moving planes has been shown to be a very powerful tool for the study of bubbling behavior of solutions.…”

An Asymptotic Analysis of Localized Solutions for Some Reaction‐Diffusion Models in Multidimensional Domains

“…Let D' be a new domain that contains D and that has the same axes of symmetry. Then, from [14], Xl = 0 for both domains. Also, from an inequality in [4], the conformal radius R(O) is larger for D' than for D. Thus, from (6.13) and (6.14), we conclude that the amplitude of the hot-spot is smaller but its spatial extent is larger for D' than for D.…”

“…Now let G(X;Xk} be the Green's function for the Laplacian with boundary condition (6.7b) and with singularity at x = Xk ED. Then, we can decompose Gas where gr is regular at x = xk' The solution to (6.7) can then be conveniently written as where (6.9a) Since each hot-spot is radially symmetric about x Xj' we obtain upon comparing (6.7c) with (6.10) that V<I>j(X j ) = ° for j = 1, ... , m. Thus, from (6.9b), the hot-spot locations x j ' for j = 1, ... , m, are to be determined from the following system: This result was derived previously in [13,14] using a complex variable method. Next, by comparing (6.7c) with (6.10), we find that j = 1, ... ,m. (6.12) Substituting (6.12) into (6.4) then yields a two-term expansion for alA).…”

“…Next, by comparing (6.7c) with (6.10), we find that j = 1, ... ,m. (6.12) Substituting (6.12) into (6.4) then yields a two-term expansion for alA). This result for aj(A), which was not obtained in [13] or [14], shows that the amplitude of a particular hot-spot depends weakly, as A --+ 0, on the Xj and on the global properties of the domain D contained in the Green's function. For a fixed value of m, it appears to be difficult to determine whether the system (6.11) has solutions for an arbitrary domain.…”

“…The qualitative feature of hot-spot solutions is that u -+ 00 as A -+ ° in a localized region near some x = Xj' for j = 1, ... , m, while u = 0(1) as A -+ ° away from these points. When b = 00, a system of equations for the hot-spot locations x j ' for j = 1, ... , m, was derived in [13,14] using the Liouville transformation, which reduces (1.3) to the study of certain complex functions. For b > ° and A -+ 0, we analyze (1.3) in a more straightforward way by using the method of matched asymptotic expansions, and we derive a new result for the amplitudes of the hot-spots.…”