11,051 research outputs found
Singular limits for the bi-laplacian operator with exponential nonlinearity in
Let be a bounded smooth domain in such that for
some integer its -th singular cohomology group with coefficients in
some field is not zero, then problem
{\Delta^{2}u-\rho^{4}k(x)e^{u}=0 & \hbox{in}\Omega,
u=\Delta u=0 & \hbox{on}\partial\Omega,
has a solution blowing-up, as , at points of , for any
given number .Comment: 30 pages, to appear in Ann. IHP Non Linear Analysi
A Symposium on the Fair Trade Laws: Part IV: Indirect Methods of Evading the Fair Trade Laws
In this paper, we build infinitely many non-radial sign-changing solutions to the critical problem (P)−Δu=|u|[Formula presented]u, in Ω,u=0, on ∂Ω on the annulus Ω:=x∈RN:a<|x|<b, N≥3. In particular, for any integer k large enough, we build a non-radial solution which look like the unique positive solution u0to (P) crowned by k negative bubbles arranged on a regular polygon with radius r0such that r0[Formula presented]u0(r0)=:maxa≤r≤br[Formula presented]u0(r)
Ancient multiple-layer solutions to the Allen-Cahn equation
We consider the parabolic one-dimensional Allen-Cahn equation The steady state , connects, as a "transition layer" the stable phases
and . We construct a solution with any given number of transition
layers between and . At main order they consist of time-traveling
copies of with interfaces diverging one to each other as .
More precisely, we find where the functions
satisfy a first order Toda-type system. They are given by
for certain explicit constants $\gamma_{jk}.
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