656 research outputs found
The two-dimensional Lazer-McKenna conjecture for an exponential nonlinearity
We consider the problem of Ambrosetti-Prodi type
\begin{equation}\label{0}\quad\begin{cases} \Delta u + e^u = s\phi_1 + h(x)
&\hbox{in} \Omega, u=0 & \hbox{on} \partial \Omega, \end{cases} \nonumber
\end{equation} where is a bounded, smooth domain in ,
is a positive first eigenfunction of the Laplacian under Dirichlet boundary
conditions and . We prove that given
this problem has at least solutions for all sufficiently large
, which answers affirmatively a conjecture by Lazer and McKenna \cite{LM1}
for this case. The solutions found exhibit multiple concentration behavior
around maxima of as .Comment: 24 pages, to appear in J. Diff. Eqn
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}.
Large mass boundary condensation patterns in the stationary Keller-Segel system
We consider the boundary value problem in
with Neumann boundary condition, where is a bounded smooth
domain in , This problem is equivalent to the
stationary Keller-Segel system from chemotaxis. We establish the existence of a
solution which exhibits a sharp boundary layer along the entire
boundary as . These solutions have large mass in
the sense that $ \int_\Omega \lambda e^{u_\lambda} \sim |\log\lambda|.
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