211 research outputs found
Existence and phase separation of entire solutions to a pure critical competitive elliptic system
We establish the existence of a positive fully nontrivial solution to
the weakly coupled elliptic system% \left\{ \begin{tabular} [c]{l}% $-\Delta
u=\mu_{1}|u|^{{2}^{\ast}-2}u+\lambda\alpha|u|^{\alpha-2}|v|^{\beta }u,$\\
$-\Delta v=\mu_{2}|v|^{{2}^{\ast}-2}v+\lambda\beta|u|^{\alpha}|v|^{\beta{-2}%
}v,$\\ $u,v\in D^{1,2}(\mathbb{R}^{N}),$% \end{tabular} \ \right. where
is the critical Sobolev exponent,
and
We show that these solutions exhibit phase separation as
and we give a precise description of their limit
domains.
If and , we prove that the system has
infinitely many fully nontrivial solutions, which are not conformally
equivalent
Spiked solutions for Schr\"odinger systems with Sobolev critical exponent: the cases of competitive and weakly cooperative interactions
In this paper we deal with the nonlinear Schr\"odinger system in dimension 4, a problem with critical
Sobolev exponent. In the competitive case ( fixed or ) or in the weakly cooperative case ( small), we
construct, under suitable assumptions on the Robin function associated to the
domain , families of positive solutions which blowup and concentrate at
different points as . This problem can be
seen as a generalization for systems of a Brezis-Nirenberg type problem.Comment: 33 page
Supercritical problems in domains with thin toroidal holes
In this paper we study the Lane-Emden-Fowler equation Here , is a smooth bounded domain in , is an
dimensional closed manifold such that
with and . We prove that,
under some symmetry assumptions, the number of sign changing solutions to
increases as goes to zero
Boundary towers of layers for some supercritical problems
We show that in some suitable torus-like domains D some supercritical
elliptic problems have an arbitrary large number of sign-changing solutions
with alternate positive and negative layers which concentrate at different
rates along a k-dimensional submanifold of the boundary of D as p approaches
2*_{N,K} from below
Multiple blow-up phenomena for the sinh-Poisson equation
We consider the sinh-Poisson equation (P)_\lambda\quad -\Delta u=\la\sinh
u\ \hbox{in}\ \Omega,\ u=0\ \hbox{on}\ \partial\Omega, where is a
smooth bounded domain in \rr^2 and is a small positive parameter.
If and is symmetric with respect to the origin, for any
integer if \la is small enough, we construct a family of solutions to
(P)_\la which blows-up at the origin whose positive mass is and
negative mass is
It gives a complete answer to an open problem formulated by Jost-Wang-Ye-Zhou
in [Calc. Var. PDE (2008) 31: 263-276]
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