242 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
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
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