630 research outputs found
Ground States for a nonlinear Schr\"odinger system with sublinear coupling terms
We study the existence of ground states for the coupled Schr\"odinger system
\begin{equation} \left\{\begin{array}{lll} \displaystyle -\Delta
u_i+\lambda_i u_i= \mu_i |u_i|^{2q-2}u_i+\sum_{j\neq i}b_{ij}
|u_j|^q|u_i|^{q-2}u_i \\ u_i\in H^1(\mathbb{R}^n), \quad i=1,\ldots, d,
\end{array}\right. \end{equation} , for ,
(the so-called "symmetric attractive case") and
. We prove the existence of a nonnegative ground state
with radially decreasing. Moreover we show that,
for , such ground states are positive in all dimensions and for all
values of the parameters
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
Existence and symmetry results for competing variational systems
In this paper we consider a class of gradient systems of type in a bounded domain
. Under suitable assumptions on and , we prove
the existence of ground-state solutions for this problem. Moreover, for ,
assuming that the domain and the potentials are radially
symmetric, we prove that the ground state solutions are foliated Schwarz
symmetric with respect to antipodal points. We provide several examples for our
abstract framework.Comment: 21 pages, 0 figure
Increasing powers in a degenerate parabolic logistic equation
The purpose of this paper is to study the asymptotic behavior of the positive
solutions of the problem as ,
where is a bounded domain and is a nonnegative function. We
deduce that the limiting configuration solves a parabolic obstacle problem, and
afterwards we fully describe its long time behavior.Comment: 15 page
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