1,721,107 research outputs found
Segregated solutions for nonlinear Schrödinger systems with weak interspecies forces
Full text linkWe find positive non-radial solutions for a system of Schrödinger equations in a weak fully attractive or repulsive regime in presence of an external radial trapping potential that exhibits a maximum or a minimum at infinity
Solutions to a cubic Schrödinger system with mixed attractive and repulsive forces in a critical regime
No full textWe study the existence of solutions to the cubic Schrödinger system {equation presented} when Ω is a bounded domain in R4, λi are positive small numbers,βij are real numbers so thatβii > 0 andβij =βji, i ≠ j. We assemble the components ui in groups so that all the interaction forcesβij among components of the same group are attractive, i.e.,βij > 0, while forces among components of different groups are repulsive or weakly attractive, i.e.,βi
A unified approach of blow-up phenomena for two-dimensional singular Liouville systems
No full textWe consider generic 2 × 2 singular Liouville systems [Equation presented here] where Ω∈ 0 is a smooth bounded domain in R2 possibly having some symmetry with respect to the origin, δ0 is the Dirac mass at 0, λ1, λ2 are small positive parameters and a, b, α1, α2 > 0. We construct a family of solutions to (∗) which blow up at the origin as λ1 → 0 and λ2 → 0 and whose local mass at the origin is a given quantity depending on a, b, α1, α2. In particular, if ab < 4 we get finitely many possible blow-up values of the local mass, whereas if ab ≥ 4 we get infinitely many. The blow-up values are produced using an explicit formula which involves Chebyshev polynomials
Blowing-up solutions for the Yamabe equation
No full textLet (M,g) be a smooth, compact Riemannian manifold of dimension N \geq 3. We consider the almost critical problem
(P_\epsilon)
-\Delta_g u+ {N-2\over 4(N-1)} Scal_g u= u^{{N+2\over N-2}+\epsilon } in} M,
u>0 in M,
where \Delta_g denotes the Laplace-Beltrami operator, Scal_g is the scalar curvature of g
and \epsilon \in R is a small parameter.
It is known that problem (P_\epsilon) does not have any blowing-up solutions when \epsilon \to 0^-, at least for N \leq 24 or in the locally conformally flat case, and this is not true anymore when \epsilon \to 0^+. Indeed, we prove that, if N \geq 7 and the manifold is not locally conformally flat, then problem (P_\epsilon) does have a family of solutions which blow-up at a maximum point of the function
\xi \to |Weyl_g(\xi)|_g
as \epsilon \to 0^+. Here Weyl_g denotes the Weyl curvature tensor of g
Nodal Solutions of the Brezis-Nirenberg Problem in Dimension 6
No full textWe show that the classical Brezis-Nirenberg problem-Delta u = u vertical bar u vertical bar + lambda u in Omega,u = 0 on partial derivative Omega,when Omega is a bounded domain in R-6 has a sign-changing solution which blows-up at a point in Omega as lambda approaches a suitable value lambda(0) > 0
Nonexistence of single blow-up solutions for a nonlinear Schrödinger equation involving critical Sobolev exponent
No full textFully nontrivial solutions to elliptic systems with mixed couplings
No full textWe study the existence of fully nontrivial solutions to the system −Δui+λiui=∑j=1lβij|uj|p|ui|p−2uiinΩ,i=1,...,l,in a bounded or unbounded domain Ω in RN,N≥3. The λi’s are real numbers, and the nonlinear term may have subcritical (1<[Formula presented]), critical (p=[Formula presented]), or supercritical growth (p>[Formula presented]). The matrix (βij) is symmetric and admits a block decomposition such that the diagonal entries βii are positive, the interaction forces within each block are attractive (i.e., all entries βij in each block are non-negative) and the interaction forces between different blocks are repulsive (i.e., all other entries are non-positive). We obtain new existence and multiplicity results of fully nontrivial solutions, i.e., solutions where every component ui is nontrivial. We also find fully synchronized solutions (i.e., ui=ciu1 for all i=2,...,l) in the purely cooperative case whenever p∈(1,2)
- …
