4,738 research outputs found
Existence of Solutions for a Modified Nonlinear Schrödinger System
We are concerned with the following modified nonlinear Schrödinger system:
-Δu+u-(1/2)uΔ(u2)=(2α/(α+β))|u|α-2|v|βu, x∈Ω, -Δv+v-(1/2)vΔ(v2)=(2β/(α+β))|u|α|v|β-2v, x∈Ω, u=0, v=0, x∈∂Ω, where α>2, β>2, α+β<2·2*, 2*=2N/(N-2) is the critical Sobolev exponent, and Ω⊂ℝN (N≥3) is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system
Concentration phenomena for critical fractional Schr\"odinger systems
In this paper we study the existence, multiplicity and concentration behavior
of solutions for the following critical fractional Schr\"odinger system
\begin{equation*} \left\{ \begin{array}{ll} \varepsilon^{2s}
(-\Delta)^{s}u+V(x) u=Q_{u}(u, v)+\frac{1}{2^{*}_{s}}K_{u}(u, v) &\mbox{ in }
\mathbb{R}^{N}\varepsilon^{2s} (-\Delta)^{s}u+W(x) v=Q_{v}(u,
v)+\frac{1}{2^{*}_{s}}K_{v}(u, v) &\mbox{ in } \mathbb{R}^{N} u, v>0 &\mbox{ in
} \R^{N}, \end{array} \right. \end{equation*} where is a
parameter, , , is the fractional Laplacian
operator, and
are positive H\"older continuous
potentials, and are homogeneous -functions having subcritical
and critical growth respectively. We relate the number of solutions with the
topology of the set where the potentials and attain their minimum
values. The proofs rely on the Ljusternik-Schnirelmann theory and variational
methods.Comment: arXiv admin note: text overlap with arXiv:1704.0060
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