1,683 research outputs found
Existence of positive ground state solutions of Schrödinger–Poisson system involving negative nonlocal term and critical exponent on bounded domain
Abstract(#br)In this paper, we prove the existence of positive ground state solutions of the Schrödinger–Poisson system involving a negative nonlocal term and critical exponent on a bounded domain. The main tools are the mountain pass theorem and the concentration compactness principle
Quantitative Properties on the Steady States to A Schr\"odinger-Poisson-Slater System
A relatively complete picture on the steady states of the following
Schrdinger-Poisson-Slater (SPS) system \begin{cases} -\Delta
Q+Q=VQ-C_{S}Q^{2}, & Q>0\text{ in }\mathbb{R}^{3}\\ Q(x)\to0, & \mbox{as
}x\to\infty,\\ -\Delta V=Q^{2}, & \text{in }\mathbb{R}^{3}\\ V(x)\to0 &
\mbox{as }x\to\infty. \end{cases}
is given in this paper: existence, uniqueness, regularity and asymptotic
behavior at infinity, where is a constant. To the author's knowledge,
this is the first uniqueness result on SPS 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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