Ground states for a zero‐mass and Coulomb–Sobolev critical Schrödinger–Poisson–Slater problem
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DOI: 10.1002/mana.70058
Abstract: In this paper, we consider the following Schrödinger–Poisson–Slater equation: −Δu+14π|x|*|u|2u=μ|u|p−2u+|u|2u,inR3,$$\begin{equation*}\hspace*{35pt} -\Delta u+ \left(\frac{1}{4\pi |x|}\ast |u|^{2}\right)u=\mu |u|^{p-2}u+|u|^{2}u,\ \ \mathrm{in} \ \mathbb {R}^{3}, \nonumber \end{equation*}$$where μ>0$\mu >0$ and 3 read more here.
Keywords: poisson slater; schr dinger; dinger poisson;
Schrödinger‐Poisson system with Hardy‐Littlewood‐Sobolev critical exponent
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DOI: 10.1002/mma.5694
Abstract: In this paper, we consider the following Schrödinger‐Poisson system: −Δu+λϕ|u|2α∗−2u=∫R3|u|2β∗|x−y|3−βdy|u|2β∗−2u,inR3,(−Δ)α2ϕ=Aα−1|u|2α∗,inR3, where parameters α,β∈(0,3),λ>0, Aα=Γ(3−α2)2απ32Γ(α2) , 2α∗=3+α , and 2β∗=3+β are the Hardy‐Littlewood‐Sobolev critical exponents. For α0, we prove the existence of nonnegative groundstate solution to… read more here.
Keywords: dinger poisson; system; hardy littlewood; littlewood sobolev ... See more keywords
Nontrivial solutions for the fractional Schrödinger‐Poisson system with subcritical or critical nonlinearities
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DOI: 10.1002/mma.6019
Abstract: In this paper, we are concerned with a class of fractional Schrödinger‐Poisson system involving subcritical or critical nonlinearities. By using the Nehari manifold and variational methods, we obtain the existence and multiplicity of nontrivial solutions. read more here.
Keywords: dinger poisson; fractional schr; critical nonlinearities; poisson system ... See more keywords
Existence and multiplicity of normalized solutions for a class of Schrödinger‐Poisson equations with general nonlinearities
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DOI: 10.1002/mma.6140
Abstract: In this paper, we study the existence and multiplicity of standing waves with prescribed L2 ‐norm Schrödinger‐Poisson equations with general nonlinearities in R3 : i∂tψ+Δψ−κ(|x|−1*|φ|2)ψ+f(ψ)=0, where κ>0 and f is superlinear and satisfies the monotonicity… read more here.
Keywords: dinger poisson; schr dinger; existence multiplicity; equations general ... See more keywords
Multibump solutions for nonlinear Schrödinger‐Poisson systems
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DOI: 10.1002/mma.6211
Abstract: In this paper, we study the following Schrödinger‐Poisson equations: −ε2Δu+V(x)u+K(x)ϕu=|u|p−2u,x∈R3,−ε2Δϕ=K(x)u2,x∈R3, where p∈(4,6) , ε>0 is a parameter and V and K satisfy the critical frequency conditions. By using variational methods and penalization arguments, we show… read more here.
Keywords: dinger poisson; schr dinger; solutions nonlinear; multibump solutions ... See more keywords
Existence and limiting behavior of minimizers for attractive Schrödinger‐Poisson systems with periodic potentials
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DOI: 10.1002/mma.6232
Abstract: We study the constrained minimizing problem of the energy functional related to attractive Schrödinger‐Poisson systems with periodic potentials: I(m)=infE(ϕ):ϕ∈H1(R3),‖ϕ‖L22=m, where E(ϕ):=12∫R3|∇ϕ(x)|2dx+12∫R3V(x)|ϕ(x)|2dx−14∬R3×R3|ϕ(x)|2|ϕ(y)|2|x−y|dxdy−1α+2∫R3|ϕ(x)|α+2dx, with m>0 , α>0 , and V is a continuous periodic potential. We first… read more here.
Keywords: poisson systems; dinger poisson; attractive schr; systems periodic ... See more keywords
On a Class of Choquard–SchröDinger–Poisson Systems Involving p$$ p $$‐Laplacian
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DOI: 10.1002/mma.70274
Abstract: In this paper, we study a class of Choquard–Schrödinger–Poisson systems with local perturbation. Unlike other existing works, this system is one coupled by Schrödinger–Poisson systems of ‐Laplacian with a Choquard equation. In addition, the nonlinearity… read more here.
Keywords: poisson systems; schr dinger; dinger poisson;
On the Existence of Ground State Solutions for Fractional Schrödinger–Poisson Systems with General Potentials and Super-quadratic Nonlinearity
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DOI: 10.1007/s00009-018-1179-8
Abstract: In this article, we are concerned with the following fractional Schrödinger–Poisson system: $$\begin{aligned} \left\{ \begin{array}{ll} (-\Delta )^{s}u+V(x)u+\phi u=f(u)&{} \quad \hbox {in}~\mathbb {R}^{3},\\ (-\Delta )^{t}\phi =u^2&{} \quad \hbox {in}~\mathbb {R}^{3},\\ \end{array} \right. \end{aligned}$$(-Δ)su+V(x)u+ϕu=f(u)inR3,(-Δ)tϕ=u2inR3,where $$03$$2s+2t>3, and $$f\in… read more here.
Keywords: dinger poisson; fractional schr; state solutions; ground state ... See more keywords
A mass- and energy-conserved DG method for the Schrödinger-Poisson equation
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DOI: 10.1007/s11075-021-01139-0
Abstract: We construct, analyze, and numerically validate a class of conservative discontinuous Galerkin (DG) schemes for the Schrödinger-Poisson equation. The proposed schemes all shown to conserve both mass and energy. For the semi-discrete DG scheme the… read more here.
Keywords: poisson equation; schr dinger; dinger poisson; mass energy ... See more keywords
FRACTIONAL SCHRÖDINGER–POISSON SYSTEM WITH SINGULARITY: EXISTENCE, UNIQUENESS, AND ASYMPTOTIC BEHAVIOR
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DOI: 10.1017/s0017089520000099
Abstract: Abstract In this paper, we consider the following fractional Schrödinger–Poisson system with singularity \begin{equation*} \left \{\begin{array}{lcl} ({-}\Delta)^s u+V(x)u+\lambda \phi u = f(x)u^{-\gamma}, &&\quad x\in\mathbb{R}^3,\\ ({-}\Delta)^t \phi = u^2, &&\quad x\in\mathbb{R}^3,\\ u>0,&&\quad x\in\mathbb{R}^3, \end{array}\right. \end{equation*} where… read more here.
Keywords: schr dinger; dinger poisson; poisson system; system singularity ... See more keywords
Multiple bound states for a class of fractional critical Schrödinger–Poisson systems with critical frequency
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DOI: 10.1063/5.0174872
Abstract: In this paper we study the fractional Schrödinger–Poisson system ε2s(−Δ)su+V(x)u=ϕ|u|2s*−3u+|u|2s*−2u,ε2s(−Δ)sϕ=|u|2s*−1,x∈R3, where s ∈ (0, 1), ɛ > 0 is a small parameter, 2s*=63−2s is the critical Sobolev exponent and V∈L32s(R3) is a nonnegative function which… read more here.
Keywords: critical frequency; schr dinger; dinger poisson;