Choquard Equation Research Articles

Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities

Abstract In this article, we study the following quasilinear equation with nonlocal nonlinearity − Δ u − κ u Δ ( u 2 ) + λ u = ( ∣ x ∣ − μ * F ( u ) ) f ( u ) , in R N , -\Delta u-\kappa u\Delta \left({u}^{2})+\lambda u=\left({| x| }^{-\mu }* F\left(u))f\left(u),\hspace{1em}\hspace{0.1em}\text{in}\hspace{0.1em}\hspace{0.33em}{{\mathbb{R}}}^{N}, where κ \kappa is a parameter, N ≥ 3 N\ge 3 , μ ∈ ( 0 , N ) \mu \in \left(0,N) , F ( t ) = ∫ 0 t f ( s ) d s F\left(t)={\int }_{0}^{t}f\left(s){\rm{d}}s , and λ \lambda is a positive constant. We are going to analyze two cases: the L 2 {L}_{2} -norm of the solution is not confirmed and the L 2 {L}_{2} -norm of the solution is prescribed. Under the almost optimal assumptions on f f , we obtain the existence of a sequence of radial solutions for two cases.

Multiplicity and concentration of solutions for a Choquard equation with critical exponential growth in $$\mathbb {R}^N$$

Multiplicity and concentration of solutions for a Choquard equation with critical exponential growth in $$\mathbb {R}^N$$

Localized nodal solutions for semiclassical Choquard equations with critical growth

Localized nodal solutions for semiclassical Choquard equations with critical growth

Normalized bound states for the Choquard equations in exterior domains

Normalized bound states for the Choquard equations in exterior domains

The higher regularity and decay estimates for positive solutions of fractional Choquard equations

AbstractIn the paper, we study the higher regularity and decay estimates for positive solutions of the following fractional Choquard equations: where , is a constant, , , , . Let be a positive solution of (). We obtain an optimal decay estimate for and reveal the relation between and the decay rate of for all permitting . We skillfully use the properties of Bessel kernel to deduce our main result by iteration. Moreover, by the decay estimates, the stronger regular estimates for are obtained.

Analytical Modeling of [001] Orientation in Silicon Trigate Rectangular Nanowire Using a Tight-Binding Model

AbstractIn the realm of electronics, the performance of Silicon Trigate Rectangular Nanowires (Si-TRNW) and the structural characteristics of <001> orientation using tight-binding models have been analyzed. The fast algorithm based on the tight-binding model for Trigate Silicon nanowires yielded a remarkable ION/IOFF ratio of 1.49 × 1010 and leakage current (ILeak or IOFF) of 3.7 × 10−17μA. Furthermore, a maximum conduction band energy level (Ecmax) of −0.003 eV and a Subthreshold Slope (SS) of 120 mV has been obtained for a channel length of 15 nm. At an energy level of 3 eV, a high Transmission coefficient, T(ε), of 4 has been attained using the E-k dispersion method. This analysis also involved the calculation of three ∆ valleys pertinent to the channel’s effectiveness in <001> orientation, with proximity nearer to 1 m0. The Schrodinger-Poisson equation has been analyzed with the Ballistic transport along the [001] z-direction in channel potential. A comparative assessment has been also performed between the lateral dimensions of rectangular nanowires with equal energy levels, utilizing both the tight-binding model and Density Functional Theory (DFT) techniques. In some high-frequency applications, a high transmission coefficient is beneficial to maximize the amount of energy or information that gets transmitted. Reducing leakage current would offer a technological pathway for performance improvement of high-frequency applications. The high ON-current (ION) has been obtained through the DFT approach between source and drain terminals is particularly desirable for applications demanding for fast switching speeds and high-performance computing. The strengths of both methods in hybrid approaches is a common strategy to achieve simulations that are both accurate and efficient. Notably, the nanowires subjected to hydrostatic strain, exhibiting enhanced mobility and exceptional electrostatic integrity, emerged as pivotal components for forthcoming technology nodes. This research augments the potential feasibility of strain-based Si nanowires, even at the 3 nm scale, in subsequent technological advancements.

Wellposedness and Scattering for Some Bi-inhomogeneous Schrödinger–Choquard Equation with Linear Damping

Wellposedness and Scattering for Some Bi-inhomogeneous Schrödinger–Choquard Equation with Linear Damping

On the Choquard equation with double critical Sobolev exponents in ℝ N

In this paper, we study the existence and nonexistence results to the Choquard equation − Δ u = ( ∫ R N | u ( y ) | 2 α ∗ | x − y | α d y ) | u | 2 α ∗ − 2 u ± | u | q − 2 u in R N , where 2 α ∗ = 2 N − α N − 2 , 0 < α < N , 1 < q ≤ 2 ∗ , 2 ∗ = 2 N N − 2 , N ≥ 3 . We first use the Pohozaev-type identity to show the nonexistence of solutions for 1 < q < 2 ∗ . When the equation has double critical exponents, i.e. q = 2 ∗ , we obtain the existence of radial ground state solutions by the Nehari manifold and Mountain pass theorem.

The Ground State Solutions to a Class of Biharmonic Choquard Equations on Weighted Lattice Graphs

The Ground State Solutions to a Class of Biharmonic Choquard Equations on Weighted Lattice Graphs

Existence and Concentration of Solutions to a Choquard Equation Involving Fractional p-Laplace via Penalization Method

Existence and Concentration of Solutions to a Choquard Equation Involving Fractional p-Laplace via Penalization Method

Existence of solutions for nonlinear biharmonic Choquard equations on weighted lattice graphs

In this paper, let G=(ZN,E,μ,ω) be a weighted lattice graph, where μ is a uniformly bounded measure and ω is a positive symmetric weight. We are concerned with the following biharmonic Choquard equation with a nonlocal nonlinearity on ZN, for any α∈(0,N),Δ2u−Δu+hu=(∑y∈ZN,y≠xF(u(y))d(x,y)N−α)f(u), where h denotes a function on ZN, f:R→R is a continuous function, F(u)=∫0uf(s)ds represents the primitive function of f, and d(x,y) stands for the distance between x and y. In particular, under some suitable conditions on h, we prove that if the nonlinearity f satisfies distinct growth conditions, then there exists a mountain-pass solution and a ground state solution respectively. As far as we know, there are no existing results for the biharmonic Choquard equation with the nonlinearity f on weighted lattice graphs.

Choquard equations with critical exponential nonlinearities in the zero mass case

&lt;p&gt;We investigate Choquard equations in $ \mathbb R^N $ driven by a weighted $ N $-Laplace operator with polynomial kernel and zero mass. Since the setting is limiting for the Sobolev embedding, we work with nonlinearities which may grow up to the critical exponential. We establish the existence of a positive solution by variational methods, complementing the analysis in &lt;sup&gt;[&lt;xref ref-type="bibr" rid="b32"&gt;32&lt;/xref&gt;]&lt;/sup&gt;, where the case of a logarithmic kernel was considered.&lt;/p&gt;

Global existence, blow-up and stability of standing waves for the Schrödinger-Choquard equation with harmonic potential

&lt;abstract&gt;&lt;p&gt;In this article, we conduct a comprehensive investigation into the global existence, blow-up and stability of standing waves for a $ L^{2} $-critical Schrödinger-Choquard equation with harmonic potential. First, by taking advantage of the ground-state solutions and scaling techniques, we obtain some criteria for the global existence and blow-up of the solutions. Second, in terms of the refined compactness argument, scaling techniques and the variational characterization of the ground state solution to the Choquard equation with $ p_{2} = 1+\frac{2+\alpha}{N} $, we explore the limiting dynamics of blow-up solutions to the $ L^{2} $-critical Choquard equation with $ L^{2} $-subcritical perturbation, including the $ L^{2} $-mass concentration and blow-up rate. Finally, the orbital stability of standing waves is investigated in the presence of $ L^{2} $-subcritical perturbation, focusing $ L^{2} $-critical perturbation and defocusing $ L^{2} $-supercritical perturbation by using variational methods. Our results supplement the conclusions of some known works.&lt;/p&gt;&lt;/abstract&gt;

Normalized ground states to the nonlinear Choquard equations with local perturbations

&lt;abstract&gt;&lt;p&gt;In this paper, we considered the existence of ground state solutions to the following Choquard equation&lt;/p&gt; &lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \begin{eqnarray*} \left\{ \begin{aligned} &amp;amp;-\Delta u = \lambda u + (I_{\alpha}\ast F(u))f(u) + \mu|u|^{q-2}u \hskip0.5cm \mbox{in} \hskip0.2cm\mathbb{R}^{N}, \\ &amp;amp; \int\limits_{\mathbb{R}^{N}}|u|^{2}dx = a &amp;gt;0, \end{aligned} \right. \end{eqnarray*} $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt; &lt;p&gt;where $ N \geq 3 $, $ I_{\alpha} $ is the Riesz potential of order $ \alpha \in (0, N) $, $ 2 &amp;lt; q \leq 2+ \frac{4}{N} $, $ \mu &amp;gt; 0 $ and $ \lambda \in \mathbb{R} $ is a Lagrange multiplier. Under general assumptions on $ F\in \mathcal{C}^{1}(\mathbb{R}, \mathbb{R}) $, for a $ L^{2} $-subcritical and $ L^{2} $-critical of perturbation $ \mu|u|^{q-2}u $, we established several existence or nonexistence results about the normalized ground state solutions.&lt;/p&gt;&lt;/abstract&gt;

Ground states for fractional Choquard equations with doubly critical exponents and magnetic fields

In this paper, we investigate the ground states for the fractional Choquard equations with doubly critical exponents and magnetic fields. We prove that the equation has a ground state solution by using the Nehari method and the Pohožaev identity. Bibliography: 12 titles.

Existence of least energy solutions for a quasilinear Choquard equation

Existence of least energy solutions for a quasilinear Choquard equation

Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent

Limit profiles and the existence of bound-states in exterior domains for fractional Choquard equations with critical exponent

Normalized solutions for the Choquard equations with critical nonlinearities

Normalized solutions for the Choquard equations with critical nonlinearities

Existence and orbital stability results for the nonlinear Choquard equation with rotation

Existence and orbital stability results for the nonlinear Choquard equation with rotation

Ground states for fractional Choquard equations with doubly critical exponents and magnetic fields

In this paper, we investigate the ground states for the fractional Choquard equations with doubly critical exponents and magnetic fields. We prove that the equation has a ground state solution by using the Nehari method and the Pokhozhaev identity.