Choquard Equation Research Articles

The Brezis–Nirenberg type double critical problem for the Choquard equation

In this paper, we study the following Choquard equation $$\begin{aligned} -\Delta u=\alpha |u|^{2^*-2}u+\beta \left( I_\mu *|u|^{2_\mu ^*}\right) |u|^{2_\mu ^* -2}u +\lambda u,\quad in\,\,\Omega , \end{aligned}$$ where $$\Omega$$ is a bounded domain of $${\mathbb {R}}^N$$ with Lipschitz boundary, $$N\ge 3,$$ $$\alpha ,\beta ,\lambda$$ are real parameters satisfying suitable conditions, $$2^* =\frac{2N}{N-2}$$ is the critical exponent for the embedding of $$H_0^1 (\Omega )$$ to $$L^p (\Omega ),$$ $$2_\mu ^* =\frac{2N-\mu }{N-2}$$ is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality. Using variational methods, we show the existence of nontrivial solutions for the Choquard equation with double critical exponents.

Effect of size quantization and quantum capacitance on the threshold voltage of a 2D nanoscale dual gate MOSFET

The size quantization effect in the channel of a 2D nanoscale MOSFET is studied using a self-consistent quantum method. Under this, Schrodinger-Poisson equations are solved for determining the electron density for 2D device channels from 3 nm × 3 nm to 100 nm × 100 nm. The lower dimension channels show a peak of the electron density at the middle whereas higher dimension channels show the accumulation of the electrons at the oxide/semiconductor interface. Also, the role of quantum capacitance on the threshold voltages of these nanoscale devices is investigated as a function of channel dimensions and electron effective masses. It is observed that not only the size but the electron effective masses dominate the conductivity of the channel for such nanoscale devices. Here, the channel electron densities are obtained using density matrix formalism. A block diagonal Hamiltonian Matrix [H] is constructed for this oxide/channel/oxide 2D structure and the channel is discretized by using the finite-difference method. This analysis is important for understanding the physics of the size quantization and its effect on the threshold voltage.

On a class of p(x)-Choquard equations with sign-changing potential and upper critical growth

Motived by the Hardy–Littlewood–Sobolev inequality for variable exponents, in this paper we use variational methods to prove the existence of a weak solution for a class of p(x)-Choquard equations with upper critical growth. Using truncation arguments and Krasnoselskii’s genus, we also show a multiplicity of solutions for a class of p(x)-Choquard equations with a nonlocal and non-degenerate Kirchhoff term. Also we show that the solutions obtained belong to $$L^{\infty }({\mathbb {R}}^{N})$$ and have polynomial decay.

Nonlinear perturbations of a periodic magnetic Choquard equation with Hardy–Littlewood–Sobolev critical exponent

In this paper, we consider the following magnetic nonlinear Choquard equation $$\begin{aligned} -(\nabla +iA(x))^2u+ V(x)u = \left( \frac{1}{|x|^{\alpha }}*|u|^{2_{\alpha }^*}\right) |u|^{2_{\alpha }^*-2} u + \lambda f(u)\ \text { in }\ \mathbb {R}^N, \end{aligned}$$ where $$2_{\alpha }^{*}=\frac{2N-\alpha }{N-2}$$ is the critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, $$\lambda >0$$ , $$N\ge 3$$ , $$0<\alpha < N$$ , $$A: \mathbb {R}^{N}\rightarrow \mathbb {R}^{N}$$ is an $$C^1$$ , $$\mathbb {Z}^N$$ -periodic vector potential and V is a continuous scalar potential given as a perturbation of a periodic potential. Considering different types of nonlinearities f, namely $$f(x,u)=\left( \frac{1}{|x|^{\alpha }}*|u|^{p}\right) |u|^{p-2} u$$ for $$(2N-\alpha )/N<p<2^{*}_{\alpha }$$ , then $$f(u)=|u|^{p-1} u$$ for $$1<p<2^*-1$$ and $$f(u)=|u|^{2^* - 2}u$$ (where $$2^*=2N/(N-2)$$ ), we prove the existence of at least one ground-state solution for this equation by variational methods if p belongs to some intervals depending on N and $$\lambda $$ .

Scattering versus blow-up beyond the threshold for the focusing Choquard equation

This note investigates the Choquard equationiu˙+Δu+(Iα⁎|u|p)|u|p−2u=0. Indeed, above the mass-energy threshold, a scattering versus finite time blow-up dichotomy is obtained in the mass super-critical and energy sub-critical regime.

Existence and concentration of positive ground states for Schrodinger-Poisson equations with competing potential functions

This article concerns the Schrodinger-Poisson equation $$\displaylines{ -\varepsilon^2\Delta u+V(x)u+K(x)\phi u=P(x)|u|^{p-1}u+Q(x) |u|^{q-1}u,\quad x\in\mathbb{R}^3,\cr -\varepsilon^2\Delta \phi=K(x)u^2,\quad x\in\mathbb{R}^3, }$$ where \(3&lt;q&lt;p&lt;5=2^{\ast}-1\). We prove that for all \(\varepsilon&gt;0\), the equation has a ground state solution. The methods used here are based on the Nehari manifold and the concentration-compactness principle. Furthermore, for \varepsilon&gt;0 small, these ground states concentrate at a global minimum point of the least energy function.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2020/78/abstr.html

Saddle solutions for the Choquard equation II

We study nodal solutions of saddle type for the Choquard equation −Δu+u=(Iα∗|u|p)|u|p−2uinRNwhere N≥3 and 2≤p<N+αN−2. The function Iα refers to the Riesz potential with order α∈(0,N). In a noncompact setting, we construct saddle solutions whose nodal domains are of conical shapes demonstrating polyhedral symmetry configurations in RN. More precisely, when N≥4, for given integers ℓ1,ℓ2≥2, this equation admits saddle solutions with its 4ℓ1ℓ2 nodal domains meeting at the origin. These among other saddle solutions not only reveal the nonlocal feature of the Choquard equation but also further complete the variational framework of constructing sign-changing solutions for the Choquard equation initiated in Ghimenti and Van Schaftingen (2016) and further developed in Xia and Wang (2019).

The Sobolev-Morawetz approach for the energy scattering of nonlinear Schrödinger-type equations with radial data

Based on recent works of Dodson-Murphy [ 12 ] and Arora-Dodson-Murphy [ 3 ], we give a unified approach for the energy scattering with radially symmetric initial data for nonlinear Schrodinger equations and nonlinear Choquard equations in any dimensions \begin{document}$ N\geq 2 $\end{document} . We also discuss its applications for other Schrodinger-type equations.

Symmetry and nonexistence results for a fractional Choquard equation with weights

<p style='text-indent:20px;'>Let <inline-formula><tex-math id="M1">\begin{document}$ u $\end{document}</tex-math></inline-formula> be a nonnegative solution to the equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ (-\Delta)^{\frac{\alpha}{2}} u = \left(\frac{1}{|x|^{n-\beta}} * |x|^a u^p \right) |x|^a u^{p-1} \quad\text{ in } \mathbb{R}^n \setminus \{0\}, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>where <inline-formula><tex-math id="M2">\begin{document}$ n \ge 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M3">\begin{document}$ 0 &lt; \alpha &lt; 2 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M4">\begin{document}$ 0 &lt; \beta &lt; n $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M5">\begin{document}$ a&gt;\max\{ -\alpha, -\frac{\alpha+\beta}{2} \} $\end{document}</tex-math></inline-formula>. By exploiting the method of scaling spheres and moving planes in integral forms, we show that <inline-formula><tex-math id="M6">\begin{document}$ u $\end{document}</tex-math></inline-formula> must be zero if <inline-formula><tex-math id="M7">\begin{document}$ 1\le p&lt;\frac{n+\beta+2a}{n-\alpha} $\end{document}</tex-math></inline-formula> and must be radially symmetric about the origin if <inline-formula><tex-math id="M8">\begin{document}$ a&lt;0 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M9">\begin{document}$ \frac{n+\beta+2a}{n-\alpha} \le p \le \frac{n+\beta+a}{n-\alpha} $\end{document}</tex-math></inline-formula>.

Existence of Multispike Positive Solutions for a Nonlocal Problem in ℝ3

In this paper, we study the following nonlinear Choquard equation −ϵ2Δu+Kxu=1/8πϵ2∫ℝ3u2y/x−ydyu,x∈ℝ3, where ϵ&gt;0 and Kx is a positive bounded continuous potential on ℝ3. By applying the reduction method, we proved that for any positive integer k, the above equation has a positive solution with k spikes near the local maximum point of Kx if ϵ&gt;0 is sufficiently small under some suitable conditions on Kx.

STABLE SOLUTIONS TO THE STATIC CHOQUARD EQUATION

This paper is concerned with the static Choquard equation $$\begin{eqnarray}-\unicode[STIX]{x1D6E5}u=\bigg(\frac{1}{|x|^{N-\unicode[STIX]{x1D6FC}}}\ast |u|^{p}\bigg)|u|^{p-2}u\quad \text{in }\mathbb{R}^{N},\end{eqnarray}$$ where $N,p&gt;2$ and $\max \{0,N-4\}&lt;\unicode[STIX]{x1D6FC}&lt;N$. We prove that if $u\in C^{1}(\mathbb{R}^{N})$ is a stable weak solution of the equation, then $u\equiv 0$. This phenomenon is quite different from that of the local Lane–Emden equation, where such a result only holds for low exponents in high dimensions. Our result is the first Liouville theorem for Choquard-type equations with supercritical exponents and $\unicode[STIX]{x1D6FC}\neq 2$.

Existence and concentration of ground state solutions for Choquard equations involving critical growth and steep potential well

In the present paper, we are interested in the following Choquard type equation −Δu+(λV(x)−μ)u=(Iα∗|u|2α∗)|u|2α∗−2u+|u|p−2uinR3,where p∈(4,6), λ∈R+, μ∈R is a constant such that the operator Lλ≔−Δ+λV(x)−μ is non-degenerate, Iα is a Riesz potential of order α∈(0,3), 2α∗=3+α is the upper critical exponent due to the Hardy–Littlewood–Sobolev inequality, the function V∈C(R3,R) is nonnegative and has a potential well Ω≔intV−1(0), additionally, the operator −Δ has a sequence of Dirichlet eigenvalues in H01(Ω) expressed as 0<μ1<μ2<⋯<μn⟶n+∞. If μ<μ1, via the Nehari manifold techniques and the Ekeland variational principle, we prove the existence and concentration of positive ground state solutions for sufficiently large λ. If μ>μ1 and μ≠μj for all j∈N+, employing the Nehari–Pankov manifold methods and the constrained minimization arguments, we obtain the existence of ground state solution for λ large enough and verify the asymptotic behaviour of ground state solutions as λ→+∞.

Potential well theory for the focusing fractional Choquard equation

This note studies the non-linear fractional Schrödinger equation iu̇−(−Δ)su+(Iα*|u|p)|u|p−2u=0. In the mass super-critical and energy sub-critical regimes, the local solutions exist globally and scatter in the energy space or blow-up in finite time if the data belong to some stable sets.

Nehari-type ground state solutions for a Choquard equation with doubly critical exponents

AbstractThis paper deals with the following Choquard equation with a local nonlinear perturbation:−Δu+u=Iα∗|u|α2+1|u|α2−1u+f(u),x∈R2;u∈H1(R2),$$\begin{array}{} \displaystyle \left\{ \begin{array}{ll} - {\it\Delta} u+u=\left(I_{\alpha}*|u|^{\frac{\alpha}{2}+1}\right)|u|^{\frac{\alpha}{2}-1}u +f(u), &amp; x\in \mathbb{R}^2; \\ u\in H^1(\mathbb{R}^2), \end{array} \right. \end{array}$$whereα∈ (0, 2),Iα: ℝ2→ ℝ is the Riesz potential andf∈ 𝓒(ℝ, ℝ) is of critical exponential growth in the sense of Trudinger-Moser. The exponentα2+1$\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$is critical with respect to the Hardy-Littlewood-Sobolev inequality. We obtain the existence of a nontrivial solution or a Nehari-type ground state solution for the above equation in the doubly critical case, i.e. the appearance of both the lower critical exponentα2+1$\begin{array}{} \displaystyle \frac{\alpha}{2}+1 \end{array}$and the critical exponential growth off(u).

Multiple nodal solutions of quadratic Choquard equations with perturbation

In this paper, we consider the Choquard equations with a local perturbation where . By using variational method and approximating approach, we prove that for any given positive integer k, the above equation has a least energy radial solution changing sign exactly k times. This solution is constructed as the limit of such solutions for the following Choquard equations as . Our result improves and extends the previous results in the literature.

Stability of ground state for the Schrödinger-Poisson equation

<p style='text-indent:20px;'>We are concerned with the stability of the ground state for the Schrödinger-Poisson equation <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ i\frac{\partial\psi}{\partial t}+\triangle\psi-(|x|^{-1}\ast|\psi|^2)\psi+|\psi|^{p-1}\psi = 0,\quad x\in \mathbb{R}^3. $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>If <inline-formula><tex-math id="M1">\begin{document}$ 2&lt;p&lt;\frac{7}{3} $\end{document}</tex-math></inline-formula> and the frequency is sufficiently large, we show that the ground state is orbitally stable.

Existence of nontrivial solutions for fractional Choquard equations with critical or supercritical growth

ABSTRACT In this paper, we study the following fractional Choquard equation with critical or supercritical growth where 0<s<1, denotes the fractional Laplacian of order s, N>2s, and . Under some suitable conditions, we prove that the equation has a nontrivial solution for small by the variational method.

Existence and stability of standing waves for the Choquard equation with partial confinement

In this paper we study the existence and orbital stability of the Choquard equation with partial confinement. This type equation originates from Frohlich and Pekar's model of the polaron, where free electrons in an ionic lattice interact with phonons associated with deformations of the lattice or with the polarisation that it creates on the medium (interaction of an electron with its own hole). On the one hand, we prove the existence of global minimizer of the associate energy functional subject to the $L^2$-constraint. On the other hand, we discuss the orbital stability and asymptotic behavior of the global minimizer.

Nontrivial solutions for nonlinear Schrödinger–Choquard equations with critical exponents

We study the nonlinear Schrödinger–Choquard equation −Δu+u=Iα∗|u|p|u|p−2u+|u|q−2u,x∈RN,where N∈N, 0<α<N, Iα denotes Riesz potential. When p=N+αN or p=N+αN−2, we get nontrivial solutions under some restrictions on N, q and α respectively. N+αN and N+αN−2 are lower and upper critical exponents in the sense of the Hardy–Littlewood–Sobolev inequality. This article extends some results of related literatures.

Charge-Based Compact Drain Current Modeling of InAs-OI-Si MOSFET Including Subband Energies and Band Nonparabolicity

In this article, we report a physics-based compact model of drain current for InAs-on-insulator MOSFETs. The quantum confinement effect has been incorporated in the proposed model by solving the 1-D Schrodinger–Poisson equations without using any empirical model parameter. The model accurately captures the variation of surface potential, charge density in the inversion layer, and subband energy levels with gate bias inside the quantum well. The conduction-band nonparabolicity effect on modification in eigenenergy, effective mass, and density of states is derived and incorporated into the proposed model. The velocity overshoot effect that originates from the quasi-ballistic nature of carrier transport is also considered in the model. The proposed drain current model has been implemented in Verilog-A to use in the SPICE environment. The model predicted results are in good agreement with the commercial device simulator results and experimental data.