Choquard Equation Research Articles

Existence of a ground state solution for Choquard equation with the upper critical exponent

In this paper, we investigate the following Choquard equation −Δu+u=(Iα∗|u|N+αN−2)|u|N+αN−2−2u+g(u)inRN,where N⩾3,α∈(0,N), Iα is the Riesz potential and g is a given function. If g satisfies the general subcritical growth conditions, we obtain the existence of a positive ground state solution.

On existence and concentration behavior of positive ground state solutions for a class of fractional Schrödinger–Choquard equations

In this paper, we first employ variational methods to show the existence of positive ground state solutions for fractional Schrodinger–Choquard equations $$\begin{aligned} \varepsilon ^{2s}(-\Delta )^{s}u+Vu=(I_{\alpha }*|u|^p)|u|^{p-2}u,~~~\mathrm {in}~~~\mathbb {R}^{N}, \end{aligned}$$ where potential $$V(x)\in C(\mathbb {R}^{N})$$ is nonnegative and bounded away from 0 as $$|x|\rightarrow \infty $$ , $$I_{\alpha }$$ is the Riesz potential of order $$\alpha \in (0,N)$$ and $$\varepsilon >0$$ is a parameter small enough. When $$V(x)\in C(\mathbb {R}^{N})$$ achieves 0 with a homogeneous behavior, we then investigate the concentration behavior of positive ground state solutions as $$\varepsilon \rightarrow 0^{+}$$ .

Infinitely many bound state solutions of Choquard equations with potentials

In this paper, we consider the following Choquard equation $$\begin{aligned} \left\{ \begin{array}{l@{\quad }l} - \Delta u+a(x)u=(I_\alpha *|u|^p)|u|^{p-2}u, &{}x\in {\mathbb {R}}^N,\\ u(x)\rightarrow 0,&{}|x|\rightarrow +\infty , \end{array}\right. {\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (\hbox {CH})} \end{aligned}$$ where $$N\ge 3, I_\alpha $$ is a Riesz potential, $$\frac{N+\alpha }{N}<p<\frac{N+\alpha }{N-2}$$ and a(x) is a given nonnegative potential function. Under some assumptions of asymptotic properties on a(x) at infinity and according to a concentration compactness argument, we obtain infinitely many solutions of (CH), whose energy can be arbitrarily large.

Existence of ground state solutions for Choquard equation involving the general upper critical Hardy-Littlewood-Sobolev nonlinear term

In this paper, we investigate the following a class of Choquard equation \begin{document}$\begin{equation*} -Δ u+u = (I_α*F(u))f(u) {\rm in} \mathbb{R}^N,\end{equation*}$ \end{document} where \begin{document}$N≥ 3,~α∈ (0,N),~I_α$\end{document} is the Riesz potential and \begin{document}$F(s) = ∈t_{0}^{s}f(t)dt$\end{document} . If \begin{document}$f$\end{document} satisfies almost necessary the upper critical growth conditions in the spirit of Berestycki and Lions, we obtain the existence of positive radial ground state solution by using the Pohožaev manifold and the compactness lemma of Strauss.

Semiclassical states for fractional Choquard equations with critical growth

In this paper, we are concerned with fractional Choquard equation \begin{document}$ \epsilon^{2α}(-Δ)^α u+V(x)u = \epsilon^{μ-3}\Bigl(\int_{\mathbb{R}^3}\frac{|u(y)|^{2_{μ,α}^*}+F(u(y))}{|x-y|^μ}dy\Bigr)\Bigl(|u|^{2_{μ,α}^*-2}u+\frac{1}{2_{μ,α}^*}f(u)\Bigr) {\rm in} \mathbb{R}^3,$ \end{document} where \begin{document} $\epsilon>0$ \end{document} is a parameter, \begin{document} $0 , \begin{document} $0 , \begin{document} $2_{μ,α}^* = \frac{6-μ}{3-2α}$ \end{document} is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality and fractional Laplace operator, \begin{document} $f$ \end{document} is a continuous subcritical term, and \begin{document} $F$ \end{document} is the primitive function of \begin{document} $f$ \end{document} . By virtue of the method of Nehari manifold and Ljusternik-Schnirelmann category theory, we prove that the equation has a ground state for \begin{document} $\epsilon$ \end{document} small enough and investigate the relation between the number of solutions and the topology of the set where \begin{document} $V$ \end{document} attains its global minimum for small \begin{document} $\epsilon$ \end{document} . We also obtain sufficient conditions for the nonexistence of ground states.

Existence and multiplicity of solutions for a quasilinear Choquard equation via perturbation method

In this paper, we consider the existence and multiplicity of solutions for the following quasilinear Choquard equation: −Δu+V(x)u−uΔ(u2)=(|x|−μ*|u|p)|u|p−2u, x∈RN, where N ≥ 3, μ∈(0,N+22), p∈(2,4N−4μN−2). Under some assumptions on V, we obtain the existence of positive solutions, negative solutions, and high-energy solutions via perturbation method.

Strong instability of standing waves for the fractional Choquard equation

Using variational methods and the potential well theory, strong instability of standing waves for a class of fractional Schrödinger-Choquard equations is established in the mass super-critical and energy sub-critical case.

Ground state solutions of Pohoz̆aev type and Nehari type for a class of nonlinear Choquard equations

In this paper, we study the autonomous Choquard equation−Δu+u=(Iα⁎F(u))f(u),inRN, where N≥3, 0<α<N, Iα is a Riesz potential, and f∈C(R,R) satisfies the general Berestycki–Lions conditions. In Sec. 2, combining constrained variational method with deformation lemma, we obtain a ground state solution of Pohoz̆aev type for the above equation. In Sec. 3, using non-Nehari manifold method, we prove that the above equation has a ground state solution of Nehari type. The results improve some ones in [12].

The Choquard Equation with Weighted Terms and Sobolev-Hardy Exponent

We study a nonlinear Choquard equation with weighted terms and critical Sobolev-Hardy exponent. We apply variational methods and Lusternik-Schnirelmann category to prove the multiple positive solutions for this problem.

Existence and qualitative properties of solutions for Choquard equations with a local term

In this paper, a nontrivial solution u∈H1(RN) to the autonomous Choquard equation with a local term −Δu+λu=(Iα∗|u|p)|u|p−2u+|u|q−2uinRNis obtained, where N≥3, α∈(0,N), λ>0 is a constant, Iα is the Riesz potential, N+αN<p<N+αN−2 and q∈(2,2∗=2NN−2). Under some further assumptions on p and q, the regularity and the Pohožaev identity of the solution are established, and then it is shown that the obtained solution is a groundstate of mountain pass type. Moreover, the positivity and symmetry of the groundstate are also considered. By using the results obtained for the autonomous equation, a positive groundstate solution for the nonautonomous equation −Δu+V(x)u=(Iα∗|u|p)|u|p−2u+|u|q−2uinRNis also found under some assumptions on V(x).

Boundary singularity of Choquard equation in the half space

The main purpose of this paper is to study the existence and non-existence of isolated singularities of positive solutions for the following Choquard equation(1){−Δu+u=Iα[up]uqinR+N,−Δu+u=0on∂R+N∖{0}, where p,q>0, N≥2, α∈(0,N) andIα[up](x)=∫R+Nu(y)p|x−y|N−αdy. Here we obtain the existence of singular solutions to (1) by considering the very weak solutions of{−Δu+u=Iα[up]uqinR+N,−Δu+u=kδ0on∂R+N, where k>0 and δ0 is Dirac mass concentrated at the origin.

Control of radiative base recombination in the quantum cascade light-emitting transistor using quantum state overlap

The concept of the quantum cascade light-emitting transistor (QCLET) is proposed by incorporating periodic stages of quantum wells and barriers in the completely depleted base–collector junction of a heterojunction bipolar transistor. The radiative band-to-band base recombination in the QCLET is shown to be controllable using the base–collector voltage bias for a given emitter–base biasing condition. A self-consistent Schrodinger–Poisson Equation model is built to validate the idea of the QCLET. A GaAs-based QCLET is designed and fabricated. Control of radiative band-to-band base recombination is observed and characterized. By changing the voltage across the quantum cascade region in the QCLET, the alignment of quantum states in the cascade region creates a tunable barrier for electrons that allows or suppresses emitter-injected electron flow from the p-type base through the quantum cascade region into the collector. The field-dependent electron barrier in the base–collector junction manipulates the effective minority carrier lifetime in the base and controls the radiative base recombination process. Under different quantum cascade region biasing conditions, the radiative base recombination is measured and analyzed.

Groundstates of the Choquard equations with a sign-changing self-interaction potential

We consider a nonlinear Choquard equation $$ -\Delta u+u= (V * |u|^p )|u|^{p-2}u \qquad \text{in }\mathbb{R}^N, $$ when the self-interaction potential $V$ is unbounded from below. Under some assumptions on $V$ and on $p$, covering $p =2$ and $V$ being the one- or two-dimensional Newton kernel, we prove the existence of a nontrivial groundstate solution $u\in H^1 (\mathbb{R}^N)\setminus\{0\}$ by solving a relaxed problem by a constrained minimization and then proving the convergence of the relaxed solutions to a groundstate of the original equation.

Limit profiles and uniqueness of ground states to the nonlinear Choquard equations

Abstract Consider nonlinear Choquard equations { - Δ ⁢ u + u = ( I α * | u | p ) ⁢ | u | p - 2 ⁢ u in ⁢ ℝ N , lim x → ∞ ⁡ u ⁢ ( x ) = 0 , \left\{\begin{aligned} \displaystyle-\Delta u+u&amp;\displaystyle=(I_{\alpha}*% \lvert u\rvert^{p})\lvert u\rvert^{p-2}u&amp;&amp;\displaystyle\phantom{}\text{in }% \mathbb{R}^{N},\\ \displaystyle\lim_{x\to\infty}u(x)&amp;\displaystyle=0,\end{aligned}\right. where I α {I_{\alpha}} denotes the Riesz potential and α ∈ ( 0 , N ) {\alpha\in(0,N)} . In this paper, we investigate limit profiles of ground states of nonlinear Choquard equations as α → 0 {\alpha\to 0} or α → N {\alpha\to N} . This leads to the uniqueness and nondegeneracy of ground states when α is sufficiently close to 0 or close to N.

Isolated singularities of positive solutions for Choquard equations in sublinear case

Our purpose of this paper is to study the isolated singularities of positive solutions to Choquard equation in the sublinear case [Formula: see text] [Formula: see text] where [Formula: see text] and [Formula: see text] is the Riesz potential, which appears as a nonlocal term in the equation. We investigate the nonexistence and existence of isolated singular solutions of Choquard equation under different range of the pair of exponent [Formula: see text]. Furthermore, we obtain qualitative properties for the minimal singular solutions of the equation.

A strongly indefinite Choquard equation with critical exponent due to the Hardy–Littlewood–Sobolev inequality

In this paper, we are concerned with the following nonlinear Choquard equation [Formula: see text] where [Formula: see text], [Formula: see text] and [Formula: see text]. If [Formula: see text] lies in a gap of the spectrum of [Formula: see text] and [Formula: see text] is of critical growth due to the Hardy–Littlewood–Sobolev inequality, we obtain the existence of nontrivial solutions by variational methods. The main result here extends and complements the earlier theorems obtained in [N. Ackermann, On a periodic Schrödinger equation with nonlocal superlinear part, Math. Z. 248 (2004) 423–443; B. Buffoni, L. Jeanjean and C. A. Stuart, Existence of a nontrivial solution to a strongly indefinite semilinear equation, Proc. Amer. Math. Soc. 119 (1993) 179–186; V. Moroz and J. Van Schaftingen, Existence of groundstates for a class of nonlinear Choquard equations, Trans. Amer. Math. Soc. 367 (2015) 6557–6579].

Existence and Concentration of Solutions For Sublinear Schrödinger-Poisson Equations

We concern the sublinear Schrodinger-Poisson equations $$\left\{ \begin{gathered} - \Delta u + \lambda V\left( x \right)u + \phi u = f\left( {x,u} \right)in{\mathbb{R}^3} \hfill \\ - \Delta \phi = {u^2}in{\mathbb{R}^3} \hfill \\ \end{gathered} \right.$$ where λ > 0 is a parameter, V ∈ C(R3,[0,+∞)), f ∈ C(R3×R,R) and V-1(0) has nonempty interior. We establish the existence of solution and explore the concentration of solutions on the set V-1(0) as λ → ∞ as well. Our results improve and extend some related works.

Existence and concentration of ground states for a Choquard equation with competing potentials

In this paper, we are concerned with the following Choquard equation in R3 that−ϵ2Δu+V(x)u=ϵμ−3[(∫R3P(y)|u(y)|p|x−y|μ)P(x)|u|p−2u+(∫R3Q(y)|u(y)|q|x−y|μ)Q(x)|u|q−2u], where ϵ>0 is a parameter, 0<μ<3, 6−μ3<q<p<6−μ, the functions V and P are positive and Q may be sign-changing. Via variational methods, we establish the existence of ground states for small ϵ, and investigate the concentration behavior of ground states and show that they concentrate at a global minimum point of the least energy function as ϵ→0.

Ground state sign-changing solutions for a class of subcritical Choquard equations with a critical pure power nonlinearity in [formula omitted

In this paper, we investigate the existence of ground state sign-changing solutions for a class of Choquard equations −△u+(1+λf(x))u=(Iα∗k|u|p)k(x)|u|p−2u+|u|2∗−2u,x∈RN,where k and f are nonnegative functions, N≥3, 2∗=2NN−2, p∈2,N+αN−2, −λ1<λ<0 and λ1 is the first eigenvalue of the equation −△u+u=λf(x)u in H1(RN). Using the sign-changing Nehari manifold, we prove that the Choquard equation has at least one ground state sign-changing solution. This paper can be regarded as the complementary work of Ghimenti and Van Schaftingen (2016), Van Schaftingen and Xia (2017).

Groundstates for a local nonlinear perturbation of the Choquard equations with lower critical exponent

We prove the existence of ground state solutions by variational methods to the nonlinear Choquard equations with a nonlinear perturbation−Δu+u=(Iα⁎|u|αN+1)|u|αN−1u+f(x,u) in RN where N≥1, Iα is the Riesz potential of order α∈(0,N), the exponent αN+1 is critical with respect to the Hardy–Littlewood–Sobolev inequality and the nonlinear perturbation f satisfies suitable growth and structural assumptions.