Choquard Type Equations Research Articles

Existence of Ground State Solutions for the Kirchhoff–Choquard Equation With Critical Exponent and External Coulomb Potential

We study the critical Kirchhoff–Choquard type equation [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text] are positive constants, [Formula: see text] denotes the Riesz potential and [Formula: see text] is the lower critical Hardy–Littlewood–Sobolev exponent. By means of the compactness-concentration principle and the squeezing energy inequality, we obtain the existence of ground state solutions of Nehari–Pohoaev type for the above problem for sufficiently small [Formula: see text].

The existence of positive ground state solutions for the Choquard type equation on groups of polynomial growth

The existence of positive ground state solutions for the Choquard type equation on groups of polynomial growth

Normalized solutions for Kirchhoff–Choquard type equations with different potentials

Normalized solutions for Kirchhoff–Choquard type equations with different potentials

The existence and concentration behavior of positive ground state solutions for a class of Choquard type equations involving local and nonlocal operators

The existence and concentration behavior of positive ground state solutions for a class of Choquard type equations involving local and nonlocal operators

Infinitely many solutions for p-fractional Choquard type equations involving general nonlocal nonlinearities with critical growth via the concentration compactness method

We prove the existence of infinitely many solutions to a fractional Choquard type equation(−Δ)psu+V(x)|u|p−2u=(K⁎g(u))g′(u)+εWW(x)f′(u)in RN involving the fractional p-Laplacian and a general convolution term with critical growth. In order to obtain infinitely many solutions, we use a type of the symmetric mountain pass lemma which gives sequences of critical values converging to zero for even functionals. To assure the (PS)c conditions, we also use a nonlocal version of the concentration compactness lemma.

Uniqueness, symmetry and convergence of positive ground state solutions of the Choquard type equation on a ball

This paper is concerned with the qualitative properties of the positive ground state solutions to the nonlocal Choquard type equation on a ball BR. First, we prove the radial symmetry of all the positive ground state solutions by using Talenti's inequality. Next we develop the Newton's Theorem and then resort to the contraction mapping principle to establish the uniqueness of the positive ground state solution. Finally, by constructing cut-off functions and applying energy comparison method, we show the convergence of the unique positive ground state solutions as the radius R→∞. Our results extend and complement the existing ones in the literature.

A Choquard type equation involving mixed local and nonlocal operators

In the present work, we are concerned with an elliptic problem involving a mixed order diffusion operator with both local and nonlocal parts and a critical nonlinearity of Hartree type. We first investigate the corresponding Hardy-Littlewood-Sobolev inequality and establish the optimal constant. Using variational methods and a Pohožaev identity we then show existence and nonexistence results for the corresponding subcritical perturbation problem.

Positive solutions to a class of Choquard type equations with a competing perturbation

In this paper, we study a class of Choquard type equations with a competing perturbation{−Δu+λV(x)u=(Iα⁎K|u|p)K|u|p−2u−|u|q−2uinRN,u∈H1(RN), where N≥3,λ>0 a parameter, V∈C(RN,R+), Iα is the Riesz potential, K(x)≥0 in RN, 1+αN<p<N+αN−2 and 2<q<2⁎=2NN−2. Unlike the existing literature, we are interested in the existence and multiplicity of positive solutions under the different relationship between p and q when the competing effect of the nonlocal term with the perturbation happens. In particular, when 2p<q, variational methods can not be applied in a standard way, even restricting the energy functional on the Nehari manifold, because Palais-Smale sequences may not be bounded. A new constraint approach developed by us recently is employed in this case.

Limit profiles for singularly perturbed Choquard equations with local repulsion

We study Choquard type equation of the form where Nge 3, I_alpha is the Riesz potential with alpha in (0,N), p>1, q>2 and varepsilon ge 0. Equations of this type describe collective behaviour of self-interacting many-body systems. The nonlocal nonlinear term represents long-range attraction while the local nonlinear term represents short-range repulsion. In the first part of the paper for a nearly optimal range of parameters we prove the existence and study regularity and qualitative properties of positive groundstates of (P_0) and of (P_varepsilon ) with varepsilon >0. We also study the existence of a compactly supported groundstate for an integral Thomas–Fermi type equation associated to (P_{varepsilon }). In the second part of the paper, for varepsilon rightarrow 0 we identify six different asymptotic regimes and provide a characterisation of the limit profiles of the groundstates of (P_varepsilon ) in each of the regimes. We also outline three different asymptotic regimes in the case varepsilon rightarrow infty . In one of the asymptotic regimes positive groundstates of (P_varepsilon ) converge to a compactly supported Thomas–Fermi limit profile. This is a new and purely nonlocal phenomenon that can not be observed in the local prototype case of (P_varepsilon ) with alpha =0. In particular, this provides a justification for the Thomas–Fermi approximation in astrophysical models of self-gravitating Bose–Einstein condensate.

Quasilinear logarithmic Choquard equations with exponential growth in [formula omitted

We consider the N-Laplacian Schrödinger equation strongly coupled with higher order fractional Poisson's equations. When the order of the Riesz potential α is equal to the Euclidean dimension N, and thus it is a logarithm, the system turns out to be equivalent to a nonlocal Choquard type equation. On the one hand, the natural function space setting in which the Schrödinger energy is well defined is the Sobolev limiting space W1,N(RN), where the maximal nonlinear growth is of exponential type. On the other hand, in order to have the nonlocal energy well defined and prove the existence of finite energy solutions, we introduce a suitable log-weighted variant of the Pohozaev-Trudinger inequality which provides a suitable functional framework where we use variational methods.

On a Kirchhoff Choquard type equation with magnetic field involving exponential critical growth in [formula omitted

This paper is concerned with the existence of solutions for a Kirchhoff Choquard type equation with magnetic field involving exponential critical growth in R2.

Groundstates for Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent

AbstractWe are concerned with a class of Choquard type equations with weighted potentials and Hardy–Littlewood–Sobolev lower critical exponent−Δu+V(x)u=Iα∗[Q(x)|u|N+αN]Q(x)|u|αN−1u,x∈RN.$$\begin{array}{} \displaystyle -{\it\Delta} u+V(x)u=\left(I_{\alpha}\ast [Q(x)|u|^{\frac{N+\alpha}{N}}]\right)Q(x)|u|^{\frac{\alpha}{N}-1}u, \quad x\in \mathbb R^N. \end{array}$$By using variational approaches, we investigate the existence of groundstates relying on the asymptotic behaviour of weighted potentials at infinity. Moreover, non-existence of non-trivial solutions is also considered. In particular, we give a partial answer to some open questions raised in [D.~Cassani, J. Van Schaftingen and J. J. Zhang, Groundstates for Choquard type equations with Hardy-Littlewood-Sobolev lower critical exponent,Proceedings of the Royal Society of Edinburgh, Section A Mathematics,150(2020), 1377–1400].

Choquard-type equation with competing coefficients

In this paper, we investigate the following Choquard-type equation−Δu+Vu=[Iα⁎(K|u|p)]K|u|p−2uinRN, where V,K∈C(RN) are two potential functions, and Iα denotes the Riesz potential. By comparing the decay rates of V and K, we obtain two results stating the existence of positive ground states to the above equation. Our proofs are heavily based on the fundamental properties of solutions. These properties are also important to study the Choquard-type equations.

Positive ground states for nonlinearly coupled Choquard type equations with lower critical exponents

We study the coupled Choquard type system with lower critical exponents \t\t\t{−Δu+λ1(x)u=μ1(Iα∗|u|N+αN)|u|αN−1u+β(Iα∗|v|N+αN)|u|αN−1u,x∈RN,−Δv+λ2(x)v=μ2(Iα∗|v|N+αN)|v|αN−1v+β(Iα∗|u|N+αN)|v|αN−1v,x∈RN,u,v∈H1(RN),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ \\textstyle\\begin{cases} -\\Delta u+\\lambda _{1}(x)u=\\mu _{1}(I_{\\alpha }* \\vert u \\vert ^{ \\frac{N+\\alpha }{N}}) \\vert u \\vert ^{\\frac{\\alpha }{N}-1}u+\\beta (I_{\\alpha }* \\vert v \\vert ^{ \\frac{N+\\alpha }{N}}) \\vert u \\vert ^{\\frac{\\alpha }{N}-1}u,\\quad x\\in {\\mathbb{R}}^{N}, \\\\ -\\Delta v+\\lambda _{2}(x)v=\\mu _{2}(I_{\\alpha }* \\vert v \\vert ^{ \\frac{N+\\alpha }{N}}) \\vert v \\vert ^{\\frac{\\alpha }{N}-1}v+\\beta (I_{\\alpha }* \\vert u \\vert ^{ \\frac{N+\\alpha }{N}}) \\vert v \\vert ^{\\frac{\\alpha }{N}-1}v,\\quad x\\in {\\mathbb{R}}^{N}, \\\\ u, v\\in H^{1}({\\mathbb{R}}^{N}), \\end{cases} $$\\end{document} where Nge 3, mu _{1}, mu _{2}, beta >0, and lambda _{1}(x), lambda _{2}(x) are nonnegative functions. The existence of at least one positive ground state of this system is proved under certain assumptions on lambda _{1}, lambda _{2}.

Symmetry and monotonicity of positive solutions for a Choquard equation with the fractional Laplacian

In this paper, we consider the Choquard-type equation with the fractional Laplacian where 0<s<1, , , is a constant and . We will give a variant decay at infinity and a variant narrow region principle, then combining the direct method of moving planes to prove the solutions of the above equation must be radially symmetric and monotone decreasing about some point in .

Vector Solutions for Linearly Coupled Choquard Type Equations with Lower Critical Exponents

The existence, nonexistence, and multiplicity of vector solutions of the linearly coupled Choquard type equations−Δu+V1xu=Iα∗uN+α/Nuα/N−1u+λv,x∈ℝN,−Δv+V2xv=Iα∗vN+α/Nvα/N−1v+λu,x∈ℝN,u,v∈H1ℝN,are proved, whereα∈0,N,N≥3,V1xV2x∈L∞ℝNare positive functions, andIαdenotes the Riesz potential.

Semiclassical states for Choquard type equations with critical growth: critical frequency case * *Minbo Yang is the corresponding author who is partially supported by NSFC (11971436, 12011530199) and ZJNSF (LD19A010001); Yanheng Ding is partially supported by NSFC (11871242); Fashun Gao is partially supported by NSFC (11901155).

In this paper we are interested in the existence of semiclassical states for the Choquard type equation where 0 < μ < N, N ⩾ 3, ɛ is a positive parameter and G is the primitive of g which is of critical growth due to the Hardy–Littlewood–Sobolev inequality. The potential function V(x) is assumed to be nonnegative with V(x) = 0 in some region of , which means it is of the critical frequency case. Firstly, we study a Choquard equation with double critical exponents and prove the existence and multiplicity of semiclassical states by the mountain-pass lemma and the genus theory. Secondly, we consider a class of critical Choquard equation without lower perturbation, by establishing a global compactness lemma for the nonlocal Choquard equation, we prove the multiplicity of high energy semiclassical states by the Lusternik–Schnirelman theory.

Symmetry of Positive Solutions to Choquard Type Equations Involving the Fractional $p$-Laplacian

We study symmetric properties of positive solutions to the Choquard type equation $$ (-\Delta )^{s}_{p} u + |x|^{a} u = \left (\frac{1}{|x|^{n-\alpha }}*u^{q} \right ) u^{r} \quad \text{in}\ \mathbb{R}^{n}, $$ where $0< s<1$ , $0<\alpha <n$ , $p\ge 2$ , $q>1$ , $r>0$ , $a\ge 0$ and $(-\Delta )^{s}_{p}$ is the fractional $p$ -Laplacian. Via a direct method of moving planes, we prove that every positive solution $u$ which has an appropriate decay property must be radially symmetric and monotone decreasing about some point, which is the origin if $a>0$ .

Ground states and non-existence results for Choquard type equations with lower critical exponent and indefinite potentials

In this paper, we study following Choquard equation with lower critical exponent: −Δu+V(x)u=(Iα∗|u|αN+1)|u|αN−1u+f(x,u),x∈RN,u∈H1(RN),where N≥1, Iα is the Riesz potential, and V:RN→R allows to be sign-changing. Under some mild assumptions imposed on the nonlinearity f, we prove that above equation has a ground state solution for the periodic case and asymptotically periodic case, respectively. The characterization of the ground states is also investigated by a direct approach that are constrained to the Nehari manifold. Moreover, we show a non-existence result for the equation via a generalized Pohožaev identity established for the non-autonomous nonlinearity f. The results extend and improve some recent ones in the literature.

Symmetry of separable functions and its applications to Choquard type equations

In this paper, we develop a new method to study the symmetry and monotonicity of solutions to elliptic equations. To be precise, we introduce concepts of separable functions in balls and in the whole space, and then investigate the qualitative properties of separable functions. We first study the axial symmetry and monotonicity of separable functions in unit circles by geometry analysis, and we prove the uniqueness of the symmetry axis for nontrivial separable functions. Then by using reduction dimension and convex analysis, we get the axial symmetry and monotonicity of separable functions in high dimensional spheres. Based on the above results on unit circles and spheres, we deduce the axial symmetry and monotonicity of separable functions in balls and the radial symmetry and monotonicity of separable functions in the whole space. Conversely, the function with axial symmetry and monotonicity in the ball domain is separable function, and the function with radial symmetry and monotonicity in the whole space is also separable function. These enable us to provide easily some examples that separable functions in balls may be just axially symmetric not radially symmetric. Finally, as applications, we obtain the axial symmetry and monotonicity of all the positive ground states to the Choquard type equations in a ball as well as the radial symmetry and monotonicity of all the positive ground states in the whole space.