Lower Critical Exponent Research Articles

Normalized Solutions for the Fractional Choquard Equations with Lower Critical Exponent and Nonlocal Perturbation

Normalized Solutions for the Fractional Choquard Equations with Lower Critical Exponent and Nonlocal Perturbation

Ground state solutions of a magnetic nonlinear Choquard equation with lower critical exponent

In this article, we establish the existence of ground state solutions for a magnetic nonlinear Choquard equation with lower critical exponent by variational methods.

Ground State Solutions of a Magnetic Nonlinear Choquard Equation with Lower Critical Exponent

Ground State Solutions of a Magnetic Nonlinear Choquard Equation with Lower Critical Exponent

Existence of ground state solutions for a class of Choquard equations with local nonlinear perturbation and variable potential

In this paper, we focus on the existence of solutions for the Choquard equation {−Δu+V(x)u=(Iα∗|u|αN+1)|u|αN−1u+λ|u|p−2u,x∈RN;u∈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} $$\\begin{aligned} \\textstyle\\begin{cases} {-}\\Delta {u}+V(x)u=(I_{\\alpha }* \\vert u \\vert ^{\\frac{\\alpha }{N}+1}) \\vert u \\vert ^{ \\frac{\\alpha }{N}-1}u+\\lambda \\vert u \\vert ^{p-2}u,\\quad x\\in \\mathbb{R}^{N}; \\\\ u\\in H^{1}(\\mathbb{R}^{N}), \\end{cases}\\displaystyle \\end{aligned}$$ \\end{document} where lambda >0 is a parameter, alpha in (0,N), Nge 3, I_{alpha }: mathbb{R}^{N}to mathbb{R} is the Riesz potential. As usual, alpha /N+1 is the lower critical exponent in the Hardy–Littlewood–Sobolev inequality. Under some weak assumptions, by using minimax methods and Pohožaev identity, we prove that this problem admits a ground state solution if lambda >lambda _{*} for some given number lambda _{*} in three cases: (i) 2< p<frac{4}{N}+2, (ii) p=frac{4}{N}+2, and (iii) frac{4}{N}+2< p<2^{*}. Our result improves the previous related ones in the literature.

Positive ground state solutions for Choquard equations with lower critical exponent and steep well potential

In this paper, we investigate the following non-autonomous Choquard equation −Δu+λV(x)−μu=Iα∗|u|2∗α|u|2∗α−2uinRN,where N≥3, λ>0, μ∈R, Iα is a Riesz potential of order α∈(0,N), 2∗α=N+αN, the function V∈C(RN,R) is nonnegative and has a potential well Ω≔intV−1(0) such that −Δ possesses a sequence of Dirichlet eigenvalues 0<μ1<μ2<⋯<μn→+∞ on Ω. Assume μ<μ1, by the Nehari manifold method we prove the existence and concentration of positive ground state solutions for λ large enough.

Nehari‐type ground state solutions for a Choquard equation with lower critical exponent and local nonlinear perturbation

This paper deals with the following Choquard equation with a local nonlinear perturbation: where N ≥ 1, α∈(0,N), λ&gt;0, 2&lt;p&lt;2∗, and is the Riesz potential. The exponent is critical with respect to the Hardy‐Littlewood‐Sobolev inequality. In the cases when , , and , respectively, we prove the above equation admits a Nehari‐type ground state solution if λ&gt;λ∗ for some given number λ∗.

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.

Ground state solutions for a Choquard equation with lower critical exponent and local nonlinear perturbation

This paper deals with the following Choquard equation with a local nonlinear perturbation: −Δu+u=Iα∗|u|αN+1|u|αN−1u+f(u),x∈RN;u∈H1(RN),where Iα:RN→R is the Riesz potential, N≥3, α∈(0,N), the exponent αN+1 is critical with respect to the Hardy–Littlewood–Sobolev inequality, and the nonlinear perturbation f is only required to satisfy some weak assumptions near 0 and ∞. Our results improve the previous related ones in the literature.

Ground state solutions for nonlinearly coupled systems of Choquard type with lower critical exponent

Ground state solutions for nonlinearly coupled systems of Choquard type with lower critical exponent

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.