Static solutions for Choquard equations with Coulomb potential and upper critical growth

2025/04/09 by Sitong Chen, Vicenţiu D. Rădulescu, Muhua Shu +1 · 2 citations
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.1007/s00208-025-03143-4

Abstract

Abstract This paper focuses on static solutions for the following Choquard equation with zero mass and Coulomb potential \beginaligned -Δ u+( (1)/(4π |x|)*u²) u=μ |u|^p-2u+(I_α *|u|^{α +3})|u|^{α +1}u, x∈ \mathbb R³, \endaligned <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mo>-</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mfenced> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>π</mml:mi> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:mfrac> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:msup> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:mfenced> <mml:mi>u</mml:mi> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>-</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> <mml:mrow> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mo>(</mml:mo> </mml:mrow> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mmultiscripts> <mml:mrow> <mml:mrow/> <mml:mo>∗</mml:mo> <mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow/> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:mmultiscripts> <mml:msup> <mml:mrow> <mml:mo>)</mml:mo> <mml:mo>|</mml:mo> <mml:mi>u</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> <mml:mi>u</mml:mi> <mml:mo>,</mml:mo> <mml:mspace/> <mml:mspace/> <mml:mspace/> <mml:mspace/> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:math> where μ gt;0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>μ</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> , (18)/(7)lt;p≤ 6 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mfrac> <mml:mn>18</mml:mn> <mml:mn>7</mml:mn> </mml:mfrac> <mml:mo>&lt;</mml:mo> <mml:mi>p</mml:mi> <mml:mo>≤</mml:mo> <mml:mn>6</mml:mn> </mml:mrow> </mml:math> , α ∈ (0, 3) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>∈</mml:mo> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mn>3</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> , α +3 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>+</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> is the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality, I_α : \mathbb R³→ \mathbb R <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>I</mml:mi> <mml:mi>α</mml:mi> </mml:msub> <mml:mo>:</mml:mo> <mml:mspace/> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:mo>→</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> is the Riesz potential, and (1)/(4π |x|) <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mrow> <mml:mn>4</mml:mn> <mml:mi>π</mml:mi> <mml:mo>|</mml:mo> <mml:mi>x</mml:mi> <mml:mo>|</mml:mo> </mml:mrow> </mml:mfrac> </mml:math> is the Coulomb potential. By carefully analyzing the intricate interplay between the power and Coulomb terms, we establish three types of variational geometries of the problem and prove the following existence results based on the behavior of p : the existence of two solutions, one being a local minimizer and the other of mountain-pass type, for an explicit range 0lt;μ lt;\mathrm Const. <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mn>0</mml:mn> <mml:mo>&lt;</mml:mo> <mml:mi>μ</mml:mi> <mml:mo>&lt;</mml:mo> <mml:mrow> <mml:mi>Const</mml:mi>

Citations

• Existence and Uniqueness of the Minimizing Solution of Choquard's Nonlinear Equation
• On the Schrödinger–Poisson–Slater System: Behavior of Minimizers, Radial and Nonradial Cases
• Nonlinear scalar field equations, I existence of a ground state
• On Gravity's role in Quantum State Reduction
• Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case
• Another look at Schrödinger equations with prescribed mass

Cited by

• Isolated singularities for elliptic equations with convolution terms in a punctured ball
• On nonlinear elliptic problems with Hardy-Littlewood-Sobolev critical exponent and Sobolev-Hardy critical exponent

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