Multiple normalized solutions for the planar Schrödinger–Poisson system with critical exponential growth

2024/02/16 by Sitong Chen, Vicenţiu D. Rădulescu, Xianhua Tang · 1 citation
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Nonlinear Differential Equations Analysis

paper · pdf · doi:10.1007/s00209-024-03432-9

Abstract

Abstract The paper deals with the existence of normalized solutions for the following Schrödinger–Poisson system with L² <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -constraint: \beginaligned \ -Δ u+λ u+μ ( log |⋅ |*u²) u=( e^u2-1-u²) u, · amp; x∈ \mathbb R²,
∫ _\mathbb R²u²\textrmdx=c, . \endaligned <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtable> <mml:mtr> <mml:mtd> <mml:mfenced> <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:mi>λ</mml:mi> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mi>μ</mml:mi> <mml:mfenced> <mml:mrow> <mml:mo>log</mml:mo> <mml:mo>|</mml:mo> <mml:mo>·</mml:mo> <mml:mo>|</mml:mo> <mml:mrow/> <mml:mo>∗</mml:mo> </mml:mrow> <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:mfenced> <mml:msup> <mml:mi>e</mml:mi> <mml:msup> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:msup> <mml:mo>-</mml:mo> <mml:mn>1</mml:mn> <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:mrow> </mml:mtd> <mml:mtd> <mml:mrow> <mml:mrow/> <mml:mi>x</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> <mml:mtr> <mml:mtd> <mml:mrow> <mml:mrow/> <mml:msub> <mml:mo>∫</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> </mml:msub> <mml:msup> <mml:mi>u</mml:mi> <mml:mn>2</mml:mn> </mml:msup> <mml:mtext>d</mml:mtext> <mml:mi>x</mml:mi> <mml:mo>=</mml:mo> <mml:mi>c</mml:mi> <mml:mo>,</mml:mo> </mml:mrow> </mml:mtd> </mml:mtr> </mml:mtable> </mml:mrow> </mml:mfenced> </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> , λ ∈ \mathbb R <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>λ</mml:mi> <mml:mo>∈</mml:mo> <mml:mi>R</mml:mi> </mml:mrow> </mml:math> will arise as a Lagrange multiplier and the nonlinearity enjoys critical exponential growth of Trudinger-Moser type. By specifying explicit conditions on the energy level c , we detect a geometry of local minimum and a minimax structure for the corresponding energy functional, and prove the existence of two solutions, one being a local minimizer and one of mountain-pass type. In particular, to catch a second solution of mountain-pass type, some sharp estimates of energy levels are proposed, suggesting a new threshold of compactness in the L² <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>L</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> -constraint. Our study extends and complements the results of Cingolani–Jeanjean (SIAM J Math Anal 51(4): 3533-3568, 2019) dealing with the power nonlinearity a|u|^p-2u <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>a</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:mi>u</mml:mi> </mml:mrow> </mml:math> in the case of agt;0 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>a</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> and pgt;4 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>p</mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>4</mml:mn> </mml:mrow> </mml:math> , which seems to be the first contribution in the context of normalized solutions. Our model presents some new difficulties due to the intricate interplay between a logarithmic convolution potential and a nonlinear term of critical exponential type and requires a novel analysis and the implementation of new ideas, especially in the compactness argument. We believe that ou

Citations

• Trudinger type inequalities in 𝐑^𝐍 and their best exponents
• Nonlinear scalar field equations, II existence of infinitely many solutions
• Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations
• Equivalent Moser type inequalities in R 2 and the zero mass case
• Normalized ground states for the NLS equation with combined nonlinearities: The Sobolev critical case
• Normalized ground states for the NLS equation with combined nonlinearities
• Axially symmetric solutions for the planar Schrödinger-Poisson system with critical exponential growth
• Multiple normalized solutions for a Sobolev critical Schrödinger equation
• Another look at Schrödinger equations with prescribed mass

Cited by

• On planar Schrodinger-Poisson systems with repulsive interactions in the mass supercritical regime

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