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<p>The paper deals with the following planar Schrödinger-Poisson system <Equation ID="Equa"> <EquationSource Format="TEX">\(\left\{ {\matrix{ { - \Delta u + \lambda u + \phi u = f(u),} \hfill &amp; {{\rm{in}}\;{\mathbb{R}^2},} \hfill \cr { - \Delta \phi = {u^2},} \hfill &amp; {{\rm{in}}\;{\mathbb{R}^2},} \hfill \cr {\int_{{\mathbb{R}^2}} {{u^2}} {\rm{d}}x = {a^2},u \in {H^1}({\mathbb{R}^2}),} \hfill } } \right.\)</EquationSource> </Equation> where <i>a</i> ∈ (0, 1), λ ∈ ℝ is an undetermined parameter which appears as a Lagrange multiplier and <i>f</i> satisfies the exponential critical growth. Under suitable conditions on <i>f</i>, we manage to establish such critical points which emerge as a local minimizer or correspond to a mountain pass. Furthermore, by using genus theory, we obtain infinitely many solutions. The proofs are based upon the reduction method by working on a natural constraint, which is introduced by Bartsch and Soave [J Funct Anal, 2017, 272: 4998–5037]. Our results complement the results made by Cingolani and Jeanjean [SIAM J Math Anal, 2019, 51: 3533–3568] where handle the Sobolev subcritical case.</p>

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Existence and multiplicity of normalized solutions for the planar Schrödinger-Poisson system with exponential critical growth

  • 期刊论文

  • 发表日期：

    2025年10月28日

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  • Xueqin Peng

摘要

The paper deals with the following planar Schrödinger-Poisson system \(\left\{ {\matrix{ { - \Delta u + \lambda u + \phi u = f(u),} \hfill & {{\rm{in}}\;{\mathbb{R}^2},} \hfill \cr { - \Delta \phi = {u^2},} \hfill & {{\rm{in}}\;{\mathbb{R}^2},} \hfill \cr {\int_{{\mathbb{R}^2}} {{u^2}} {\rm{d}}x = {a^2},u \in {H^1}({\mathbb{R}^2}),} \hfill } } \right.\) where a ∈ (0, 1), λ ∈ ℝ is an undetermined parameter which appears as a Lagrange multiplier and f satisfies the exponential critical growth. Under suitable conditions on f, we manage to establish such critical points which emerge as a local minimizer or correspond to a mountain pass. Furthermore, by using genus theory, we obtain infinitely many solutions. The proofs are based upon the reduction method by working on a natural constraint, which is introduced by Bartsch and Soave [J Funct Anal, 2017, 272: 4998–5037]. Our results complement the results made by Cingolani and Jeanjean [SIAM J Math Anal, 2019, 51: 3533–3568] where handle the Sobolev subcritical case.