{"title":"On the number of normalized solutions for a fractional Schrödinger problem with logarithmic nonlinearity","authors":"Xiaolu Lin, Shenzhou Zheng","doi":"10.1016/j.cnsns.2025.108618","DOIUrl":null,"url":null,"abstract":"In this paper, we investigate the multiplicity and concentration of normalized solutions to a fractional logarithmic Schrödinger problem <mml:math altimg=\"si1.svg\" display=\"block\"><mml:mrow><mml:msup><mml:mrow><mml:mi>ɛ</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:msup><mml:mrow><mml:mrow><mml:mo>(</mml:mo><mml:mo>−</mml:mo><mml:mi>Δ</mml:mi><mml:mo>)</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mi>s</mml:mi></mml:mrow></mml:msup><mml:mi>u</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">+</mml:mo><mml:mi>V</mml:mi><mml:mrow><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mi>u</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">=</mml:mo><mml:mi>λ</mml:mi><mml:mi>u</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">+</mml:mo><mml:mi>u</mml:mi><mml:mo>log</mml:mo><mml:msup><mml:mrow><mml:mi>u</mml:mi></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mspace width=\"1em\"></mml:mspace><mml:mtext>in</mml:mtext><mml:mspace width=\"1em\"></mml:mspace><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:math> with the prescribed mass <mml:math altimg=\"si2.svg\" display=\"inline\"><mml:mrow><mml:msub><mml:mrow><mml:mo>∫</mml:mo></mml:mrow><mml:mrow><mml:msup><mml:mrow><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup></mml:mrow></mml:msub><mml:msup><mml:mrow><mml:mrow><mml:mo>|</mml:mo><mml:mi>u</mml:mi><mml:mo>|</mml:mo></mml:mrow></mml:mrow><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:msup><mml:mi mathvariant=\"normal\">d</mml:mi><mml:mi>x</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">=</mml:mo><mml:mi>a</mml:mi><mml:msup><mml:mrow><mml:mi>ɛ</mml:mi></mml:mrow><mml:mrow><mml:mi>N</mml:mi></mml:mrow></mml:msup><mml:mspace width=\"1em\"></mml:mspace><mml:mtext>with</mml:mtext><mml:mspace width=\"1em\"></mml:mspace><mml:mi>a</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>, where <mml:math altimg=\"si3.svg\" display=\"inline\"><mml:mrow><mml:mi>ɛ</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">></mml:mo><mml:mn>0</mml:mn></mml:mrow></mml:math>, <mml:math altimg=\"si4.svg\" display=\"inline\"><mml:mrow><mml:mi>λ</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">∈</mml:mo><mml:mi mathvariant=\"double-struck\">R</mml:mi></mml:mrow></mml:math> is unknown and appears as a Lagrange multiplier. By the minimization method combined with penalization technique and Ljusternik–Schnirelmann theory, we prove the multiplicity of normalized solutions where the numbers of normalized solutions are linked to the topology of the set where potential <mml:math altimg=\"si5.svg\" display=\"inline\"><mml:mi>V</mml:mi></mml:math> attains its minimum. Moreover, the concentration and decay of normalized solutions are analyzed in the end. The above properties of our normalized solutions are also new, even for <mml:math altimg=\"si6.svg\" display=\"inline\"><mml:mrow><mml:mi>s</mml:mi><mml:mo linebreak=\"goodbreak\" linebreakstyle=\"after\">=</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:math>.","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"1 1","pages":""},"PeriodicalIF":3.4000,"publicationDate":"2025-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Communications in Nonlinear Science and Numerical Simulation","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1016/j.cnsns.2025.108618","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper, we investigate the multiplicity and concentration of normalized solutions to a fractional logarithmic Schrödinger problem ɛ2s(−Δ)su+V(x)u=λu+ulogu2inRN with the prescribed mass ∫RN|u|2dx=aɛNwitha>0, where ɛ>0, λ∈R is unknown and appears as a Lagrange multiplier. By the minimization method combined with penalization technique and Ljusternik–Schnirelmann theory, we prove the multiplicity of normalized solutions where the numbers of normalized solutions are linked to the topology of the set where potential V attains its minimum. Moreover, the concentration and decay of normalized solutions are analyzed in the end. The above properties of our normalized solutions are also new, even for s=1.
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The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity.
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Topics of interest:
Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity.
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