含自聚焦核的薛定谔方程解的存在性

Abstract

Abstract:

We are interested in the sign-changing solutions for the equation $- \Delta u + u = {Q_n}(x){\left| u \right|^{p - 2}}u$ in $\mathbb{R}^N$, where $Q_n$ are concrete bounded functions with a self-focusing core ${{\rm supp}\{Q_n^+>0\}}$ that shrinks to a finite set of points as $n\rightarrow \infty$. We prove the existence of least energy sign-changing solutions that change sign only once for any $n>0$ if the self-focusing core shrinks to one or two points. Additionally, the least energy sign-changing solutions may also concentrate and converge to the solution of some limit equation. Furthermore, we utilize a penalty function distinct from that in Fang and Wang [Fang X D, Wang Z Q. Calc Var, 2020, 59(4): Art 129] to construct localized, bounded, sign-changing solutions of concentration type.

Key words: sign-changing solutions, Schrödinger equations, existence

CLC Number:

O175.29

Junyuan Liu, Xinran Wei. Sign-Changing Solutions for Schrödinger Equations with Shrinking Self-Focusing Core[J].Acta mathematica scientia,Series A, 2026, 46(5): 1721-1741.

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References 27

[1]	Ackermann N, Szulkin A. A concentration phenomenon for semilinear elliptic equations. Arch Ration Mech Anal, 2013, 207: 1075-1089 doi: 10.1007/s00205-012-0589-1
[2]	Ambrosetti A, Arcoya D, Gámez J L. Asymmetric bound states of differential equations in nonlinear optics. Rend Sem Mat Univ Padova, 1998, 100: 231-247
[3]	Ambrosetti A, Badiale M, Cingolani S. Semiclassical states of nonlinear Schrödinger equations. Arch Ration Mech Anal, 1997, 140: 285-300 doi: 10.1007/s002050050067
[4]	Ambrosetti A, Felli V, Malchiodi A. Ground states of nonlinear Schrödinger equations with potentials vanishing at infinity. J Eur Math Soc, 2005, 7: 117-144 doi: 10.4171/jems
[5]	Ambrosetti A, Malchiodi A, Ruiz D. Bound states of nonlinear Schrödinger equations with potentials vanishing at infinity. J Anal Math, 2006, 98: 317-348 doi: 10.1007/BF02790279
[6]	Ambrosetti A, Wang Z Q. Nonlinear Schrödinger equations with vanishing and decaying potentials. Differ Integral Equ, 2005, 18: 1321-1332
[7]	Bartsch T, Weth T, Willem M. Partial symmetry of least energy nodal solutions to some variational problems. Anal Math, 2005, 96: 1-18 doi: 10.1007/BF01209151
[8]	Bonheure D, Gomes J M, Habets P. Multiple positive solutions of superlinear elliptic problems with sign-changing weight. J Differential Equations, 2005, 214(1): 36-64 doi: 10.1016/j.jde.2004.08.009
[9]	Buryak A V, Trapani P D, Skryabin D V, Trillo S. Optical solitons due to quadratic nonlinearities: from basic physics to futuristic applications. Phys Rep, 2002, 370: 63-235 doi: 10.1016/S0370-1573(02)00196-5
[10]	Byeon J, Wang Z Q. Standing waves with a critical frequency for nonlinear Schrödinger equations. Arch Ration Mech Anal, 2002, 165: 295-316 doi: 10.1007/s00205-002-0225-6
[11]	Byeon J, Wang Z Q. Standing waves with a critical frequency for nonlinear Schrödinger equations II. Calc Var, 2003, 18(2): 207-219 doi: 10.1007/s00526-002-0191-8
[12]	Cao D M, Noussair E S. Multi-bump standing waves with a critical frequency for nonlinear Schrdinger equations. J Differential Equations, 2004, 203(2): 292-312 doi: 10.1016/j.jde.2004.05.003
[13]	Cao D M, Peng S J. Multi-bump bound states of Schrödinger equations with a critical frequency. Mathematische Annalen, 2006, 336(4): 925-948 doi: 10.1007/s00208-006-0021-y
[14]	Castro A, Cossio J, Neuberger J. A sign-changing solution for a superlinear Dirichlet problem. Rocky Mt J Math, 1997, 27: 1041-1053
[15]	Cui Z Y, Shuai W. Sign-changing solutions for {K}irchhoff-type equations with indefinite nonlinearities. Z Angew Math Phys, 2023, 74(4): Art 150 doi: 10.1007/s00033-023-02031-8
[16]	Del Pino M, Felmer P. Local mountain passes for semilinear elliptic problems in unbounded domains. Calc Var, 1996, 4: 121-137 doi: 10.1007/BF01189950
[17]	Del Pino M, Felmer P. Multi-peak bound states for nonlinear Schrödinger equations. Ann Inst H Poincaré, 1998, 15(2): 127-149
[18]	Del Pino M, Felmer P. Semi-classical states for nonlinear Schrödinger equations. J Funct Anal, 1997, 149: 245-265 doi: 10.1006/jfan.1996.3085
[19]	Dror N, Malomed B A. Solitons supported by localized nonlinearities in periodic media. Phys Rev A, 2011, 83: Art 033828
[20]	Fang X D, Wang Z Q. Limiting profile of solutions for Schrödinger equations with shrinking self-focusing core. Calc Var, 2020, 59(4): Art 129
[21]	Floer A, Weinstein A. Nonspreading wave packets for the cubic Schrödinger equations with a bounded potential. J Func Anal, 1996, 69: 397-408 doi: 10.1016/0022-1236(86)90096-0
[22]	Girão P M, Gomes J M. Multibump nodal solutions for an indefinite superlinear elliptic problem. J Differential Equations, 2009, 247(4): 1001-1012 doi: 10.1016/j.jde.2009.04.018
[23]	Jeanjean L, Tanaka K. Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities. Calc Var, 2004, 21: 287-318
[24]	Kartashov Y V, Malomed B A, Torner L. Solitons in nonlinear lattices. Rev Mod Phys, 2011, 83: 247-305 doi: 10.1103/RevModPhys.83.247
[25]	López-Gómez J. Varying bifurcation diagrams of positive solutions for a class of indefinite superlinear boundary value problems. Trans Amer Math Soc, 2000, 352(4): 1825-1858 doi: 10.1090/tran/2000-352-04
[26]	Rabinowitz P H. On a class of nonlinear Schrödinger equations. Z Angew Math Phys, 1992, 43: 270-291 doi: 10.1007/BF00946631
[27]	Wang X. On concentration of positive bound states of nonlinear Schrödinger equations. Commun Math Phys, 1993, 153: 229-244 doi: 10.1007/BF02096642

Sign-Changing Solutions for Schrödinger Equations with Shrinking Self-Focusing Core

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