Normalized concentrating solutions to nonlinear elliptic problems

JOURNAL OF DIFFERENTIAL EQUATIONS

卷:275

Article

Pellacci, Benedetta¹  Pistoia, Angela²  Vaira, Giusi¹  Verzini, Gianmaria³ 

[1] Univ Campania Luigi Vanvitelli, Dipartimento Matemat & Fis, Via A Lincoln 5, I-81100 Caserta, Italy

[2] Sapienza Univ Roma, Dipartimento SBAI, Via Antonio Scapa 16, I-00161 Rome, Italy

[3] Politecn Milan, Dipartimento Matemat, Pza Leonardo da Vinci 32, I-20133 Milan, Italy

关键词: Nonlinear Schrodinger equation;    Mean Field Games;    Singularly perturbed problems;    Lyapunov-Schmidt reduction;   

DOI  :  10.1016/j.jde.2020.11.003

来源: Elsevier

【 摘 要 】

We prove the existence of solutions (lambda, v) is an element of R x H-1(Omega) of the elliptic problem {-Lambda nu + (V(x)+ lambda)nu = nu(p) in Omega, nu > 0, integral(nu 2)(Omega) dx = rho. Any vsolving such problem (for some lambda) is called a normalized solution, where the normalization is settled in L-2(Omega). Here Omega is either the whole space R-N or a bounded smooth domain of R-N, in which case we assume V equivalent to 0 and homogeneous Dirichlet or Neumann boundary conditions. Moreover, 1 < p < N+2/N- 2 if N >= 3 and p > 1 if N = 1, 2. Normalized solutions appear in different contexts, such as the study of the Nonlinear Schrodinger equation, or that of quadratic ergodic Mean Field Games systems. We prove the existence of solutions concentrating at suitable points of Omega as the prescribed mass rho is either small (when p< 1 + 4/N) or large (when p = 1 + 4N) or it approaches some critical threshold (when p = 1 + 4N). (c) 2020 Elsevier Inc. All rights reserved.

【 授权许可】

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