We prove the existence of solutions (λ,v)∈R×H1(Ω) of the elliptic problem {−Δv+(V(x)+λ)v=vp in Ω,v>0,∫Ωv2dx=ρ. Any v solving such problem (for some λ) is called a normalized solution, where the normalization is settled in L2(Ω). Here Ω is either the whole space RN or a bounded smooth domain of RN, in which case we assume V≡0 and homogeneous Dirichlet or Neumann boundary conditions. Moreover, [Formula presented] if N≥3 and p>1 if N=1,2. Normalized solutions appear in different contexts, such as the study of the Nonlinear Schrödinger equation, or that of quadratic ergodic Mean Field Games systems. We prove the existence of solutions concentrating at suitable points of Ω as the prescribed mass ρ is either small (when [Formula presented]) or large (when [Formula presented]) or it approaches some critical threshold (when [Formula presented]).

Normalized concentrating solutions to nonlinear elliptic problems / Pellacci, B.; Pistoia, A.; Vaira, G.; Verzini, G.. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 275:(2021), pp. 882-919. [10.1016/j.jde.2020.11.003]

Normalized concentrating solutions to nonlinear elliptic problems

Pistoia A.
;
2021

Abstract

We prove the existence of solutions (λ,v)∈R×H1(Ω) of the elliptic problem {−Δv+(V(x)+λ)v=vp in Ω,v>0,∫Ωv2dx=ρ. Any v solving such problem (for some λ) is called a normalized solution, where the normalization is settled in L2(Ω). Here Ω is either the whole space RN or a bounded smooth domain of RN, in which case we assume V≡0 and homogeneous Dirichlet or Neumann boundary conditions. Moreover, [Formula presented] if N≥3 and p>1 if N=1,2. Normalized solutions appear in different contexts, such as the study of the Nonlinear Schrödinger equation, or that of quadratic ergodic Mean Field Games systems. We prove the existence of solutions concentrating at suitable points of Ω as the prescribed mass ρ is either small (when [Formula presented]) or large (when [Formula presented]) or it approaches some critical threshold (when [Formula presented]).
2021
Lyapunov-Schmidt reduction; Mean Field Games; Nonlinear Schrödinger equation; Singularly perturbed problems
01 Pubblicazione su rivista::01a Articolo in rivista
Normalized concentrating solutions to nonlinear elliptic problems / Pellacci, B.; Pistoia, A.; Vaira, G.; Verzini, G.. - In: JOURNAL OF DIFFERENTIAL EQUATIONS. - ISSN 0022-0396. - 275:(2021), pp. 882-919. [10.1016/j.jde.2020.11.003]
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11573/1680262
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