KIT, Institut für Analysismandel/media/talkhalle.pdf · Karlsruhe Institut für Technologie KIT – Universität des Landes Baden-Württemberg und nationales Forschungszentrum in

Karlsruhe Institut für Technologie

KIT – Universität des Landes Baden-Württemberg undnationales Forschungszentrum in der Helmholtz-Gemeinschaft www.kit.edu

Vector solutions of a cubic nonlinear Helmholtz systemjoint work with D. Scheider (KIT)

Rainer MandelHalle, May 15th 2019

KIT, Institut für Analysis

Karlsruhe Institut für TechnologieOutline

1 Nonlinear Helmholtz equations

2 Nonlinear Helmholtz systems: Variational methods

3 Nonlinear Helmholtz systems: Bifurcations

2/20 15.05.2019 Rainer Mandel - Helmholtz systems KIT, Institut für Analysis


NLH equations NLH systems: Variational methods NLH systems: Bifurcations

Outline







Helmholtz vs. Schrödinger

−∆u + u = ±|u|p−2u in Rn

”Nonlinear Schrödinger equation”

−∆u − u = ±|u|p−2u in Rn

”Nonlinear Helmholtz equation”

Schrödinger HelmholtzSpectrum 0 < σ(−∆ + 1) 0 ∈ σ(−∆ − 1)

# Nodal domains < ∞ ∞

Decay exponential ∼ |x |1−n

2

Spaces L2(Rn) L2n

n−1 +ε(Rn)Radial sol. distinct continuum




Helmholtz vs. Schrödinger

−∆u + u = ±|u|p−2u in Rn

”Nonlinear Schrödinger equation”

−∆u − u = ±|u|p−2u in Rn

”Nonlinear Helmholtz equation”

Schrödinger HelmholtzSpectrum 0 < σ(−∆ + 1) 0 ∈ σ(−∆ − 1)

# Nodal domains < ∞ ∞

Decay exponential ∼ |x |1−n

2

Spaces L2(Rn) L2n

n−1 +ε(Rn)Radial sol. distinct continuum




NLH equation: Dual variational methods

−∆u − u = ±|u|p−2u in Rn

1 Dual formulationu = ±R(|u|p−2u)

2 Substitution v := |u|p−2u

|v |p′−2v = ±Rv

3 Variational structure

J(v) :=1p′

∫Rn|v |p

′

∓12

∫Rn

vRv

4 Critical Point Theorem ; Nontrivial solutions





−∆u − u = ±|u|p−2u in Rn



|v |p′−2v = ±Rv


J(v) :=1p′

∫Rn|v |p

′

∓12

∫Rn

vRv






−∆u − u = ±|u|p−2u in Rn



|v |p′−2v = ±Rv


J(v) :=1p′

∫Rn|v |p

′

∓12

∫Rn

vRv






−∆u − u = ±|u|p−2u in Rn



|v |p′−2v = ±Rv


J(v) :=1p′

∫Rn|v |p

′

∓12

∫Rn

vRv






−∆u − u = ±|u|p−2u in Rn



|v |p′−2v = ±Rv


J(v) :=1p′

∫Rn|v |p

′

∓12

∫Rn

vRv


G. Evequoz, T. Weth: Dual variational methods and nonvanishing for the nonlinearHelmholtz equation, Adv. Math. 280 (2015), 690–728.





−∆u − u = ±|u|p−2u in Rn



|v |p′−2v = ±Rv


J(v) :=1p′

∫Rn|v |p

′

∓12

∫Rn

vRv


Question: How to define R? Need: (−∆ − 1)Rf = f .





Computation:

−∆uε − (1 + iε)uε = f ⇔ (|ξ|2 − 1 − iε)uε(ξ) = f (ξ)

⇔ uε = F −1

f| · |2 − 1 − iε

Theorem (Gutiérrez, 2004)

Let n ≥ 3 and 2(n+1)n−1 ≤ p ≤ 2n

n−2 . Then:

f ∈ Lp′(Rn) ⇒ Rf := limε→0

Re(uε) ∈ Lp(Rn)





Tomas-Stein-exponent 2(n+1)n−1

‖g‖L2(Sn−1) ≤ Cp‖g‖p′ for p ≥2(n + 1)

n − 1

‖g‖L2(Sn−1) ≤ Cp‖g‖p′ for p >2n

n − 1if g is radial

Theorem

Let n ≥ 3 and 4n2

(n−1)(2n−1)≤ p ≤ 2n

n−2 . Then:

f ∈ Lp′

rad (Rn) ⇒ Rf = limε→0

Re(uε) ∈ Lprad (Rn)





Theorem (Evequoz-Weth (2015), MMP (2016))

Assume 2(n+1)n−1 < p < 2n

n−2 . Then ∃u ∈ Lp(Rn),u , 0 s.t.

−∆u − u = ±|u|p−2u in Rn.

Moreover, u attains the dual mountain pass level c > 0.

Regularity: u ∈ C2,α(Rn)

Decay: |u(x)| ≤ (1 + |x |)(1−n)/2 if p > 3n−1n−1

Small (non-)radial solutions via fixed point techniquesResults for periodic coefficient functions. . .




Outline







NLH systems: Variational methods

−∆u − µu = (u2 + bv2)u in Rn,

−∆v − νv = (v2 + bu2)v in Rn

µ, ν > 0

Question:Dual formulation?Fully nontrivial solutions: u , 0 and v , 0?





−∆u − µu = (|u|p/2 + b|v |p/2)|u|p/2−2u in Rn,

−∆v − νv = (|v |p/2 + b|u|p/2)|v |p/2−2v in Rn,

µ, ν > 0,2(n + 1)

n − 1< p <

2nn − 2

, hence p , 4

Question:Dual formulation?Fully nontrivial solutions: u , 0 and v , 0?






−∆v − νv = (|v |p/2 + b|u|p/2)|v |p/2−2v in Rn

Propositionf is differentiable, strictly convex and co-finite iff 0 ≤ b ≤ p − 1.







u = Rµ(|u|p/2 + b|v |p/2)|u|p/2−2u

)v = Rν

(|v |p/2 + b|u|p/2)|v |p/2−2v

)








u = Rµ(|u|p/2 + b|v |p/2)|u|p/2−2u

)= Rµ

(fu(u, v)

),

v = Rν(|v |p/2 + b|u|p/2)|v |p/2−2v

)= Rν

(fv (u, v)

),

where f (u, v) =1p|u|p +

1p|v |p +

2bp|u|p/2|v |p/2








u = Rµ(|u|p/2 + b|v |p/2)|u|p/2−2u

)= Rµ

(fu(u, v)

),

v = Rν(|v |p/2 + b|u|p/2)|v |p/2−2v

)= Rν

(fv (u, v)

),

where f (u, v) =1p|u|p +

1p|v |p +

2bp|u|p/2|v |p/2






Proposition (Legendre-transform)If f is differentiable, strictly convex and co-finite then there ish ∈ C1(R2,R2) such that ∇h = (∇f )−1 given by

h : R2 → R2, (s, t) 7→ sup(s,t)∈R2

(ss + t t − f (s, t)

).

u = Rµ(fu(u, v)

)v = Rν

(fv (u, v)

) 0≤b≤p−1⇔

hu(u, v) = Rµ(u)

hv (u, v) = Rν(v)

Dual functional: With h(u, v) ≈ |u|p′

+ |v |p′

J(u, v) :=

∫Rn

h(u, v) −12

∫Rn

uRµu + vRνv .





Theorem (Mandel, Scheider (2017))

Assume 2(N+1)N−1 < p < 2N

N−2 , 0 ≤ b ≤ p − 1 and µ, ν > 0.Then ∃u, v ∈ Lp(Rn), (u, v) , (0,0) s.t.


−∆v − νv = (|v |p/2 + b|u|p/2)|v |p/2−2v in Rn.

Moreover, (u, v) attains the dual mountain pass level cµν > 0.

Question: Are these solutions (dual ground states) fully nontrivial?





Theorem (Mandel, Scheider (2017))Under the above assumptions:

2 < p < 4 ⇒ u , 0 and v , 0 provided b > 0

p ≥ 4 ⇒ u , 0 and v , 0 provided b > 2p−2

2 − 1, µ ≈ ν

Open: Find (many) nontrivial solutions for. . . the cubic system. . . b < [0,p − 1]

. . . µ ≈ ν





Theorem (Mandel, Scheider (2017))Under the above assumptions:

2 < p < 4 ⇒ u , 0 and v , 0 provided b > 0

p ≥ 4 ⇒ u , 0 and v , 0 provided b > 2p−2

2 − 1, µ ≈ ν

Open: Find (many) nontrivial solutions for. . . the cubic system. . . b < [0,p − 1]

. . . µ ≈ ν




Outline







NLH systems: Bifurcations

−∆u − µu = (u2 + bv2)u in R3,

−∆v − νv = (v2 + bu2)v in R3

Question: Global bifurcation from a semitrivial solution family

T := {(u0,0,b) : b ∈ R} ?




NLH systems: BifurcationsClassical formulation:

−∆u − µu = (u2 + bv2)u in R3,

−∆v − νv = (v2 + bu2)v in R3





(−∆ − µ)(u − u0) = (u2 + bv2)u − u30 in R3,

−∆v − νv = (v2 + bu2)v in R3

Dual formulation:

u − u0 = Rµ

((u2 + bv2)u − u3

0

),

v = Rν

((v2 + bu2)v

)





(−∆ − µ)(u − u0) = (u2 + bv2)u − u30 in R3,

−∆v − νv = (v2 + bu2)v in R3

Dual formulation:

u − u0 = Rµ

((u2 + bv2)u − u3

0

),

v = Rν

((v2 + bu2)v

)





(−∆ − µ)(u − u0) = (u2 + bv2)u − u30 in R3,

−∆v − νv = (v2 + bu2)v in R3

Dual formulation:

u − u0 = Rµ

((u2 + bv2)u − u3

0

),

v = Rν

((v2 + bu2)v

)Dual formulation:

u − u0 =cos(

√µ| · |)

4π| · |∗((u2 + bv2)u − u3

0

),

v =cos(

√ν| · |)

4π| · |∗((v2 + bu2)v

)17/20 15.05.2019 Rainer Mandel - Helmholtz systems KIT, Institut für Analysis




(−∆ − µ)(u − u0) = (u2 + bv2)u − u30 in R3,

−∆v − νv = (v2 + bu2)v in R3

Dual formulation:

u − u0 = Rµ

((u2 + bv2)u − u3

0

),

v = Rν

((v2 + bu2)v

)Extended dual formulation: τ, ω = asymptotic phase parameters

u − u0 =

(cos(

√µ| · |)

4π| · |+ cot(τ)

sin(√µ| · |)

4π| · |

)∗((u2 + bv2)u − u3

0

),

v =

(cos(

√ν| · |)

4π| · |+ cot(ω)

sin(√µ| · |)

4π| · |

)∗((v2 + bu2)v





(−∆ − µ)(u − u0) = (u2 + bv2)u − u30 in R3,

−∆v − νv = (v2 + bu2)v in R3

Dual formulation:

u − u0 = Rµ

((u2 + bv2)u − u3

0

),

v = Rν

((v2 + bu2)v

)Extended dual formulation: τ, ω = asymptotic phase parameters

u − u0 = Rτµ

((u2 + bv2)u − u3

0

),

v = Rων

((v2 + bu2)v




NLH systems: BifurcationsFunctional analytical setting

Xq :=

{w ∈ Crad(R3,R) | sup

x∈R3(1 + |x |2)

q2 |w(x)| < ∞

}.

Proposition(i) Rτµ,Rων : X3 → X1 are compact(ii) Linearized operator has simple kernels at (u0,0,bk (ω)) (k ∈ Z)

(iii) Solutions (u, v ,b) ∈ X1 × X1 × R satisfy as |x | → ∞

u(x) − u0(x) = c1sin(|x |

√µ + τ)

|x |+ O

(1|x |2

)v(x) = c2

sin(|x |√ν + ω)

|x |+ O

(1|x |2




TheoremAssumptions: µ, ν > 0, u0 ∈ X1 \ {0} a scalar solution.Then for all b ∈ R an unbounded continuum in X1 ×X1 ×R consistingof solutions bifurcates from the semitrivial solution (u0,0,b) ∈ T .

Proof.For τ, ω ∈ [0, π) solve

w = Rτµ

(((w + u0)2 + bv2)(w + u0) − u3

0

),

v = Rων

((v2 + bu2)v

)Smooth compact perturbation of the identity in X1 × X1 × R

Identification of bifurcation points (u0,0,bk (ω)) (k ∈ Z)

Crandall-Rabinowitz Theorem ; Local bifurcationKrasnoselski-Rabinowitz Theorem ; Global bifurcationb ∈ R iff b = bk (ω) for some k ∈ Z, ω ∈ [0, π) �






w = Rτµ

(((w + u0)2 + bv2)(w + u0) − u3

0

),

v = Rων

((v2 + bu2)v









w = Rτµ

(((w + u0)2 + bv2)(w + u0) − u3

0

),

v = Rων

((v2 + bu2)v









w = Rτµ

(((w + u0)2 + bv2)(w + u0) − u3

0

),

v = Rων

((v2 + bu2)v









w = Rτµ

(((w + u0)2 + bv2)(w + u0) − u3

0

),

v = Rων

((v2 + bu2)v







Global characterization of Ck via “Asymptotic phase”:

ωv :=1√ν

∫ ∞

0(v(r)2 + bu(r)2)v(r) sin2(φ(r)

√ν) dr

where φ′(r) = 1 +1ν

(v(r)2 + bu(r)2)v(r) sin2(φ(r)√ν), φ(0) = 0.

Proposition(i) ωv = ω + kπ for all (u, v ,b) ∈ Ck near (u0,0,bk (ω))

(ii) ωv ∈ ω + Zπ for all (u, v ,b) ∈ Ck with v , 0(iii) v 7→ ωv is continuous on X1

Open question: Does ωv = ω + kπ hold on the whole Ck?

Thank you for listening!




Global characterization of Ck via “Asymptotic phase”:

ωv :=1√ν

∫ ∞

0(v(r)2 + bu(r)2)v(r) sin2(φ(r)

√ν) dr

where φ′(r) = 1 +1ν

(v(r)2 + bu(r)2)v(r) sin2(φ(r)√ν), φ(0) = 0.

Proposition(i) ωv = ω + kπ for all (u, v ,b) ∈ Ck near (u0,0,bk (ω))

(ii) ωv ∈ ω + Zπ for all (u, v ,b) ∈ Ck with v , 0(iii) v 7→ ωv is continuous on X1

Open question: Does ωv = ω + kπ hold on the whole Ck?

Thank you for listening!


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KIT, Institut für Analysismandel/media/talkhalle.pdf · Karlsruhe Institut für Technologie KIT – Universität des Landes Baden-Württemberg und nationales Forschungszentrum in