solutions to gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.alexis.gelfand.pdf · 2016. 9....
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Solutions to Gelfand problems
Alexis MolinoU. Granada (Spain)
International Center for Advanced Studies
Buenos Aires27 September 2016
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 2: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/2.jpg)
Contents
1 Classical Gelfand Problem−∆u = λ f (u) , in Ω,u = 0 , on ∂Ω,
2 Elliptic eq. with natural growth in the quadratic gradient term−∆u + g(u)|∇u|2 = λ f (u) , in Ω,u = 0 , on ∂Ω,
M, Gelfand type problem for singular quadratic quasilinear equations, Nonlinear Differential Equations and Applications
NoDEA (2016).
3 1-homogeneous p-laplacian−∆N
p u = λ f (u) , in Ω,u = 0 , on ∂Ω,
Work in preparation with J. Carmona and J.D. Rossi.
Alexis Molino (U. Granada) Solutions to Gelfand problems
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Classical Gelfand Problem
Existence of positive solutions−∆u = λeu , in Ω,u = 0 , on ∂Ω,
(1)
• Model for the thermal reaction process in a combustible.• I.M. Gelfand, Some problems in the theory of quasilinear
equations, Amer. Math. Soc. Transl. (1963).
• No solutions for λ large. Even in the weak sense.• Leray-Schauder continuation methods. Existence of an
unbounded continuum of solutions.
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 4: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/4.jpg)
Classical Gelfand Problem
Existence of positive solutions−∆u = λeu , in Ω,u = 0 , on ∂Ω,
(1)
• Model for the thermal reaction process in a combustible.• I.M. Gelfand, Some problems in the theory of quasilinear
equations, Amer. Math. Soc. Transl. (1963).• No solutions for λ large. Even in the weak sense.• Leray-Schauder continuation methods. Existence of an
unbounded continuum of solutions.
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 5: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/5.jpg)
Classical Gelfand Problem
Ω = B1(0): Gidas-Ni-Niremberg→ solutions are radially, symmetricand satisfy
−u ′′ − N−1r u ′ = λeu , r ∈ [0,1),
u ′(0) = u(1) = 0 ,(2)
where u(r) := u(|x |).• 1 ≤ N ≤ 2: There exists λ∗ > 0 such that (2) has exactly one
solution for λ = λ∗ and exactly two solutions for each λ ∈ (0, λ∗).• 2 < N < 10: Eq. (2) has a continuum of solutions which oscillates
around the line λ = 2(N − 2); with the amplitude of oscillationstending to zero, as u(0) = ‖u‖∞ →∞.
• N ≥ 10: Eq. (2) has a unique solution for each λ ∈ (0,2(N − 2))and no solutions for λ ≥ 2(N − 2): Moreover, ‖u‖∞ →∞ asλ→ 2(N − 2).
Alexis Molino (U. Granada) Solutions to Gelfand problems
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Classical Gelfand Problem
Classical Gelfand type problems−∆u = λ f (u) , in Ω,u > 0 , in Ω,u = 0 , on ∂Ω,
(Gλ)
Ω ⊂ RN bounded, smooth, λ > 0, f (0) > 0, derivable, increasing,
convex and superlinear(
lims→∞f (s)
s=∞
).
(f (u) = eu, (1 + u)p, 1(1−u)k . . . with p > 1, k > 0).
Crandall-Rabinowitz (1973)
There exists a positive number λ∗ called the extremal parameter suchthat• If λ < λ∗ the problem (Gλ) admits a minimal bounded solution wλ.• If λ > λ∗ the problem (Gλ) admits no solution.
Even more, the sequence wλ is increasing.
Alexis Molino (U. Granada) Solutions to Gelfand problems
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Classical Gelfand Problem
Classical Gelfand type problems−∆u = λ f (u) , in Ω,u > 0 , in Ω,u = 0 , on ∂Ω,
(Gλ)
Ω ⊂ RN bounded, smooth, λ > 0, f (0) > 0, derivable, increasing,
convex and superlinear(
lims→∞f (s)
s=∞
).
(f (u) = eu, (1 + u)p, 1(1−u)k . . . with p > 1, k > 0).
Crandall-Rabinowitz (1973)
There exists a positive number λ∗ called the extremal parameter suchthat• If λ < λ∗ the problem (Gλ) admits a minimal bounded solution wλ.• If λ > λ∗ the problem (Gλ) admits no solution.
Even more, the sequence wλ is increasing.
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 8: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/8.jpg)
Classical Gelfand Problem
What happens when λ = λ∗ ?
Let u∗(x) := limλ→λ∗wλ(x)
Crandall-Rabinowitz (1975) f (u) = (1 + u)p, p > 1
The extremal solution u∗ is bounded in dimensions
N < 4 + 2p
p − 1+ 4√
pp − 1
for any domain Ω.
Mignot-Puel (1980) f (u) = eu
• The extremal solution u∗ is bounded in dimensions N ≤ 9 for anydomain Ω.
• When N ≥ 10, u∗(x) = log 1|x |2 is the singular extremal solution for
Ω = B1(0).
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 9: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/9.jpg)
Classical Gelfand Problem
What happens when λ = λ∗ ?
Let u∗(x) := limλ→λ∗wλ(x)
Crandall-Rabinowitz (1975) f (u) = (1 + u)p, p > 1
The extremal solution u∗ is bounded in dimensions
N < 4 + 2p
p − 1+ 4√
pp − 1
for any domain Ω.
Mignot-Puel (1980) f (u) = eu
• The extremal solution u∗ is bounded in dimensions N ≤ 9 for anydomain Ω.
• When N ≥ 10, u∗(x) = log 1|x |2 is the singular extremal solution for
Ω = B1(0).
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 10: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/10.jpg)
Classical Gelfand Problem
What happens when λ = λ∗ ?
Let u∗(x) := limλ→λ∗wλ(x)
Crandall-Rabinowitz (1975) f (u) = (1 + u)p, p > 1
The extremal solution u∗ is bounded in dimensions
N < 4 + 2p
p − 1+ 4√
pp − 1
for any domain Ω.
Mignot-Puel (1980) f (u) = eu
• The extremal solution u∗ is bounded in dimensions N ≤ 9 for anydomain Ω.
• When N ≥ 10, u∗(x) = log 1|x |2 is the singular extremal solution for
Ω = B1(0).
Alexis Molino (U. Granada) Solutions to Gelfand problems
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Classical Gelfand Problem
Brézis-Vázquez (1997)What is the regularity of u∗ (depending on dimension N) for a generalnonlinearities f ?
• u∗ is bounded for N ≤ 3, G. Nedev (2000).• u∗ is bounded for N = 4, S. Villegas (2013).• Still unknown for 5 ≤ N ≤ 9 ! There are some results....
The stability condition, i.e., uλ satisfies∫Ω|∇φ|2 ≥ λ
∫Ω
f ′(uλ)φ2, ∀φ ∈ C∞c (Ω),
plays an important role in order to prove the existence a regularity ofu∗.
uλ stable solution→ ‖uλ‖∞ ≤ C.
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 12: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/12.jpg)
Classical Gelfand Problem
Brézis-Vázquez (1997)What is the regularity of u∗ (depending on dimension N) for a generalnonlinearities f ?
• u∗ is bounded for N ≤ 3, G. Nedev (2000).• u∗ is bounded for N = 4, S. Villegas (2013).• Still unknown for 5 ≤ N ≤ 9 ! There are some results....
The stability condition, i.e., uλ satisfies∫Ω|∇φ|2 ≥ λ
∫Ω
f ′(uλ)φ2, ∀φ ∈ C∞c (Ω),
plays an important role in order to prove the existence a regularity ofu∗.
uλ stable solution→ ‖uλ‖∞ ≤ C.
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 13: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/13.jpg)
Classical Gelfand Problem
Brézis-Vázquez (1997)What is the regularity of u∗ (depending on dimension N) for a generalnonlinearities f ?
• u∗ is bounded for N ≤ 3, G. Nedev (2000).• u∗ is bounded for N = 4, S. Villegas (2013).• Still unknown for 5 ≤ N ≤ 9 ! There are some results....
The stability condition, i.e., uλ satisfies∫Ω|∇φ|2 ≥ λ
∫Ω
f ′(uλ)φ2, ∀φ ∈ C∞c (Ω),
plays an important role in order to prove the existence a regularity ofu∗.
uλ stable solution→ ‖uλ‖∞ ≤ C.
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 14: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/14.jpg)
Elliptic eq. with natural growth in the quadratic gradient term
The Problem:Existence positive solutions of
−∆u + g(u)|∇u|2 = λ f (u) , in Ω,u = 0 , on ∂Ω,
where:• λ > 0 , f is "strictly increasing” and derivable in [0,∞) with
f (0) > 0.• g is a positive function such that lim sups→∞ g(s) <∞ and
e−G(s) ∈ L1(1,∞), being G(s) =∫ s
0 g(t)dt .We are thinking in:
f (s) = es, (1 + s)p , ... g(s) = c,1sγ, ... c > 0, γ ∈ [0,1).
Alexis Molino (U. Granada) Solutions to Gelfand problems
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Elliptic eq. with natural growth in the quadratic gradient term
why g(u)|∇u|2 ?Consider the functional J : W 1,2
0 (Ω)→ R
J(v) =12
∫Ω
a(x)|∇v |2 −∫
Ωf v ,
0 < α ≤ a(x) ≤ β, f ∈ L2N
N+2 (Ω). ∃ u ∈W 1,20 (Ω) minimum. Moreover, u
is a solution of the E-L equation
v ∈W 1,20 (Ω) : −div(a(x)∇v) = f (x).
Consider the functional
J(v) =12
∫Ω
a(v)|∇v |2 −∫
Ωf v .
The E-L equation associated is
−div(a(v)∇v) +12
a′(v)|∇v |2 = f .
Alexis Molino (U. Granada) Solutions to Gelfand problems
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Elliptic eq. with natural growth in the quadratic gradient term
why g(u)|∇u|2 ?Consider the functional J : W 1,2
0 (Ω)→ R
J(v) =12
∫Ω
a(x)|∇v |2 −∫
Ωf v ,
0 < α ≤ a(x) ≤ β, f ∈ L2N
N+2 (Ω). ∃ u ∈W 1,20 (Ω) minimum. Moreover, u
is a solution of the E-L equation
v ∈W 1,20 (Ω) : −div(a(x)∇v) = f (x).
Consider the functional
J(v) =12
∫Ω
a(v)|∇v |2 −∫
Ωf v .
The E-L equation associated is
−div(a(v)∇v) +12
a′(v)|∇v |2 = f .
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 17: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/17.jpg)
Elliptic eq. with natural growth in the quadratic gradient term
why g(u)|∇u|2 ?Consider the functional J : W 1,2
0 (Ω)→ R
J(v) =12
∫Ω
a(x)|∇v |2 −∫
Ωf v ,
0 < α ≤ a(x) ≤ β, f ∈ L2N
N+2 (Ω). ∃ u ∈W 1,20 (Ω) minimum. Moreover, u
is a solution of the E-L equation
v ∈W 1,20 (Ω) : −div(a(x)∇v) = f (x).
Consider the functional
J(v) =12
∫Ω
a(v)|∇v |2 −∫
Ωf v .
The E-L equation associated is
−div(a(v)∇v) +12
a′(v)|∇v |2 = f .
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 18: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/18.jpg)
Elliptic eq. with natural growth in the quadratic gradient term
Elliptic eq. with natural growth in the quadratic gradient term
u ∈W 1,20 (Ω) : −∆u + g(u)|∇u|2 = h(x ,u)
• Invariant to non-linear changes of variable v = F (u) (F ∈ C1).• Non existence result when the growth in ∇u is faster than quadratic at infinity (Serrin’69).• Existence for g continuous by Boccardo-Murat-Puel ’82 and an extensive literature since
them: Abdellaoui, Arcoya, Bensoussan, Dall’Aglio, Gallouët, Peral, Giachetti, Segura deLeón,...
• Extensions to systems of elliptic partial differential equations, unbounded domains,parabolic case.
• In recent years the case g singular appears, whose simple model is g(s) = 1sγ . (Arcoya,
Boccardo, Carmona, Leonori, Martínez-Aparicio, Rossi,...).• Several applications: growth patterns in clusters and fronts of solidification, growth of
tumors, flame propagation,...(Kardar, Parisi, Zhang, Berestychi, Kamin, ...).
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 19: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/19.jpg)
Elliptic eq. with natural growth in the quadratic gradient term
Elliptic eq. with natural growth in the quadratic gradient term
u ∈W 1,20 (Ω) : −∆u + g(u)|∇u|2 = h(x ,u)
• Invariant to non-linear changes of variable v = F (u) (F ∈ C1).
• Non existence result when the growth in ∇u is faster than quadratic at infinity (Serrin’69).• Existence for g continuous by Boccardo-Murat-Puel ’82 and an extensive literature since
them: Abdellaoui, Arcoya, Bensoussan, Dall’Aglio, Gallouët, Peral, Giachetti, Segura deLeón,...
• Extensions to systems of elliptic partial differential equations, unbounded domains,parabolic case.
• In recent years the case g singular appears, whose simple model is g(s) = 1sγ . (Arcoya,
Boccardo, Carmona, Leonori, Martínez-Aparicio, Rossi,...).• Several applications: growth patterns in clusters and fronts of solidification, growth of
tumors, flame propagation,...(Kardar, Parisi, Zhang, Berestychi, Kamin, ...).
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 20: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/20.jpg)
Elliptic eq. with natural growth in the quadratic gradient term
Elliptic eq. with natural growth in the quadratic gradient term
u ∈W 1,20 (Ω) : −∆u + g(u)|∇u|2 = h(x ,u)
• Invariant to non-linear changes of variable v = F (u) (F ∈ C1).• Non existence result when the growth in ∇u is faster than quadratic at infinity (Serrin’69).
• Existence for g continuous by Boccardo-Murat-Puel ’82 and an extensive literature sincethem: Abdellaoui, Arcoya, Bensoussan, Dall’Aglio, Gallouët, Peral, Giachetti, Segura deLeón,...
• Extensions to systems of elliptic partial differential equations, unbounded domains,parabolic case.
• In recent years the case g singular appears, whose simple model is g(s) = 1sγ . (Arcoya,
Boccardo, Carmona, Leonori, Martínez-Aparicio, Rossi,...).• Several applications: growth patterns in clusters and fronts of solidification, growth of
tumors, flame propagation,...(Kardar, Parisi, Zhang, Berestychi, Kamin, ...).
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 21: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/21.jpg)
Elliptic eq. with natural growth in the quadratic gradient term
Elliptic eq. with natural growth in the quadratic gradient term
u ∈W 1,20 (Ω) : −∆u + g(u)|∇u|2 = h(x ,u)
• Invariant to non-linear changes of variable v = F (u) (F ∈ C1).• Non existence result when the growth in ∇u is faster than quadratic at infinity (Serrin’69).• Existence for g continuous by Boccardo-Murat-Puel ’82 and an extensive literature since
them: Abdellaoui, Arcoya, Bensoussan, Dall’Aglio, Gallouët, Peral, Giachetti, Segura deLeón,...
• Extensions to systems of elliptic partial differential equations, unbounded domains,parabolic case.
• In recent years the case g singular appears, whose simple model is g(s) = 1sγ . (Arcoya,
Boccardo, Carmona, Leonori, Martínez-Aparicio, Rossi,...).• Several applications: growth patterns in clusters and fronts of solidification, growth of
tumors, flame propagation,...(Kardar, Parisi, Zhang, Berestychi, Kamin, ...).
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 22: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/22.jpg)
Elliptic eq. with natural growth in the quadratic gradient term
Elliptic eq. with natural growth in the quadratic gradient term
u ∈W 1,20 (Ω) : −∆u + g(u)|∇u|2 = h(x ,u)
• Invariant to non-linear changes of variable v = F (u) (F ∈ C1).• Non existence result when the growth in ∇u is faster than quadratic at infinity (Serrin’69).• Existence for g continuous by Boccardo-Murat-Puel ’82 and an extensive literature since
them: Abdellaoui, Arcoya, Bensoussan, Dall’Aglio, Gallouët, Peral, Giachetti, Segura deLeón,...
• Extensions to systems of elliptic partial differential equations, unbounded domains,parabolic case.
• In recent years the case g singular appears, whose simple model is g(s) = 1sγ . (Arcoya,
Boccardo, Carmona, Leonori, Martínez-Aparicio, Rossi,...).• Several applications: growth patterns in clusters and fronts of solidification, growth of
tumors, flame propagation,...(Kardar, Parisi, Zhang, Berestychi, Kamin, ...).
Alexis Molino (U. Granada) Solutions to Gelfand problems
![Page 23: Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9. 27. · 2 Elliptic eq. with natural growth in the quadratic gradient term ˆ u + g(u)jruj2](https://reader033.vdocuments.site/reader033/viewer/2022060917/60aa6bd8a552cc78954eea63/html5/thumbnails/23.jpg)
Elliptic eq. with natural growth in the quadratic gradient term
Elliptic eq. with natural growth in the quadratic gradient term
u ∈W 1,20 (Ω) : −∆u + g(u)|∇u|2 = h(x ,u)
• Invariant to non-linear changes of variable v = F (u) (F ∈ C1).• Non existence result when the growth in ∇u is faster than quadratic at infinity (Serrin’69).• Existence for g continuous by Boccardo-Murat-Puel ’82 and an extensive literature since
them: Abdellaoui, Arcoya, Bensoussan, Dall’Aglio, Gallouët, Peral, Giachetti, Segura deLeón,...
• Extensions to systems of elliptic partial differential equations, unbounded domains,parabolic case.
• In recent years the case g singular appears, whose simple model is g(s) = 1sγ . (Arcoya,
Boccardo, Carmona, Leonori, Martínez-Aparicio, Rossi,...).• Several applications: growth patterns in clusters and fronts of solidification, growth of
tumors, flame propagation,...(Kardar, Parisi, Zhang, Berestychi, Kamin, ...).
Alexis Molino (U. Granada) Solutions to Gelfand problems
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Elliptic eq. with natural growth in the quadratic gradient term
The Problem:Existence positive solutions of
−∆u + g(u)|∇u|2 = λ f (u) , in Ω,u = 0 , on ∂Ω,
where:• λ > 0, f derivable in [0,∞) with f (0) > 0 and increasing conditon
f ′(s)− g(s)f (s) > 0.• lim sups→∞ g(s) <∞, e−G(s) ∈ L1(1,∞), being G(s) =
∫ s0 g(t)dt .
Alexis Molino (U. Granada) Solutions to Gelfand problems
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Elliptic eq. with natural growth in the quadratic gradient term
Some Definitions
We recall that a function 0 < u ∈W 1,20 (Ω) is a (weak) solution if
g(u)|∇u|2, f (u) ∈ L1(Ω) and it satisfies∫Ω∇u∇φ+
∫Ω
g(u) |∇u|2 φ =
∫Ωλ f (u)φ, (3)
for all test function φ ∈W 1,20 (Ω) ∩ L∞(Ω).
• If u ≤ v for every another solution v , we say that u is minimal.• If u belongs to L∞(Ω) we say that u is bounded (or regular).• Let u be a solution, we say that u is stable if
f ′(u)− g(u)f (u) ∈ L1loc(Ω) and∫
Ω|∇φ|2 ≥ λ
∫Ω
(f ′(u)− g(u)f (u))φ2
holds for every φ ∈ C∞c (Ω).Alexis Molino (U. Granada) Solutions to Gelfand problems
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Elliptic eq. with natural growth in the quadratic gradient term
Theorem (Crandall-Rabinowitz version)There exists 0 < λ∗ <∞ such that there is a bounded minimal solutionwλ for every λ < λ∗ and no solution for λ > λ∗. Moreover, thesequence wλ is increasing respect to λ.
Theorem (Existence and regularity extremal solution)
Set h(s) := e−G(s)f (s) and assume that lims→∞sh ′(s)h(s) ∈ (1,∞]. Then
• u∗(x) := limλ→λ∗wλ(x) is solution.• u∗ is bounded whenever
N <4 + 2(µ+ α) + 4
√µ+ α
1 + α,
being α = lims→∞g(s)h(s)
h ′(s), µ = lims→∞
(h ′(s))2
h ′′(s)h(s).
Alexis Molino (U. Granada) Solutions to Gelfand problems
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Elliptic eq. with natural growth in the quadratic gradient term
Theorem (Crandall-Rabinowitz version)There exists 0 < λ∗ <∞ such that there is a bounded minimal solutionwλ for every λ < λ∗ and no solution for λ > λ∗. Moreover, thesequence wλ is increasing respect to λ.
Theorem (Existence and regularity extremal solution)
Set h(s) := e−G(s)f (s) and assume that lims→∞sh ′(s)h(s) ∈ (1,∞]. Then
• u∗(x) := limλ→λ∗wλ(x) is solution.• u∗ is bounded whenever
N <4 + 2(µ+ α) + 4
√µ+ α
1 + α,
being α = lims→∞g(s)h(s)
h ′(s), µ = lims→∞
(h ′(s))2
h ′′(s)h(s).
Alexis Molino (U. Granada) Solutions to Gelfand problems
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Elliptic eq. with natural growth in the quadratic gradient term
Corollary (Exponential version)
Let g(s) = c > 0 and f (s) = e(c+1)s. Then,• There exists 0 < λ∗ <∞ such that there is a bounded minimal
solution wλ of−∆u + c|∇u|2 = λe(c+1)u , in Ω,u = 0 , on ∂Ω,
for every λ < λ∗ and no solution for λ > λ∗.• u∗ is solution.• u∗ is bounded whenever
N <4 + 2(1 + c) + 4
√1 + c
1 + c,
(α = c, µ = 1).
Alexis Molino (U. Granada) Solutions to Gelfand problems
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Elliptic eq. with natural growth in the quadratic gradient term
Corollary (Mignot-Puel version)
Let g(s) = c > 0 and f (s) = ecs(1 + s)p, p > 1. Then,• There exists 0 < λ∗ <∞ such that there is a bounded minimal
solution wλ of−∆u + c|∇u|2 = λecu(1 + u)p , in Ω,u = 0 , on ∂Ω,
for every λ < λ∗ and no solution for λ > λ∗.• u∗ is solution.• u∗ is bounded whenever
N < 4 + 2p
p − 1+ 4√
pp − 1
,
(α = 0, µ = pp−1).
Alexis Molino (U. Granada) Solutions to Gelfand problems
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1-homogeneous p-laplacian
Gelfand problem for 1-homogeneous p-laplacian−∆N
p u = λ f (u) , in Ω,u = 0 , on ∂Ω,
f is a general continuous nonlinearity that verifies: f (0) > 0, increasingand f (s)
s ≥ k > 0.
where Ω ⊂ RN is a regular bounded domain, p ∈ [2,∞] and theoperator ∆N
p is the called 1-homogeneous p-laplacian defined by
∆Np u :=
1p − 1
|∇u|2−p div(|∇u|p−2∇u
)= α∆u + β∆∞u, (4)
being
α =1
p − 1, β =
p − 2p − 1
and ∆∞u =∇u|∇u|
·(
D2u∇u|∇u|
)is the 1-homogeneous infinity laplacian.
Alexis Molino (U. Granada) Solutions to Gelfand problems
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1-homogeneous p-laplacian
Main difficulties• Operator in no-divergence form→ framework of viscosity
solutions.
• Singular at |∇u| = 0→ semicontinuous envelopes.• Lack of uniform ellipticity→ extensive literature can not be applied.• Lack of variational structure→ stable solutions make no sense.
Alexis Molino (U. Granada) Solutions to Gelfand problems
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1-homogeneous p-laplacian
Main difficulties• Operator in no-divergence form→ framework of viscosity
solutions.• Singular at |∇u| = 0→ semicontinuous envelopes.
• Lack of uniform ellipticity→ extensive literature can not be applied.• Lack of variational structure→ stable solutions make no sense.
Alexis Molino (U. Granada) Solutions to Gelfand problems
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1-homogeneous p-laplacian
Main difficulties• Operator in no-divergence form→ framework of viscosity
solutions.• Singular at |∇u| = 0→ semicontinuous envelopes.• Lack of uniform ellipticity→ extensive literature can not be applied.
• Lack of variational structure→ stable solutions make no sense.
Alexis Molino (U. Granada) Solutions to Gelfand problems
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1-homogeneous p-laplacian
Main difficulties• Operator in no-divergence form→ framework of viscosity
solutions.• Singular at |∇u| = 0→ semicontinuous envelopes.• Lack of uniform ellipticity→ extensive literature can not be applied.• Lack of variational structure→ stable solutions make no sense.
Alexis Molino (U. Granada) Solutions to Gelfand problems
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1-homogeneous p-laplacian
Results so far...
Theorem 1There exists a positive extremal parameter λ∗ = λ∗ (Ω,N,p) such that:• If λ < λ∗, problem admits a minimal positive solution wλ.• If λ > λ∗, problem has no positive solution.
Moreover, the branch of minimal solutions wλ is increasing with λ.In the case of a ball, Ω = Br , the minimal solution is radial.
Theorem 2For every fixed p ∈ [2,∞], there exists an unbounded continua ofsolutions C that emanates from λ = 0,u = 0, with C ⊂ [0, λ∗]× C(Ω),being λ∗ the extremal parameter. Moreover, for every fixed small λthere exists a continua of solutions D ⊂ [2,∞]× C(Ω), with
‖u‖∞ ≤ C, ∀p ∈ [2,∞].
Alexis Molino (U. Granada) Solutions to Gelfand problems
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1-homogeneous p-laplacian
Results so far...
Theorem 1There exists a positive extremal parameter λ∗ = λ∗ (Ω,N,p) such that:• If λ < λ∗, problem admits a minimal positive solution wλ.• If λ > λ∗, problem has no positive solution.
Moreover, the branch of minimal solutions wλ is increasing with λ.In the case of a ball, Ω = Br , the minimal solution is radial.
Theorem 2For every fixed p ∈ [2,∞], there exists an unbounded continua ofsolutions C that emanates from λ = 0,u = 0, with C ⊂ [0, λ∗]× C(Ω),being λ∗ the extremal parameter. Moreover, for every fixed small λthere exists a continua of solutions D ⊂ [2,∞]× C(Ω), with
‖u‖∞ ≤ C, ∀p ∈ [2,∞].
Alexis Molino (U. Granada) Solutions to Gelfand problems
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1-homogeneous p-laplacian
Results so far...
Theorem 1There exists a positive extremal parameter λ∗ = λ∗ (Ω,N,p) such that:• If λ < λ∗, problem admits a minimal positive solution wλ.• If λ > λ∗, problem has no positive solution.
Moreover, the branch of minimal solutions wλ is increasing with λ.In the case of a ball, Ω = Br , the minimal solution is radial.
Theorem 2For every fixed p ∈ [2,∞], there exists an unbounded continua ofsolutions C that emanates from λ = 0,u = 0, with C ⊂ [0, λ∗]× C(Ω),being λ∗ the extremal parameter. Moreover, for every fixed small λthere exists a continua of solutions D ⊂ [2,∞]× C(Ω), with
‖u‖∞ ≤ C, ∀p ∈ [2,∞].
Alexis Molino (U. Granada) Solutions to Gelfand problems
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Thank you for your attention !
Alexis Molino (U. Granada) Solutions to Gelfand problems