Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9....

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

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.


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.



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.



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).



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.



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).



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.


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).



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 .




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, ...).




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









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,...








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








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 .



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


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).



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).



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).


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.



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.




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.




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.




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.



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,∞].


Thank you for your attention !


Solutions to Gelfand problemsicas.unsam.edu.ar/talks/2016.09.27.Alexis.gelfand.pdf · 2016. 9....

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