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.

Alexis Molino (U. Granada) Solutions to Gelfand problems

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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Alexis Molino (U. Granada) Solutions to Gelfand problems