Perturbations of singular solutions to Gelfand s problem p.1

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1 Perturbations of singular solutions to Gelfand s problem Juan Dávila (Universidad de Chile) In celebration of the 60th birthday of Ireneo Peral February 2007 collaboration with Louis Dupaigne (Université de Amiens) Perturbations of singular solutions to Gelfand s problem p.1

2 The problem Let Ω R N be an open bounded set with smooth boundary, λ > 0. Perturbations of singular solutions to Gelfand s problem p.2

3 The problem Let Ω R N be an open bounded set with smooth boundary, λ > 0. Consider the equation u = λe u u = 0 in Ω on Ω Perturbations of singular solutions to Gelfand s problem p.2

4 The problem Let Ω R N be an open bounded set with smooth boundary, λ > 0. Consider the equation u = λe u u = 0 in Ω on Ω Some references: Gelfand (1963) Some problems in the theory of quasilinear equations. Section 15 due to Barenblatt. Liouville (1853) Sur l équation aux différences partielles d2 log λ dudv ± λ 2a = 0. 2 Perturbations of singular solutions to Gelfand s problem p.2

5 Chandrasekhar (1939, 1957) An introduction to the study of stellar structure. (N = 3) Frank-Kamenetskii (1955) Diffusion and heat exchange in chemical kinetics. Bebernes et Eberly (1989) Mathematical problems from combustion theory. (N = 2, 3) Perturbations of singular solutions to Gelfand s problem p.3

6 Basic properties Proposition. There is 0 < λ < + such that: if 0 λ < λ then there exists a classical solution, Perturbations of singular solutions to Gelfand s problem p.4

7 Basic properties Proposition. There is 0 < λ < + such that: if 0 λ < λ then there exists a classical solution, if λ = λ there exists a unique weak solution u Perturbations of singular solutions to Gelfand s problem p.4

8 Basic properties Proposition. There is 0 < λ < + such that: if 0 λ < λ then there exists a classical solution, if λ = λ there exists a unique weak solution u if λ > λ no solution exists (even in the weak sense). Perturbations of singular solutions to Gelfand s problem p.4

9 Basic properties Proposition. There is 0 < λ < + such that: if 0 λ < λ then there exists a classical solution, if λ = λ there exists a unique weak solution u if λ > λ no solution exists (even in the weak sense). Moreover, if 0 λ < λ there is a unique minimal solution u λ, which is smooth and characterized by λ e u λ ϕ 2 ϕ 2 ϕ C0 (Ω). Ω Ω Perturbations of singular solutions to Gelfand s problem p.4

10 Basic properties Definition: u L 1 (Ω) is a weak solution if dist(x, Ω)e u L 1 (Ω) and Ω u( ζ) = λ Ω eu ζ ζ C 2 (Ω), ζ Ω = 0. Perturbations of singular solutions to Gelfand s problem p.5

11 Basic properties Definition: u L 1 (Ω) is a weak solution if dist(x, Ω)e u L 1 (Ω) and Ω u( ζ) = λ Ω eu ζ ζ C 2 (Ω), ζ Ω = 0. Other properties: u = lim λ λ u λ and λ Ω e u ϕ 2 Ω ϕ 2 ϕ C 0 (Ω). Perturbations of singular solutions to Gelfand s problem p.5

12 The question We know that there exist λ such that if 0 λ < λ there is a classical solution, if λ = λ there is a unique weak solution u if λ > λ there is no solution. Perturbations of singular solutions to Gelfand s problem p.6

13 The question We know that there exist λ such that if 0 λ < λ there is a classical solution, if λ = λ there is a unique weak solution u if λ > λ there is no solution. Is u a classical solution? Perturbations of singular solutions to Gelfand s problem p.6

14 Bifurcation diagram for Ω = B 1 Joseph-Lundgren (72): Perturbations of singular solutions to Gelfand s problem p.7

15 Bifurcation diagram for Ω = B 1 Joseph-Lundgren (72): In particular Theorem. If Ω = B 1 and N 9 then u is classical, and if N 10 then u = 2 log x, λ = 2(N 2). Perturbations of singular solutions to Gelfand s problem p.7

16 General domains Theorem. (Crandall-Rabinowitz (75), Mignot-Puel (80)) If Ω is a smooth bounded domain in R N and N 9 then u is classical. Perturbations of singular solutions to Gelfand s problem p.8

17 Stability and Sobolev s inequalities Let u = u λ. Multiplying the equation by e 2ju 1 and integrating λ e u (e 2ju 1) = u e 2ju Ω Ω = 2j e 2ju u 2 = 2 (e ju ) 2. Ω j Ω Perturbations of singular solutions to Gelfand s problem p.9

18 Stability and Sobolev s inequalities Let u = u λ. Multiplying the equation by e 2ju 1 and integrating λ e u (e 2ju 1) = u e 2ju Ω Ω = 2j e 2ju u 2 = 2 (e ju ) 2. Ω j Ω From stability with ϕ = e ju 1 we have λ e u (e ju 1) 2 (e ju 1) 2. Ω Ω Perturbations of singular solutions to Gelfand s problem p.9

19 Hence 2 j Ω e u (e ju 1) 2 Ω e u (e 2ju 1) Perturbations of singular solutions to Gelfand s problem p.10

20 Hence 2 e u (e ju 1) 2 e u (e 2ju 1) j Ω Ω ( ) 2 j 1 e (2j+1)u lower order terms Ω Perturbations of singular solutions to Gelfand s problem p.10

21 Hence 2 e u (e ju 1) 2 e u (e 2ju 1) j Ω Ω ( ) 2 j 1 e (2j+1)u lower order terms Ω Conclusion: if j < 2, q = 2j + 1 then u L q C where C is independent of λ. Perturbations of singular solutions to Gelfand s problem p.10

22 Hence 2 e u (e ju 1) 2 e u (e 2ju 1) j Ω Ω ( ) 2 j 1 e (2j+1)u lower order terms Ω Conclusion: if j < 2, q = 2j + 1 then u L q C where C is independent of λ. By elliptic estimates and Sobolev s inequality u W 2,q L if q > N/2, which works if N 9. Perturbations of singular solutions to Gelfand s problem p.10

23 General nonlinearities Let g : [0, ) [0, ) be a C 2, positive, increasing function satisfying: lim s + g(u)/u = +. Consider u = λg(u) in Ω u = 0 on Ω Perturbations of singular solutions to Gelfand s problem p.11

24 General nonlinearities Let g : [0, ) [0, ) be a C 2, positive, increasing function satisfying: lim s + g(u)/u = +. Consider u = λg(u) in Ω u = 0 on Ω Cabré (06) If Ω R N is strictly convex and N 4 then u is classical. Cabré-Capella (06) If Ω = B 1 and N 9 then u is classical. Nedev (00) If Ω R N is any bounded domain and N 3 then u is classical. Perturbations of singular solutions to Gelfand s problem p.11

25 Other operators { p u = λg(u) in Ω u = 0 on Ω Perturbations of singular solutions to Gelfand s problem p.12

26 Other operators { p u = λg(u) in Ω u = 0 on Ω Clément-de Figueiredo-Mitidieri (96), Jacobsen and Schmitt (02) Considered the radial case with exponential and power nonlinearities. Perturbations of singular solutions to Gelfand s problem p.12

27 Other operators { p u = λg(u) in Ω u = 0 on Ω Clément-de Figueiredo-Mitidieri (96), Jacobsen and Schmitt (02) Considered the radial case with exponential and power nonlinearities. García-Azorero-Peral-Puel (94) In general domains, if g(u) = e u the extremal solution is bounded if N < p + 4p/(p 1) and this condition is optimal. Perturbations of singular solutions to Gelfand s problem p.12

28 Other operators { p u = λg(u) in Ω u = 0 on Ω Clément-de Figueiredo-Mitidieri (96), Jacobsen and Schmitt (02) Considered the radial case with exponential and power nonlinearities. García-Azorero-Peral-Puel (94) In general domains, if g(u) = e u the extremal solution is bounded if N < p + 4p/(p 1) and this condition is optimal. Cabré-Sanchón (06) Consider general domains and power type nonlinearities. Perturbations of singular solutions to Gelfand s problem p.12

29 The case Ω = B 1, N 10 Lemma. (Brezis-Vázquez (97)) If u H 1 0(Ω) is a solution u L (Ω) for some λ which is stable, then λ = λ and u = u. Perturbations of singular solutions to Gelfand s problem p.13

30 The case Ω = B 1, N 10 Lemma. (Brezis-Vázquez (97)) If u H 1 0(Ω) is a solution u L (Ω) for some λ which is stable, then λ = λ and u = u. Application to the case Ω = B 1. Note that u = 2 log x, λ = 2(N 2) is a singular solution. Perturbations of singular solutions to Gelfand s problem p.13

31 The case Ω = B 1, N 10 Lemma. (Brezis-Vázquez (97)) If u H 1 0(Ω) is a solution u L (Ω) for some λ which is stable, then λ = λ and u = u. Application to the case Ω = B 1. Note that u = 2 log x, λ = 2(N 2) is a singular solution. Let ϕ C 0 (B 1 ). Then ϕ 2 λ e u ϕ 2 = B 1 B 1 B 1 ϕ 2 λ B1 ϕ 2 x 2. Perturbations of singular solutions to Gelfand s problem p.13

32 The case Ω = B 1, N 10 Lemma. (Brezis-Vázquez (97)) If u H 1 0(Ω) is a solution u L (Ω) for some λ which is stable, then λ = λ and u = u. Application to the case Ω = B 1. Note that u = 2 log x, λ = 2(N 2) is a singular solution. Let ϕ C 0 (B 1 ). Then ϕ 2 λ e u ϕ 2 = B 1 B 1 B 1 ϕ 2 λ B1 ϕ 2 x 2. Hardy s inequality: if N 3 (N 2) 2 RN ϕ 2 4 x ϕ 2 ϕ C 2 0 (R N ). R N Perturbations of singular solutions to Gelfand s problem p.13

33 The case Ω = B 1, N 10 By Hardy s inequality u = 2 log x is stable if 2(N 2) (N 2)2 4 N 10. Perturbations of singular solutions to Gelfand s problem p.14

34 More precise question If N 10 and Ω is a bounded smooth domain, is u singular? Perturbations of singular solutions to Gelfand s problem p.15

35 More precise question If N 10 and Ω is a bounded smooth domain, is u singular? Remark: if Ω = B 1 \ B 1/2 then u is always classical (any N). Perturbations of singular solutions to Gelfand s problem p.15

36 More precise question If N 10 and Ω is a bounded smooth domain, is u singular? Remark: if Ω = B 1 \ B 1/2 then u is always classical (any N). Suppose N 10 and Ω is a bounded smooth convex domain. Is u singular? Perturbations of singular solutions to Gelfand s problem p.15

37 Perturbations of a ball Let ψ C 2 (B; R N ) and Ω t = {x + tψ(x) : x B}. Perturbations of singular solutions to Gelfand s problem p.16

38 Perturbations of a ball Let ψ C 2 (B; R N ) and Ω t = {x + tψ(x) : x B}. Consider u = λe u u = 0 in Ω t on Ω t Perturbations of singular solutions to Gelfand s problem p.16

39 Perturbations of a ball Let ψ C 2 (B; R N ) and Ω t = {x + tψ(x) : x B}. Consider u = λe u u = 0 in Ω t on Ω t Theorem. (D.-Dupaigne) If N 4 there exists δ > 0 such that if t < δ then there is a singular solution λ(t), u(t) such that u(t) log 1 x ξ t 2 + λ(t) 2(N 2) 0 L as t 0, where ξ t B. Perturbations of singular solutions to Gelfand s problem p.16

40 Corollary. If N 11 and t is small then u is singular. Moreover there is ξ t B such that 1 u (t) log x ξ t 2 + λ (t) 2(N 2) 0 L as t 0. Perturbations of singular solutions to Gelfand s problem p.17

41 There is a singular solution (λ(t), u(t)) such that u(t) log 1 x ξ t 2 + λ(t) 2(N 2) 0 L Perturbations of singular solutions to Gelfand s problem p.18

42 There is a singular solution (λ(t), u(t)) such that u(t) log 1 x ξ t 2 + λ(t) 2(N 2) 0 L Since N 11 we have 2(N 2) < (N 2)2 4. Then for small t λ(t)e u(t) 2 log 1 x ξ t L (N 2) 2 4. Perturbations of singular solutions to Gelfand s problem p.18

43 There is a singular solution (λ(t), u(t)) such that u(t) log 1 x ξ t 2 + λ(t) 2(N 2) 0 L Since N 11 we have 2(N 2) < (N 2)2 4. Then for small t λ(t)e 1 u(t) 2 log x ξ t L (N 2) 2 4. For ϕ C0 (Ω t ), by Hardy s inequality: λ(t) Ωt e u(t) ϕ 2 (N 2)2 4 RN ϕ 2 x 2 R N ϕ 2. Perturbations of singular solutions to Gelfand s problem p.18

44 There is a singular solution (λ(t), u(t)) such that u(t) log 1 x ξ t 2 + λ(t) 2(N 2) 0 L Since N 11 we have 2(N 2) < (N 2)2 4. Then for small t λ(t)e 1 u(t) 2 log x ξ t L (N 2) 2 4. For ϕ C0 (Ω t ), by Hardy s inequality: λ(t) Ωt e u(t) ϕ 2 (N 2)2 4 RN ϕ 2 x 2 R N ϕ 2. By the lemma of Brezis-Vázquez we conclude u (t) = u(t), λ (t) = λ(t). Perturbations of singular solutions to Gelfand s problem p.18

45 Consider u = λ(1 + u) p in Ω t u = 0 on Ω t where Ω t is a C 2 perturbation of the ball, p > 1. Perturbations of singular solutions to Gelfand s problem p.19

46 Consider u = λ(1 + u) p in Ω t u = 0 on Ω t where Ω t is a C 2 perturbation of the ball, p > 1. Theorem. If N 11 and p > p p p 1 then for t small the extremal solution is singular. Perturbations of singular solutions to Gelfand s problem p.19

47 Consider u = λ(1 + u) p in Ω t u = 0 on Ω t where Ω t is a C 2 perturbation of the ball, p > 1. Theorem. If N 11 and p > p p p 1 then for t small the extremal solution is singular. It is known that for any domain, if N 10, or N 11 and p < p p p 1 then u is classical. Perturbations of singular solutions to Gelfand s problem p.19

48 Linearization The proof is by linearization around the singular solution 2 log x, λ = 2(N 2). Perturbations of singular solutions to Gelfand s problem p.20

49 Linearization The proof is by linearization around the singular solution 2 log x, λ = 2(N 2). We change variables y = x + tψ(x), x B 1 and define Then v(x) = u(x + tψ(x)). y u = x v + L t v where L t is a small second order operator. Perturbations of singular solutions to Gelfand s problem p.20

50 Linearization We look for a solution of the form v(x) = log where c = 2(N 2). 1 x ξ 2 + φ, λ = c + µ, Perturbations of singular solutions to Gelfand s problem p.21

51 Linearization We look for a solution of the form v(x) = log where c = 2(N 2). Then we need to solve φ L t φ 1 x ξ 2 + φ, λ = c + µ, c x ξ 2φ = c 1 φ) + x ξ 2(eφ ( 1 ) + L t log x ξ 2 φ = log 1 x ξ 2 on B. µ x ξ 2eφ in B Perturbations of singular solutions to Gelfand s problem p.21

52 A simple case Ω t is an ellipsoid, v(x) = u(x, (1 t)x N ), x = (x, x N ), and then ξ = 0. Perturbations of singular solutions to Gelfand s problem p.22

53 A simple case Ω t is an ellipsoid, v(x) = u(x, (1 t)x N ), x = (x, x N ), and then ξ = 0. Then the equation becomes φ t 2 2 φ x 2 N + µ φ = 0 + t 2 2 u 0 x 2eφ x 2 N on B. where u 0 (x) = 2 log x. c c x 2φ = 1 φ) x 2(eφ in B Perturbations of singular solutions to Gelfand s problem p.22

54 Comments Consider the linear operator c x ξ 2 Perturbations of singular solutions to Gelfand s problem p.23

55 Comments Consider the linear operator c x ξ 2 If N 9 the operator is not coercive in H 1 0. Perturbations of singular solutions to Gelfand s problem p.23

56 Comments Consider the linear operator c x ξ 2 If N 9 the operator is not coercive in H 1 0. If c (N 2)2 4, which holds if N 10, this operator is coercive. Perturbations of singular solutions to Gelfand s problem p.23

57 Comments Consider the linear operator c x ξ 2 If N 9 the operator is not coercive in H 1 0. If c (N 2)2 4, which holds if N 10, this operator is coercive. Typically solutions are singular at ξ, with a behavior x ξ α for some α > 0. Perturbations of singular solutions to Gelfand s problem p.23

58 Comments Consider the linear operator c x ξ 2 If N 9 the operator is not coercive in H 1 0. If c (N 2)2 4, which holds if N 10, this operator is coercive. Typically solutions are singular at ξ, with a behavior x ξ α for some α > 0. This functional setting is not useful since the nonlinear term that appears in the right hand side, c namely x ξ (e φ 1 φ), is too strong. 2 Perturbations of singular solutions to Gelfand s problem p.23

59 Right inverse for the linear operator Lemma. Let N 4, h,g be such that x ξ 2 h, x ξ 2 g L (B), h > 0, w smooth Perturbations of singular solutions to Gelfand s problem p.24

60 Right inverse for the linear operator Lemma. Let N 4, h,g be such that x ξ 2 h, x ξ 2 g L (B), h > 0, w smooth Consider φ c x ξ 2φ = g + µ 0h + φ = w where V i,ξ are explicit. on B N i=1 µ i V i,ξ in B Perturbations of singular solutions to Gelfand s problem p.24

61 Right inverse for the linear operator Lemma. Let N 4, h,g be such that x ξ 2 h, x ξ 2 g L (B), h > 0, w smooth Consider φ c x ξ 2φ = g + µ 0h + φ = w on B N i=1 µ i V i,ξ in B where V i,ξ are explicit. If ξ is small enough there is a solution φ, µ 0,..., µ N such that φ L + µ i C h x ξ 2 g L Perturbations of singular solutions to Gelfand s problem p.24

62 Right inverse for the linear operator The same result is true for the linear operator φ c x ξ 2φ L tφ if ξ and t are small enough. Perturbations of singular solutions to Gelfand s problem p.25

63 The nonlinear problem Using the linear lemma and the fixed point theorem we obtain φ, µ 0,..., µ N such that Perturbations of singular solutions to Gelfand s problem p.26

64 The nonlinear problem Using the linear lemma and the fixed point theorem we obtain φ, µ 0,..., µ N such that c φ L t φ x ξ 2φ = c 1 φ) x ξ 2(eφ where u ξ = log 1 x ξ 2 + µ 0 1 x ξ 2eφ + L t u ξ + φ = u ξ B N i=1 µ i V i,ξ Perturbations of singular solutions to Gelfand s problem p.26

65 Reduction We have to show that there is a choice of ξ such that µ 1,..., µ N = 0. Perturbations of singular solutions to Gelfand s problem p.27

66 Reduction We have to show that there is a choice of ξ such that µ 1,..., µ N = 0. Multiplying the equation by suitable test functions and integrating we reach a system of equations of the form F (ξ, t) = 0 Perturbations of singular solutions to Gelfand s problem p.27

67 Reduction We have to show that there is a choice of ξ such that µ 1,..., µ N = 0. Multiplying the equation by suitable test functions and integrating we reach a system of equations of the form F (ξ, t) = 0 This can be solved by the implicit function theorem. Perturbations of singular solutions to Gelfand s problem p.27

68 The linear lemma We consider only the operator c x 2. Perturbations of singular solutions to Gelfand s problem p.28

69 The linear lemma We consider only the operator c x 2. Let N 3, g C (B \ {0}) be a radial function such that x 2 g L (B). Then the equation φ c x 2φ = g in B φ = 0 on B has a solution in L (B) if and only if B gw 0 = 0 Perturbations of singular solutions to Gelfand s problem p.28

70 The linear lemma We consider only the operator c x 2. Let N 3, g C (B \ {0}) be a radial function such that x 2 g L (B). Then the equation φ c x 2φ = g in B φ = 0 on B has a solution in L (B) if and only if B gw 0 = 0 where W 0 = r α+ r α and α ± = N 2 (N 2) 2 ± 2 4 c. Perturbations of singular solutions to Gelfand s problem p.28

71 The linear lemma We consider only the operator c x 2. Let N 3, g C (B \ {0}) be a radial function such that x 2 g L (B). Then the equation φ c x 2φ = g in B φ = 0 on B has a solution in L (B) if and only if B gw 0 = 0 where W 0 = r α+ r α and α ± = N 2 (N 2) 2 ± 2 4 c. Moreover φ L C x 2 g L and this solution is unique. Perturbations of singular solutions to Gelfand s problem p.28

72 Idea of the proof: the condition is necessary W 0 = r α+ r α is in the kernel of the linear operator W 0 c x 2W 0 = 0 in B W 0 = 0 on B Perturbations of singular solutions to Gelfand s problem p.29

73 Idea of the proof: the condition is necessary W 0 = r α+ r α is in the kernel of the linear operator W 0 c x 2W 0 = 0 in B W 0 = 0 on B If φ is bounded one may justify the integration by parts ) ) gw 0 = ( φ c x 2φ W 0 = φ ( W 0 c x 2W 0 = 0 Perturbations of singular solutions to Gelfand s problem p.29

74 Idea of the proof: the condition is sufficient Construction of a solution: we seek φ(r) that solves φ c x 2 φ = g: φ + N 1 r φ + c r2φ = g Perturbations of singular solutions to Gelfand s problem p.30

75 Idea of the proof: the condition is sufficient Construction of a solution: we seek φ(r) that solves φ c x 2 φ = g: φ + N 1 r φ + c r2φ = g Then φ(r) = 1 α α + r 0 s((s/r)α (s/r) α+ )g(s) ds Perturbations of singular solutions to Gelfand s problem p.30

76 Idea of the proof: the condition is sufficient Construction of a solution: we seek φ(r) that solves φ c x 2 φ = g: φ + N 1 r φ + c r2φ = g Then φ(r) = 1 r α α + 0 (s/r) s((s/r)α α+ )g(s) ds φ(r) = r2 S N 1 B W 0(x)g(rx) dx Since g(x) C/ x 2 we have φ L. Since B W 0g = 0 we have φ(1) = 0. Perturbations of singular solutions to Gelfand s problem p.30

77 Non radial case We decompose φ in a Fourier series φ(x) = k φ k(r)ϕ k (θ) where r > 0, θ S N 1, and ϕ k are the eigenfunctions of on the sphere S N 1 : S N 1ϕ k = λ k ϕ k. Perturbations of singular solutions to Gelfand s problem p.31

78 Non radial case We decompose φ in a Fourier series φ(x) = k φ k(r)ϕ k (θ) where r > 0, θ S N 1, and ϕ k are the eigenfunctions of on the sphere S N 1 : S N 1ϕ k = λ k ϕ k. Then φ c x 2 φ = g in B is equivalent to φ k N 1 φ k c λ k φ r r 2 k = g k Perturbations of singular solutions to Gelfand s problem p.31

79 Non radial case We decompose φ in a Fourier series φ(x) = k φ k(r)ϕ k (θ) where r > 0, θ S N 1, and ϕ k are the eigenfunctions of on the sphere S N 1 : S N 1ϕ k = λ k ϕ k. Then φ c x 2 φ = g in B is equivalent to φ k N 1 φ k c λ k φ r r 2 k = g k If c λ k 0 the equation has a bounded solution without requiring orthogonality conditions. If c λ k > 0 orthogonality conditions are required (with respect to elements in the kernel ). Perturbations of singular solutions to Gelfand s problem p.31

80 c = 2(N 2) λ 0 = 0 λ 1 =... = λ N = N 1 λ k 2N, k N + 1 and N 4 yields Numbers... Perturbations of singular solutions to Gelfand s problem p.32

81 c = 2(N 2) λ 0 = 0 λ 1 =... = λ N = N 1 λ k 2N, k N + 1 and N 4 yields Numbers... c λ k > 0 for k = 0,..., N c λ k 0 for k N + 1 So N + 1 conditions are required to have a bounded solution. Perturbations of singular solutions to Gelfand s problem p.32

82 Related work Caffarelli-Hardt-Simon (84) Construction of singular minimal surfaces which are not cones (by perturbation of minimal cones). Perturbations of singular solutions to Gelfand s problem p.33

83 Related work Caffarelli-Hardt-Simon (84) Construction of singular minimal surfaces which are not cones (by perturbation of minimal cones). Pacard (92,93) Existence of singular solutions to u = e u and u p (without boundary conditions) Perturbations of singular solutions to Gelfand s problem p.33

84 Related work Caffarelli-Hardt-Simon (84) Construction of singular minimal surfaces which are not cones (by perturbation of minimal cones). Pacard (92,93) Existence of singular solutions to u = e u and u p (without boundary conditions) Mazzeo-Pacard (96) construct solutions to u = u p with singularities on points ( N N 2 < p < N+2 k N 2 ) or manifolds ( k 2 < p < k+2 k 2, k is de codimension). Perturbations of singular solutions to Gelfand s problem p.33

85 Related work Caffarelli-Hardt-Simon (84) Construction of singular minimal surfaces which are not cones (by perturbation of minimal cones). Pacard (92,93) Existence of singular solutions to u = e u and u p (without boundary conditions) Mazzeo-Pacard (96) construct solutions to u = u p with singularities on points ( N N 2 < p < N+2 k N 2 ) or manifolds ( k 2 < p < k+2 k 2, k is de codimension). Rebai (96,99) u = e u in a ball in dimension 3, also multiple singularities N 10 (without boundary condition) Perturbations of singular solutions to Gelfand s problem p.33

86 A variant Consider u = λe u in B u = ψ on B where ψ is a smooth function. Perturbations of singular solutions to Gelfand s problem p.34

87 Consider A variant u = λe u u = ψ in B on B where ψ is a smooth function. Theorem. (D.-Dupaigne) If N 4 and ψ is small enough (in C 2,α ) then there exists ξ B and a singular solution λ, u such that u log 1 x ξ 2 L (B). Perturbations of singular solutions to Gelfand s problem p.34

88 Consider A variant u = λe u u = ψ in B on B where ψ is a smooth function. Theorem. (D.-Dupaigne) If N 4 and ψ is small enough (in C 2,α ) then there exists ξ B and a singular solution λ, u such that u log 1 x ξ 2 L (B). Note that ξ depends on ψ. Perturbations of singular solutions to Gelfand s problem p.34

89 The case N = 3 Consider u = λe u in B u = ψ on B where ψ is a smooth function. Perturbations of singular solutions to Gelfand s problem p.35

90 The case N = 3 Consider u = λe u u = ψ where ψ is a smooth function. in B on B Theorem. (Matano, Rebai (99)) If N = 3 there is δ > 0 such that for any ψ C 2,α < δ and any ξ < δ there is a singular solution λ, u such that u log 1 x ξ 2 L (B). Perturbations of singular solutions to Gelfand s problem p.35

91 Isolated singularities in dimension 3 The function u(r, θ) = log(1/r 2 )+log(2/λ)+2ω(θ) r > 0, θ S 2 is a singular solution in R 3 if and only if S 2ω + e 2ω 1 = 0 in S 2. Smooth solutions form a 3 dimensional manifold. Perturbations of singular solutions to Gelfand s problem p.36

92 Isolated singularities in dimension 3 The function u(r, θ) = log(1/r 2 )+log(2/λ)+2ω(θ) r > 0, θ S 2 is a singular solution in R 3 if and only if S 2ω + e 2ω 1 = 0 in S 2. Smooth solutions form a 3 dimensional manifold. Bidaut-Veron Veron (91) describe all possible behaviors of smooth solutions to u = λe u in B 1 \ {0} (with an isolated singularity) in dimension 3, such that e u C x 2. Perturbations of singular solutions to Gelfand s problem p.36

93 Isolated singularities in N = 3 In dimension 3, if e u C x 2. Perturbations of singular solutions to Gelfand s problem p.37

94 Isolated singularities in N = 3 In dimension 3, if e u C x 2. then either 0 is a removable singularity, Perturbations of singular solutions to Gelfand s problem p.37

95 Isolated singularities in N = 3 In dimension 3, if e u C x 2. then either 0 is a removable singularity, or lim x 0 (u(x) γ x 1 ) exists for some γ < 0 and u = λe u + 4πγδ 0, Perturbations of singular solutions to Gelfand s problem p.37

96 Isolated singularities in N = 3 In dimension 3, if e u C x 2. then either 0 is a removable singularity, or lim x 0 (u(x) γ x 1 ) exists for some γ < 0 and u = λe u + 4πγδ 0, or there exists ω solution to S 2ω + e 2ω 1 = 0 in S 2 such that lim r + u(rθ) log(1/r2 ) = ω(θ), θ S 2. Perturbations of singular solutions to Gelfand s problem p.37

97 Is u singular for all Ω convex? Let ε > 0, x = (y, z) R N, y R N 1, z RN 2. Perturbations of singular solutions to Gelfand s problem p.38

98 Is u singular for all Ω convex? Let ε > 0, x = (y, z) R N, y R N 1, z RN 2. Ω ε = {(y, εz) : (y, z) B 1 R N }. ε R N 2 R N 1 Perturbations of singular solutions to Gelfand s problem p.38

99 Is u singular for all Ω convex? Let ε > 0, x = (y, z) R N, y R N 1, z RN 2. Ω ε = {(y, εz) : (y, z) B 1 R N }. ε R N 2 R N 1 Theorem. (Dancer (93)) Suppose N 2 9. If ε is small then u ε is classical. Note that N = N 1 + N 2 may be larger than 10. Perturbations of singular solutions to Gelfand s problem p.38

100 Is u singular for all Ω convex? Let ε > 0, x = (y, z) R N, y R N 1, z RN 2. Ω ε = {(y, εz) : (y, z) B 1 R N }. ε R N 2 R N 1 Theorem. (Dancer (93)) Suppose N 2 9. If ε is small then u ε is classical. Note that N = N 1 + N 2 may be larger than 10. The result is still true if Ω is a smooth bounded strictly convex. Perturbations of singular solutions to Gelfand s problem p.38

101 What happens if N 2 10? Consider a torus Ω ε in R N with cross-section a ball of radius ε > 0 in R N 2, N = N 1 + N 2, N 1 1. Perturbations of singular solutions to Gelfand s problem p.39

102 What happens if N 2 10? Consider a torus Ω ε in R N with cross-section a ball of radius ε > 0 in R N 2, N = N 1 + N 2, N 1 1. If N 2 11 then for ε small the extremal solution is singular. Perturbations of singular solutions to Gelfand s problem p.39

103 Singularities at infinity Consider u + u p = 0, u > 0 in Ω = R N \ D, u = 0 on D, lim u(x) = 0 x + where p > N+2 N 2 and D is a smooth bounded open set such that Ω is connected. Perturbations of singular solutions to Gelfand s problem p.40

104 Singularities at infinity Consider u + u p = 0, u > 0 in Ω = R N \ D, u = 0 on D, lim u(x) = 0 x + where p > N+2 N 2 and D is a smooth bounded open set such that Ω is connected. Theorem. (D.-del Pino-Musso-Wei) If N 3 and p > N+2 N 2 then there are infinitely many solutions, that have slow decay u(x) x 2 p 1 as x +. Perturbations of singular solutions to Gelfand s problem p.40

105 If λ > λ there is no solution. Perturbations of singular solutions to Gelfand s problem p.41

106 If λ > λ there is no solution. If λ = λ there is a unique solution. Perturbations of singular solutions to Gelfand s problem p.41

107 If λ > λ there is no solution. If λ = λ there is a unique solution. Suppose λ < λ and let u λ be the minimal solution. Then u (u u λ ) = λ e u (u u λ ) Ω Ω u λ (u u λ ) = λ e u λ (u u λ ) Ω Ω Perturbations of singular solutions to Gelfand s problem p.41

108 If λ > λ there is no solution. If λ = λ there is a unique solution. Suppose λ < λ and let u λ be the minimal solution. Then u (u u λ ) = λ e u (u u λ ) Ω Ω u λ (u u λ ) = λ e u λ (u u λ ) Ω Ω Hence Ω (u u λ ) 2 = λ Ω (e u e u λ )(u u λ ). Perturbations of singular solutions to Gelfand s problem p.41

109 We have Ω (u u λ) 2 = λ Ω (eu e u λ )(u u λ) Perturbations of singular solutions to Gelfand s problem p.42

110 We have Ω (u u λ) 2 = λ Ω (eu e u λ )(u u λ) Since u is stable Ω (u u λ) 2 λ Ω eu (u u λ ) 2 Perturbations of singular solutions to Gelfand s problem p.42

111 We have Ω (u u λ) 2 = λ Ω (eu e u λ )(u u λ) Since u is stable Ω (u u λ) 2 λ Ω eu (u u λ ) 2 It follows that (e u + e u (u λ u) e u λ )(u u λ ) 0. Ω Perturbations of singular solutions to Gelfand s problem p.42

112 We have Ω (u u λ) 2 = λ Ω (eu e u λ )(u u λ) Since u is stable Ω (u u λ) 2 λ Ω eu (u u λ ) 2 It follows that (e u + e u (u λ u) e u λ )(u u λ ) 0. Ω By convexity the integrand is non-positive. This implies u = u λ but u L while u λ is classical. Perturbations of singular solutions to Gelfand s problem p.42