Blow-up solutions for critical Trudinger-Moser equations in R 2

Transcription

1 Blow-up solutions for critical Trudinger-Moser equations in R 2 Bernhard Ruf Università degli Studi di Milano

2 The classical Sobolev embeddings We have the following well-known Sobolev inequalities: let Ω R N bounded, and Set Then, for N > 2: S q := H0 1 (Ω) : Sobolev space 2 = 2N N 2 sup u q dx u 2 1 Ω (critical exponent) < +, q < 2 compact, attained < +, q = 2 non compact, not attained = +, q > 2

3 The borderline case N = 2 The Trudinger-Moser inequality: S. Pohozaev (1965), N. Trudinger (1967): u H0 1 (Ω) e u2 dx < + Sharpened by J. Moser (1971): T α := sup e αu2 dx u 2 1 Ω Ω < +, α < 4π compact, attained < +, α = 4π non compact, attained? = +, α > 4π

4 The result of L. Carleson S.-Y. Alice Chang For Ω = B R (0): T 4π is attained Surprising, if compared to Sobolev case! Proof: Assume not attained: then maximizing sequence {u n } concentrates determine lim e 4πu2 n = (1 + e) Ω : non-compactness level n Ω show (by explicit example) that sup e 4πu2 > (1 + e) Ω : u 2 1 Ω contradiction! Generalizations: Struwe (1988): Ω near B R (0) Flucher (1992): general bounded Ω R 2

5 Associated differential equation An extremal u for T α is a critical point of the functional e u2 on S α = {u H0 1 (Ω) ; u 2 2 = α} namely Ω D Sα [ ] e u2 = 0 Ω that is { u = λ u e u 2 in Ω u = 0 on Ω, λ = α Ω u2 e u2

6 Two related (but not equivalent) problems: (T α ) u = αu eu2 Ω u2 e u2 in Ω, u = 0 on Ω and (P λ ) u = λ u e u2 in Ω, u = 0 on Ω Solutions of (P λ ) correspond to critical points of the free energy J λ (u) = u 2 λ e u2 dx Ω Ω

7 Some existence results (T α ) has (positive) solution, by Carleson-Chang, Flucher Existence of solutions for: (P λ ) u = λue u2 in Ω, u = 0 on Ω Adimurthi (1990) : a positive solution u λ exists, for 0 < λ < λ 1. Result valid for more general nonlinearities: suppose that f (t) = h(t) e t2, h C(R), and set F (t) := 0 f (s)ds. Assume H1) 0 < F (s) 1 2f (s)s, s R \ {0} H2) lim sup s 0 2F (s) s 2 < λ 1 H3) lim inf s h(s)s > 2 d 2 h(t) e t2 0, d: radius of largest ball B d Ω

8 Theorem: (de Figueiredo-Miyagaki-R., 1995). Assume H1) - H3). Then the critical growth equation u = f (u), in Ω, u = 0 on Ω has a positive solution Proof: (similar to Brezis-Nirenberg result for u = λu + u 2 1 ) use mountain-pass theorem by Ambrosetti-Rabinowitz determine non-compactness level use H3) and Moser sequence to show that mountain-pass level stays below non-compactness level

9 This existence result is almost sharp: de Figueiredo-R. (1995): Let Ω = B 1 (0). There exists δ > 0 such that if lim sup h(s)s < δ s then (P λ ) has no positive solution. On the other hand: topology helps to get solutions (cf. Coron, 1984; Bahri-Coron, 1988): Struwe (2000): For large class of critical nonlinearities (which includes above) there exists positive solution on suitable non-contractible domains

10 Loss of compactness: Quantization for (PS)-sequences Recall Struwe s result for Brezis-Nirenberg functional: I λ (u) = 1 u 2 λ u u 2, Ω R N, N 3 Ω For a sequence λ n λ 0, and a (PS)-sequence {u n }, i.e. I λ n (u n ) 0 and I λn (u n ) c, one has Ω I λn (u n ) = I λ0 (u 0 ) + k S N + o(1), for some k 1 where u 0 a critical point of I λ0, and S N is a positive constant Ω

11 Bubbling in the Trudinger-Moser case (P λ ) Druet (2006): Suppose that {u λ } is sequence of solutions of (P λ ) with uniformly bounded energy, i.e. (P λ ) u λ = λ u λ e u2 λ in Ω, uλ = 0 on Ω with Jλ (u λ ) = then for some integer k 0 Ω u λ 2 λ Ω e u2 λ (D k ) J λ (u λ ) k 4π, as λ 0, that is: bubbling in k points. c,

12 Previous results on single point bubbling: k = 1 Adimurthi-Struwe (2000) Adimurthi-Druet (2004): in addition: The solution u λ has unique isolated maximum which blows up around x 0 (as λ 0), where x 0 is critical point of Robin s function. Let G(x, y) be Green s function of the problem x G(x, y) = 8πδ y (x), x Ω ; G(x, y) = 0, x Ω and H(x, y) its regular part: H(x, y) = 4 log H(ξ, ξ): Robin s function of Ω 1 G(x, y) x y

13 Existence of bubbling solutions Recall results for: Liouville equation in 2d : u = λe u in Ω, u = 0 on Ω Concentration as λ 0 or λ = µ Ω eu (also on compact manifolds and with singular weights) Brezis, Merle, Nagasaki, Suzuki, Li, Shafrir, Weston, Ma, Wei, Tarantello, Struwe, Nolasco, Baraket, Pacard, Del Pino, Kowalczyk, Musso, Esposito, Grossi, Pistoia, Bartolucci, Chen, Lin, Chang, Gursky, Yang, Malchiodi, Robert... Del Pino-Kowalczyk-Musso (2005): If Ω is not simply connected, then solutions with k bubbling points exist for each given k 1. Blowing-up takes place at critical points of ψ k (ξ) := k i=1 H(ξ j, ξ j ) i j G(ξ i, ξ j ).

14 Existence of solutions to (P λ ) with Property (D k )? Bubbling solutions: For k distinct points ξ 1,..., ξ k Ω and k positive numbers m 1,..., m k R consider the function ϕ k (ξ, m) = k 2mj 2 (b + log mj 2 ) + mj 2 H(ξ j, ξ j ) m i m j G(ξ i, ξ j ) i j j=1 where b = 2 log 8 2

15 Stable critical point situation (SCS) We say that ϕ k has a stable critical point situation, if for some open set Λ with Λ { (ξ, m) Ω k R k + ξ i ξ j if i j } there exists δ > 0 such that for any g C 1 (Λ) with g C 1 (Λ) < δ ϕ k + g has a critical point in Λ.

16 Important facts: ϕ 1 (ξ, m) satisfies (SCS), with Λ a neighborhood of its minimum set. ϕ 1 (ξ, m) = 2m 2 (b + log m 2 ) + m 2 H(ξ, ξ) ϕ 2 (ξ, m) satisfies (SCS) whenever Ω is not simply connected. 2 ϕ 2 (ξ, m) = 2mj 2 (b + log mj 2 ) + mj 2 H(ξ j, ξ j ) 2m 1 m 2 G(ξ 1, ξ 2 ). j=1 We believe that (SCS) holds for any k 2 (if Ω is not simply connected).

17 Theorem 1. (M. del Pino - M. Musso - R., JFA, 2009) Assume that ϕ k (ξ, m) has (SCS). Then there exist solutions u λ to (P λ ) which blow up around k points ξ j as λ 0, where ϕ k (ξ, m) 0 away from the points ξ j the solutions u λ take the form (1) u λ (x) = k λ m j [ G(x, ξ j ) + o(1) ] Furthermore j=1 J λ (u λ ) = 4πk + λ [ Ω + 8π ϕ k (ξ, m) + o(1) ]. This result applies for k = 1: there exists bubbling solution near minimizer of H(ξ, ξ) k = 2: there exists solution with two bubbles, provided that Ω is not simply connected.

18 Argument of the proof: construction of approximate solution, based on standard bubble linearize equation in this approximate solution finite dimensional variational reduction via Lyapunov-Schmidt yields finite-dimensional functional f k which is C 1 -close to ϕ k (ξ, m) critical points of f k yield the solution, and the information on the location of the bubbles This method has been successfully applied by numerous authors to problems in the critical growth range: Bahri, Coron, Yanyan Li, Rey, Ni, Wei, Takagi, del Pino, Felmer, Musso, Kowalczyk, Pistoia, Grossi, Esposito, Ge, Jing, Baraket, Pacard,...

19 Standard bubbles Let ξ j Ω, j = 1,..., k, with ξ j ξ k for j k. Fix an index j and define γ j and ε j by the relations γ 2 j := 1 4m 2 j λ, ε2 j := 2m 2 j e 1 4m j 2λ so that γ j and ε j 0, as λ 0. Let us write u(x) = γ j + 1 ( ) x ξj v. 2γ j Then u = λue u2 v = (1 + v 2γ 2 j ε j )e v 2 4γ j 2 e v

20 Limiting equation v = (1 + v 2γ 2 j )e v 2 4γ j 2 e v For γ j + we get the limiting Liouville equation (2) v = e v, in R 2 The radial solutions of (2) are given by : ω µ ( y ) = log 8µ 2 (µ 2 + y 2, µ > 0 : standard bubble ) 2

21 Approximate solution Let ξ 1,..., ξ k Ω, m 1,..., m k > 0 be given. We build an approximate solution U λ (ξ, m). First approximation: Want that first approximation U λ is such that for some µ j > 0, and for x close to ξ j [ U λ (x) γ j + 1 ( x ξj ) ] ω µj =: W j (x) 2γ j ε j Add to W j a harmonic function such that boundary condition zero is satisfied: that is U j (x) = W j (x) λ m j [log 2m 2 j + log 8µ 2 j + H j (x)], H j = 0 in Ω, H j (x) = log 1 (ε 2 j µ2 j + x ξ j 2 ) 2 on Ω ;

22 Define U λ (ξ, m) := k j=1 U j Choose the µ j s such that U λ (x) W j (x) for x close to ξ j, for all j U λ are approximate solutions, with error R λ (x) = U λ + λu λ e U2 λ Introduce suitable L -weighted norm so that R λ Cλ Goal: find solution u of the form u = U λ + φ such that φ is small for suitable points ξ i and m i

23 Linearization in U λ : denote f (s) = se s2 φ + f (U λ )φ = R [f (U λ + φ) f (U λ ) f (U λ )φ] =: R N(φ) If L λ := + f (U λ ) were boundedly invertible: invert L λ, use contraction principle: done! Note: far from the points ξ j : f (U λ ) = O(λ), hence L λ = + f (U λ ) = + O(λ) is small perturbation of away from the concentration points near the points ξ j : f (U λ ) 1 e ω ε 2 j. j Hence L λ is approximately superposition of the linear operators L j (φ) = φ + 1 ε 2 j e ω j φ

24 Kernel of L Thus: L λ is nontrivial perturbation of near the ξ j, while essentially equal to on most of the domain. Note: L j (φ) = φ + 1 ε 2 j e ω j φ = 0 has bounded solutions, due to natural invariances of (3) ω + e ω = 0 Let ω µ (y) : explicit solution of (3), then z 0j (x) = µ ω µj (y), z 1j = x1 ω µj (x), z 2j = x2 ω µj (x), j = 1,..., k satisfy Z + e ωµ j Z = 0

25 Projection onto kernel of L Hence: Z ij (x) := z ij ( x ξj ε j ), i = 0, 1, 2 are bounded solutions of L j (Z) = 0 on R 2 ; in fact, they are all bounded solutions (Bakaret-Pacard, 1998) Consider projected version of equation L λ φ = R N(φ), i.e. projection onto the almost kernel given by above functions, multiplied by cut-off functions ζ j 2 k L λ (φ) = R N(φ) + c ij Z ij ζ j, in Ω (4) i=0 j=1 φ = 0, on Ω Ω Z ijζ j φ = 0, for all i, j Then: (4) has unique solution φ(ξ, m), with φ Cλ.

26 Final step: Find ( ξ, m) such that (5) c ij ( ξ, m) = 0, for all i = 0, 1, 2; j = 1,, k Put problem in variational form: ( ( (6) (ξ, m) J λ λ Uλ (ξ, m) + φ(ξ, m) )) =: T λ (ξ, m) Fact: if ( ξ, m) is critical point of (6), then (5) holds Energy computation at first approximation: T λ (ξ, m) = 4πk λ Ω + λ 8π ϕ k (ξ, m) + O(λ 2 ), where ϕ k is the functional in statement of Theorem 1. By the (SCS) property: T λ has a critical point.

27 Back to the Moser functional ( i.e. the case (T α ) ) Recall: J α (u) := Ω with associated equation e u2 dx on S α = {u H 1 0 (Ω) : u 2 2 = α} sup J α (u) attained, if α 4π u S α (T α ) u = α u eu2 Ω u2 e u2 in Ω, u = 0 on Ω

28 Existence of critical points in the Supercritical Trudinger-Moser case Numerical evidence (Monahan, 1971): For Ω = B 1 (0) R 2, the Trudinger-Moser functional (7) J α (u) = e u2 dx, u 2 2 = α admits local maximum and second critical value in the Ω supercritical regime, i.e. for α > 4π (and near 4π) Struwe (1988): the global maximum of J α can be continued as a local maximum for values 4π < α < α 1 and there exists a second solution u α to (T α ), defined for α in a dense subset of (4π, α 1 ).

29 u(p) u(α) 1 2 p 4π α Approaching criticality in the Sobolev and the Moser case

30 Lamm, Robert, Struwe (2009) A second solution u α of (T α ) is defined on the entire interval (4π, α 1 ). Proof: Asymptotic analysis of a geometric heat flow associated to the Trudinger Moser functional and a min-max argument. Little qualitative information on the second solution The mentioned numerical evidence (in the case of a ball) suggests existence of a branch of solutions of (T α ) that blows up as α 4π.

31 Theorem 2. (M. del Pino, M. Musso, R.) There is a positive solution u α for (T α ) for 4π < α < α 1 such that u α blows-up around a point ξ α, and away from ξ α : u α (x) = (α 4π) 1 4 G(x, ξα ) + o(1). Besides H(ξ α, ξ α ) min H(ξ, ξ). ξ

32 Theorem 3. (M. del Pino, M. Musso, R.) Assume that Ω is not simply connected. Then there is a positive solution u α to (T α ) for 8π < α < α 2 which blows up around two points ξ 1, ξ 2 Ω, and away from them: u α (x) = a (α 8π) 1 4 [ m1 G(x, ξ 1 ) + m 2 G(x, ξ 1 ) ] + o(1). where ϕ 2 (ξ, m) 0 as α 8π, with 2 ϕ 2 (ξ, m) = 2(b + mj 2 ) log mj 2 + mj 2 H(ξ j, ξ j ) 2m 1 m 2 G(ξ 1, ξ 2 ) j=1 b = 2 log 8 2.

33 We believe: If Ω is not simply connected and k 4π < α < α k (Ω), then there is a positive solution u α of u + α u eu2 Ω u2 e u2 = 0 in Ω, u = 0 on Ω with k blow-up points ξ j. Away from them: where with ϕ k (ξ, m) = u α (x) = a (α 4kπ) 1 4 ϕ k (ξ, m) 0 k m j G(x, ξ j ) + o(1) j=1 as α 8π k 2mj 2 (b + log mj 2 ) + mj 2 H(ξ j, ξ j ) m i m j G(ξ j, ξ j ). i j j=1

34 This is true if Ω R 2 is an annulus: Theorem 4. (M. del Pino, M. Musso, R.) If Ω = {x R 2 : 0 < a < x < b}, then for 4π k < α < α k there exists positive solution u α to (T α ) - invariant under rotations by angle 2π k - blows up around k points (lying on vertices of regular polygon), as α 4πk

35 Proofs of Theorems 2-4: Idea similar to proof of Theorem 1, however different in technical details and estimates