Electronic Journal of Differential Equations, Vol. 2022 (2022), No. 14, pp. 1–10. ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu COMPACTNESS OF THE SET OF SOLUTIONS TO ELLIPTIC EQUATIONS IN 2 DIMENSIONS SAMY SKANDER BAHOURA Abstract. We study the behavior of solutions to elliptic equations in 2 di- mensions. In particular, we show that the set of solutions is compact under a Lipschitz condition. 1. Introduction Let us define the operator eLε := ∆ + ε(x1∂1 + x2∂2) = div[aε(x)∇] aε(x) , with aε(x) = eε|x| 2/2. We consider the equation −∆u− ε(x1∂1u+ x2∂2u) = −Lεu = V eu in Ω ⊂ R2, u = 0 in ∂Ω , (1.1) where Ω is a starshaped set, u ∈W 1,1 0 (Ω), eu ∈ L1(Ω), 0 ≤ V ≤ b, 1 ≥ ε ≥ 0. For ε = 0 equation (1.1) has been studied by many authors with and without the boundary condition. This equation also has been studied in Riemann surfaces; see [1]–[20], where one can find some existence and compactness results. Also we have a nice formulation in the sense of the distributions of this problem in [7]. Among the known results we find the following Theorem. Theorem 1.1 (Brezis-Merle [6]). If (ui) and (Vi) are two sequences of functions in problem (1.1) with ε = 0, and 0 < a ≤ Vi ≤ b < +∞, then for all compact subset K of Ω it holds sup K ui ≤ c, with c depending on a, b,K and Ω. We can find an interior estimate if we assume a = 0, but we need an assumption on the integral of eui . 2020 Mathematics Subject Classification. 35J60, 35B44, 35B45. Key words and phrases. Blow-up; compactness; boundary; elliptic equation; Lipschitz condition; starshaped domain. ©2022. This work is licensed under a CC BY 4.0 license. Submitted September 4, 2021. Published February 21, 2022. 1 2 S. S. BAHOURA EJDE-2022/14 Theorem 1.2 (Brezis-Merle [6]). Let (ui) and (Vi) two sequences of functions in problem (1.1) with 0 ≤ Vi ≤ b < +∞ and ∫ Ω euidy ≤ C. Then, for all compact subset K of Ω it holds sup K ui ≤ c, with c depending on b, C,K and Ω. The condition ∫ Ω euidy ≤ C is a necessary in Problem (1.1) as showed by the following statement for ε = 0. Theorem 1.3 (Brezis-Merle [6]). There are sequences (ui) and (Vi) in problem (1.1) with 0 ≤ Vi ≤ b < +∞, ∫ Ω euidy ≤ C, such that supΩ ui → +∞. To obtain Theorems 1.1 and 1.2 Brezis and Merle used an inequality [6, Theorem 1] obtained by an approximation argument, Fatou’s lemma, and the maximum principle in W 1,1 0 (Ω), which arises from Kato’s inequality. Also this weak form of the maximum principle is used to prove the local uniform boundedness result by comparing a certain function and the Newtonian potential. We refer the reader to [5] for information about the weak form of the maximum principle. Note that for problem (1.1), by using the Pohozaev identity, we can prove that∫ Ω eui is uniformly bounded when 0 < a ≤ Vi ≤ b < +∞, ‖∇Vi‖L∞ ≤ A, and Ω starshaped. When a = 0 and ∇ log Vi is uniformly bounded, we can find a uniform bound for ∫ Ω Vie ui . Ma-Wei [17] proved that those results remain valid for all open sets not neces- sarily starshaped when a > 0. Chen-Li [9] proved that if a = 0, ∫ Ω eui is uniformly bounded, and ∇ log Vi is uniformly bounded, then (ui) is bounded near the bound- ary and we have directly the compactness result for the problem (1.1). Ma-Wei [17] extend this result in the case where a > 0. When ε = 0 and if we assume V more regular we can have another type of estimates called sup + inf type inequalities. It was proved by Shafrir [19] that, if (ui), (Vi) are two sequences of solutions to Problem (1.1), without assumption on the boundary and 0 < a ≤ Vi ≤ b < +∞, then it holds C (a b ) sup K ui + inf Ω ui ≤ c = c(a, b,K,Ω). We find in [10] the explicit value C(a/b) = √ a/b. In his proof, Shafrir [19] used the blow-up function, the Stokes formula and an isoperimetric inequality. Chen-Lin [10] used the blow-up analysis combined with some geometric type inequality for obtaining the integral curvature. Now, if we assume (Vi) is uniformly Lipschitzian with constant A, then C(a/b) = 1 and c = c(a, b, A,K,Ω) see Brezis-Li-Shafrir [4]. This result was extended for Hölderian sequences (Vi) by Chen-Lin [10]. Also we have in [15], an extension of the Brezis-Li-Shafrir result to compact Riemannian surfaces without boundary. One can see in [17] an explicit form, (8πm,m ∈ N∗ exactly), for the numbers in front of the Dirac masses when the solutions blow-up. Here the notion of isolated EJDE-2022/14 ELLIPTIC EQUATIONS IN 2 DIMENSIONS 3 blow-up point is used. Also one can find in [11] refined estimates near the isolated blow-up points and the bubbling behavior of the blow-up sequences. Here we study the behavior of the blow-up points on the boundary, and give a compactness result with Lipschitz condition. Note that our problem is an extension of the Brezis-Merle Problem. Brezis-Merle Problem [6]. Suppose that Vi → V in C0(Ω̄) with 0 ≤ Vi, and consider a sequence of solutions (ui) of (1.1) relative to (Vi) such that∫ Ω eui dx ≤ C. Is it possible to have ‖ui‖L∞ ≤ C = C(b, C, V,Ω) ? Here we give a blow-up analysis on the boundary when Vi are nonnegative and bounded (similar to the prescribed curvature when ε = 0). On the other hand, if we add the assumption that these functions (similar to the prescribed curvature) are uniformly Lipschitzian, we have a compactness of the solutions of problem (1.1) for ε small enough. (In particular we can take a sequence of εi tending to 0). For the behavior of the blow-up points on the boundary, the following condition is sufficient, 0 ≤ Vi ≤ b, The condition Vi → V in C0(Ω̄) is not necessary. But for the compactness of the solutions we add the condition ‖∇Vi‖L∞ ≤ A. Our main results read as follows. Theorem 1.4. Assume that maxΩ ui → +∞, where (ui) are solutions of (1.1) with ε = εi and 0 ≤ Vi ≤ b, ∫ Ω eui dx ≤ C, εi → 0 . Then, after passing to a subsequence, there are a function u, a number N ∈ N, and N points x1, . . . , xN ∈ ∂Ω, such that ∂νui → ∂νu+ N∑ j=1 αjδxj , αj ≥ 4π, in the sense of measures on ∂Ω, and ui → u in C1 loc(Ω̄− {x1, . . . , xN}). Theorem 1.5. Assume that (ui) are solutions of (1.1) with ε = εi, and 0 ≤ Vi ≤ b, ‖∇Vi‖L∞ ≤ A, ∫ Ω eui ≤ C, εi → 0. Then ‖ui‖L∞ ≤ c(b, A,C,Ω) . 4 S. S. BAHOURA EJDE-2022/14 2. Proofs of main results Proof of Theorem 1.4. First we remark that −∆ui = εi(x1∂1ui + x2∂2ui) + Vie ui ∈ L1(Ω) in Ω ⊂ R2, ui = 0 in ∂Ω. (2.1) and ui ∈W 1,1 0 (Ω). By [6, Corollary 1] we have eui ∈ Lk(Ω) for all k > 2 and the elliptic estimates of Agmon and the Sobolev embedding see [1] imply that ui ∈W 2,k(Ω) ∩ C1,ε(Ω̄). Also we remark that for two positive constants Cq = C(q,Ω) and C1 = C1(Ω), we have ‖∇ui‖Lq ≤ Cq‖∆ui‖L1 ≤ (C ′q + εC1‖∇ui‖L1), ∀i and 1 < q < 2. (see [7]). Thus, if ε > 0 is small enough and by Holder’s inequality, ‖∇ui‖Lq ≤ C ′′q , ∀i and 1 < q < 2. Step 1: Interior estimate. First we consider the equation −∆wi = εi(x1∂1ui + x2∂2ui) ∈ Lq, 1 < q < 2 in Ω ⊂ R2, wi = 0 in ∂Ω. (2.2) If we consider vi as the Newtonnian potential of εi(x1∂1ui + x2∂2ui), we have vi ∈ C0(Ω̄), ∆(wi − vi) = 0. By the maximum principle wi − vi ∈ C0(Ω̄) and thus wi ∈ C0(Ω̄). Also we have by elliptic estimates that wi ∈W 2,1+ε ⊂ L∞, and we can write the equation of the Problem as −∆(ui − wi) = Ṽie ui−wi in Ω ⊂ R2, ui − wi = 0 in ∂Ω, (2.3) with 0 ≤ Ṽi = Vie wi ≤ b̃, ∫ Ω eui−wi ≤ C̃. We apply the Brezis-Merle theorem to ui−wi to have ui−wi ∈ L∞loc(Ω), and, thus ui ∈ L∞loc(Ω). Step2: Boundary estimate. Let ∂νui be the inner derivative of ui. By the maximum principle ∂νui ≥ 0. Then we have∫ ∂Ω ∂νuidσ ≤ C. We have the existence of a nonnegative Radon measure µ such that∫ ∂Ω ∂νuiφdσ → µ(φ), ∀φ ∈ C0(∂Ω). We take an x0 ∈ ∂Ω such that µ(x0) < 4π. Set B(x0, ε) ∩ ∂Ω := Iε. We choose a function ηε such that ηε ≡ 1, on Iε, 0 < ε < δ/2, ηε ≡ 0, outside I2ε, EJDE-2022/14 ELLIPTIC EQUATIONS IN 2 DIMENSIONS 5 0 ≤ ηε ≤ 1, ‖∇ηε‖L∞(I2ε) ≤ C0(Ω, x0) ε . We take a η̃ε such that −∆η̃ε = 0 in Ω ⊂ R2, η̃ε = ηε in ∂Ω. Remark 2.1. We use the following steps in the construction of η̃ε, taking a cutoff function η0 in B(0, 2) or in B(x0, 2): (1) We set ηε(x) = η0(|x− x0|/ε) in the case of the unit disk it is sufficient. (2) Or, in the general case: we use a chart (f, Ω̃) with f(0) = x0 and we take µε(x) = η0(f(|x|/ε)) to have connected sets Iε and we take ηε(y) = µε(f −1(y)). Because f, f−1 are Lipschitz, |f(x)−x0| ≤ k2|x| ≤ 1 for |x| ≤ 1/k2 and |f(x)−x0| ≥ k1|x| ≥ 2 for |x| ≥ 2/k1 > 1/k2, the support of η is in I(2/k1)ε. ηε ≡ 1, on f(I(1/k2)ε), 0 < ε < δ/2, ηε ≡ 0, outside f(I(2/k1)ε), 0 ≤ ηε ≤ 1, ‖∇ηε‖L∞(I(2/k1)ε) ≤ C0(Ω, x0) ε . (3) Also, we can take: µε(x) = η0(|x|/ε) and ηε(y) = µε(f −1(y)), we extend it by 0 outside f(B1(0)). We have f(B1(0)) = D1(x0), f(Bε(0)) = Dε(x0) and f(B+ ε ) = D+ ε (x0) with f and f−1 smooth diffeomorphism. ηε ≡ 1, on the connected set Jε = f(Iε), 0 < ε < δ/2, ηε ≡ 0, outside J ′ε = f(I2ε), 0 ≤ ηε ≤ 1, ‖∇ηε‖L∞(J′ ε) ≤ C0(Ω, x0) ε . And H1(J ′ε) ≤ C1H1(I2ε) = C14ε, because f is Lipschitz. Here H1 is the Hausdorff measure. We solve the Dirichlet Problem ∆η̄ε = ∆ηε in Ω ⊂ R2, η̄ε = 0 in ∂Ω. and finally we set η̃ε = −η̄ε + ηε. Also, by the maximum principle and the elliptic estimates we have ‖∇η̃ε‖L∞ ≤ C(‖ηε‖L∞ + ‖∇ηε‖L∞ + ‖∆ηε‖L∞) ≤ C1 ε2 , with C1 depending on Ω. As we said in the beginning, see also [3, 7, 13, 20], we have ‖∇ui‖Lq ≤ Cq, ∀i, 1 < q < 2. We deduce from the above estimate that, (ui) converge weakly in W 1,q 0 (Ω), almost everywhere to a function u ≥ 0 and ∫ Ω eu < +∞ (by Fatou lemma). Also, Vi 6 S. S. BAHOURA EJDE-2022/14 weakly converge to a nonnegative function V in L∞. The function u is in W 1,q 0 (Ω) solution of −∆u = V eu ∈ L1(Ω) in Ω ⊂ R2, u = 0 in ∂Ω. According to [6, Ccorollary 1], we have eku ∈ L1(Ω), k > 1. By the elliptic esti- mates, we have u ∈W 2,k(Ω) ∩ C1,ε(Ω̄). We denote by f · g the inner product of any two vectors f and g of R2. Then we can write −∆((ui − u)η̃ε) = (Vie ui − V eu)η̃ε − 2∇(ui − u) · ∇η̃ε + εi(∇ui · x)η̃ε. (2.4) We use the interior estimate in Brezis-Merle [6]. Step 1: Estimate of the integral of the first term of the right-hand side of (2.4). We use Green’s formula between η̃ε and u, to obtain∫ Ω V euη̃ε dx = ∫ ∂Ω ∂νuηε ≤ Cε = O(ε) (2.5) then we have −∆ui − εi∇ui · x = Vie ui in Ω ⊂ R2, u = 0 in ∂Ω. We use Green’s formula between ui and η̃ε to have∫ Ω Vie ui η̃ε dx = ∫ ∂Ω ∂νuiηεdσ − εi ∫ Ω (∇ui · x)η̃ε = ∫ ∂Ω ∂νuiηεdσ + o(1) → µ(ηε) ≤ µ(J ′ε) ≤ 4π − ε0, ε0 > 0 (2.6) From (2.5) and (2.6) we have that for all ε > 0 there is i0 such that, for i ≥ i0,∫ Ω |(Vieui − V eu)η̃ε| dx ≤ 4π − ε0 + Cε (2.7) Step 2.1: Estimate of integral of the second term of the right hand side of (2.4). Let Σε = {x ∈ Ω, d(x, ∂Ω) = ε3} and Ωε3 = {x ∈ Ω, d(x, ∂Ω) ≥ ε3}, ε > 0. Then, for ε small enough, Σε is an hypersurface. The measure of Ω− Ωε3 is k2ε 3 ≤ meas(Ω− Ωε3) = µL(Ω− Ωε3) ≤ k1ε 3. Remark 2.2. For the unit ball B̄(0, 1), our new manifold is B̄(0, 1− ε3). To prove this fact, we consider consider d(x, ∂Ω) = d(x, z0), z0 ∈ ∂Ω, which implies that (d(x, z0))2 ≤ (d(x, z))2 for all z ∈ ∂Ω. This is equivalent to (z−z0)·(2x−z−z0) ≤ 0 for all z ∈ ∂Ω. Let us consider a chart around z0 and γ(t) a curve in ∂Ω, we have (γ(t)−γ(t0) · (2x−γ(t)−γ(t0)) ≤ 0 if we divide by (t− t0) (with the sign and tend t to t0), we have γ′(t0) · (x− γ(t0)) = 0. This implies that x = z0 − sν0 where ν0 is the outward normal of ∂Ω at z0) From the above remark, we can say that S = {x, d(x, ∂Ω) ≤ ε} = {x = z0 − sνz0 , z0 ∈ ∂Ω, −ε ≤ s ≤ ε}. It is sufficient to work on ∂Ω. Let us consider charts (z,D = B(z, 4εz), γz) with z ∈ ∂Ω such that ∪zB(z, εz) is cover of ∂Ω . One can extract a finite cover EJDE-2022/14 ELLIPTIC EQUATIONS IN 2 DIMENSIONS 7 (B(zk, εk)), k = 1, . . . ,m, by the area formula the measure of S ∩ B(zk, εk) is less than a kε (a ε-rectangle). For the reverse inequality, it is sufficient to consider one chart around one point of the boundary). We write∫ Ω |∇(ui−u)·∇η̃ε| dx = ∫ Ωε3 |∇(ui−u)·∇η̃ε| dx+ ∫ Ω−Ωε3 |∇(ui−u)·∇η̃ε| dx. (2.8) Step 2.1.1: Estimate of ∫ Ω−Ωε3 |∇(ui − u) · ∇η̃ε| dx. First, we know from elliptic estimates that ‖∇η̃ε‖L∞ ≤ C1/ε 2, C1 depends on Ω. We know that (|∇ui|)i is bounded in Lq, 1 < q < 2, we can extract from this sequence a subsequence which converge weakly to h ∈ Lq. But, we know that we have locally the uniform convergence to |∇u| (by Brezis-Merle’s theorem), then, h = |∇u| a.e. Let q′ be the conjugate of q. We have that for all f ∈ Lq′(Ω),∫ Ω |∇ui|f dx→ ∫ Ω |∇u|f dx If we take f = 1Ω−Ωε3 , for each ε > 0 there exists i1 = i1(ε) ∈ N, such that i ≥ i1 implies ∫ Ω−Ωε3 |∇ui| ≤ ∫ Ω−Ωε3 |∇u|+ ε3. Then, for i ≥ i1(ε),∫ Ω−Ωε3 |∇ui| ≤ meas(Ω− Ωε3)‖∇u‖L∞ + ε3 = ε3(k1‖∇u‖L∞ + 1) = O(ε3). Thus, we obtain∫ Ω−Ωε3 |∇(ui − u) · ∇η̃ε| dx ≤ εC1(2k1‖∇u‖L∞ + 1) = O(ε) (2.9) The constant C1 does not depend on ε but on Ω. Step 2.1.2: Estimate of ∫ Ωε3 |∇(ui − u) · ∇η̃ε| dx. We know that, Ωε ⊂⊂ Ω, and (because of Brezis-Merle’s interior estimates) ui → u in C1(Ωε3). We have ‖∇(ui − u)‖L∞(Ωε3 ) ≤ ε3, for i ≥ i3. We write∫ Ωε3 |∇(ui − u) · ∇η̃ε| dx ≤ ‖∇(ui − u)‖L∞(Ωε3 )‖∇η̃ε‖L∞ = C1ε = O(ε) for i ≥ i3. For ε > 0, and i ∈ N, with i ≥ i′, we have∫ Ω |∇(ui − u) · ∇η̃ε| dx ≤ εC1(2k1‖∇u‖L∞ + 2) = O(ε) (2.10) From (2.7) and (2.10), for ε > 0, there is i′′ such that i ≥ i′′, we have∫ Ω |∆[(ui−u)η̃ε]|dx ≤ 4π−ε0 +ε2C1(2k1‖∇u‖L∞ +2+C) = 4π−ε0 +O(ε) (2.11) Now we choose ε > 0 small enough to have a good estimate of (2.4). Indeed, we have −∆[(ui − u)η̃ε] = gi,ε textinΩ ⊂ R2, (ui − u)η̃ε = 0 in ∂Ω. 8 S. S. BAHOURA EJDE-2022/14 with ‖gi,ε‖L1(Ω) ≤ 4π − ε0/2. We can use [6, Theorem 1] to conclude that there are q ≥ q̃ > 1 such that∫ Vε(x0) eq̃|ui−u| dx ≤ ∫ Ω eq|ui−u|η̃ε dx ≤ C(ε,Ω), where, Vε(x0) is a neighborhood of x0 in Ω̄. Here we have used that in a neighbor- hood of x0 by the elliptic estimates, 1− Cε ≤ η̃ε ≤ 1. Thus, for each x0 ∈ ∂Ω− {x̄1, . . . , x̄m} there is ε0 > 0, q0 > 1 such that∫ B(x0,ε0) eq0ui dx ≤ C, ∀i. By elliptic estimates see [14], we have ‖ui‖C1,θ[B(x0,ε)] ≤ c3 ∀i. We have proved that there is a finite number of points x̄1, . . . , x̄m such that the sequence (ui) is locally uniformly bounded in C1,θ, (θ > 0) on Ω̄− {x̄1, . . . , x̄m}. Proof of theorem 1.5. The Pohozaev identity gives∫ ∂Ω 1 2 (x·ν)(∂νui) 2dσ+ε ∫ Ω (x·∇ui)2 dx+ ∫ ∂Ω (x·ν)Vie uidσ = ∫ Ω (x·∇Vi+2Vi)e ui dx . We use the boundary condition, that Ω is starshaped, and that ε > 0 to have∫ ∂Ω (∂νui) 2 dx ≤ c0(b, A,C,Ω). (2.12) Thus we can use the weak convergence in L2(∂Ω) to have a subsequence ∂νui, such that ∫ ∂Ω ∂νuiφdx→ ∫ ∂Ω ∂νuφ dx, ∀φ ∈ L2(∂Ω), Thus, αj = 0, j = 1, . . . , N and (ui) is uniformly bounded. Remark 2.3. If we assume the open set bounded starshaped and Vi uniformly Lipschitzian and between two positive constants we can bound, by using the inner normal derivative ∫ Ω eui . If we assume the open set bounded starshaped and ∇ log Vi uniformly bounded, by the previous Pohozaev identity (we consider the inner normal derivative) one can bound ∫ Ω Vie ui uniformly. One can consider the problem on the unit ball and an ellipse. These two problems are different, because: (1) if we use a linear transformation, (y1, y2) = (x1/a, x2/b), the Laplcian is not invariant under this map. (2) If we use a conformal transformation, by a Riemann theorem, the quantity x · ∇u is not invariant under this map. We can not use, after using those transformations, the Pohozaev identity. EJDE-2022/14 ELLIPTIC EQUATIONS IN 2 DIMENSIONS 9 3. A counterexample We start with the notation of the counterexample of Brezis and Merle. The domain Ω is the unit ball centered in x0 = (1, 0). Consider zi (obtained by the variational method), such that −∆zi − εi(x− x0) · ∇zi = −L̃εi(zi) = fεi , with Dirichlet condition. By the regularity theorem, zi ∈ C1(Ω̄). Then we have ‖fεi‖1 = 4πA. Thus by the duality theorem of Stampacchia or Brezis-Strauss, we have ‖∇zi‖q ≤ Cq, 1 ≤ q < 2. We solve −∆wi = εi(x− x0) · ∇zi, with Dirichlet boundary condition. By elliptic estimates, wi ∈ C1(Ω̄) and wi ∈ C0(Ω̄) uniformly. By the maximum principle we have zi − wi ≡ ui. Where ui is the function of the counterexemple of Brezis Merle. Then we write −∆zi − εi(x− x0) · ∇zi = fεi = Vie zi . Thus, we have ∫ Ω ezi ≤ C1, 0 ≤ Vi ≤ C2, zi(ai) ≥ ui(ai)− C3 → +∞, ai → O . To have a counterexample on the unit disk, we do a translation x→ x− x0 in the previous counterexample. Acknowledgments. 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Phys. 37 (1996), no. 8, 3769–3796. Samy Skander Bahoura Department of Mathematics, Pierre et Marie Curie University, 4 Place Jussieu, 75005, Paris, France Email address: samybahoura@gmail.com 1. Introduction Brezis-Merle Problem b5 2. Proofs of main results Proof of Theorem ?? Proof of theorem ?? 3. A counterexample Acknowledgments References