DSpace at VNU: Solutions of elliptic problems of p-Laplacian type in a cylindrical symmetric domain tài liệu, giáo án, b...
Acta Math Hungar., 135 (1–2) (2012), 42–55 DOI: 10.1007/s10474-011-0163-6 First published online October 6, 2011 SOLUTIONS OF ELLIPTIC PROBLEMS OF p-LAPLACIAN TYPE IN A CYLINDRICAL SYMMETRIC DOMAIN N T CHUNG1,∗ and H Q TOAN2 Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Vietnam e-mail: ntchung82@yahoo.com Faculty of Mathematics, Mechanics and Informatics, University of Natural Sciences, Vietnam National University, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam e-mail: hq toan@yahoo.com (Received February 27, 2011; revised May 16, 2011; accepted May 16, 2011) Abstract We consider the p-Laplacian type elliptic problem − div (a(x, ∇u)) = h(x)|u|q−2 u + g(x) u=0 in Ω, in ∂Ω, where Ω = Ω1 × Ω2 ⊂ RN is a bounded domain having cylindrical symmetry, Ω1 ⊂ Rm is a bounded regular domain and Ω2 is a k-dimensional ball of radius R, centered in the origin and m + k = N , m 1, k Under some suitable conditions on the functions a and h, using variational methods we prove that the problem has at least one resp at least two solutions in two cases: g = and g = Introduction In this article, we are concerned with nonlinear elliptic equations in which the divergence form operator − div a(x, ∇u) is involved Such operators appear in many nonlinear diffusion problems, in particular the mathematical modeling of non-Newtonian fluids The p-Laplacian operator Δp u = − div |∇u|p−2 ∇u is a special case of the operator − div a(x, ∇u) We refer the readers to some recent works [8,12,13] on elliptic problems of p-Laplacian type in the general case ∗ Corresponding author Key words and phrases: elliptic problem, p-Laplacian type, cylindrical symmetric domain, mountain pass theorem 2010 Mathematics Subject Classification: 35J65, 35J20 0236-5294/$ 20.00 c 2011 Akad´ emiai Kiad´ o, Budapest, Hungary SOLUTIONS OF ELLIPTIC PROBLEMS OF p-LAPLACIAN TYPE 43 In [8], P De N´apoli et al studied the Dirichlet problem of the form − div a(x, ∇u) = f (x, u) in Ω, (1.1) u=0 on ∂Ω, where Ω ⊂ RN (N 3) is a bounded domain with smooth boundary, the function a : Ω × RN → RN satisfies the condition (1.2) a(x, ξ) C + |ξ|p−1 for all ξ ∈ RN , a.e x ∈ Ω Under condition of Ambrosetti–Rabinowitz type, i.e., there exists μ > p such that t < μF (x, t) := μ f (x, s) ds f (x, t)t for all t ∈ R\{0}, and a.e x ∈ Ω, the authors showed the existence of at least one solution for (1.1) using the arguments of classical mountain pass type [1] In [12], A Krist´ aly et al studied (1.1) with condition (1.2) in the case when f (x, t) = λf (t) is (p − 1)-sublinear at infinity, i.e., lim t→∞ f (t) = |t|p−1 It is worth observing that in this case the nonlinear term f does not satisfy the Ambrosetti–Rabinowitz type condition Using the three critical point theorem by G Bonanno [4], the authors proved that problem (1.1) has at least three weak solutions If Ω = RN , problem (1.1) with condition (1.2) was studied in [13], in which M Mih˘ ailescu gave some sufficient conditions on the nonlinearities to obtain some existence and multiplicity results In a recent paper, D M Duc et al [10] have studied (1.1) in the case when the function a satisfies (1.3) a(x, ξ) C θ(x) + σ(x)|ξ|p−1 for all ξ ∈ RN , a.e x ∈ Ω, in which θ and σ are two non-negative measurable functions satisfying p p−1 θ ∈ L (Ω), σ ∈ L1loc (Ω), and σ(x) for a.e x ∈ Ω It is clear that condition (1.3) is weaker than (1.2) This interesting idea comes from the mountain pass theorem for weakly continuously differentiable functionals in [9] Using this result, they then showed in [10] that problem (1.1) has at least one weak solution provided that the functions a and f satisfy some further suitable conditions Regarding some extensions of [10], the readers may consult in [7,14–16,19], in which the authors studied the existence of solutions for problem (1.1) with condition (1.3) and subcritical nonlinearities relying essentially on the compactness embedding W01,p (Ω) → Lq (Ω), Acta Mathematica Hungarica 135, 2012 44 N T CHUNG and H Q TOAN p q < p = NN−p If q = p (critical case) or q > p (superciritical case) the compactness of this embedding is no longer valid, so variational arguments cannot be applied as usual Generally speaking, some kinds of geometric and topological properties of the domain lead to the solvability of elliptic problems; for example, the symmetry of the domain could improve Sobolev embeddings Motivated by the recent results on the effect of the topology of the domain in [5,6,20], in the present work, we will investigate (1.1) with condition (1.3) in the case when Ω = Ω1 × Ω2 ⊂ RN is a bounded domain having cylindrical symmetry, Ω1 ⊂ Rm is a bounded regular domain and Ω2 is a k-dimensional ball of radius R, centered in the origin and m + k = N , and m 1, k Using variational methods, we prove some existence and multiplicity results for (1.1) in the critical and supercritical cases q p So the results introduced here are better than those of [10] and its previous extensions Let < p < N , and consider a : Ω × RN → RN , a = a(x, ξ) the continuous derivative with respect to ξ of the continuous function A : Ω × RN → R, A = A(x, ξ), that is, a(x, ξ) = ∂A(x,ξ) ∂ξ Suppose that a and A satisfy the following hypotheses: (A1) a(x, 0) = A(x, 0) = for a.e x ∈ Ω (A2) a(x, ξ) C θ(x) + σ(x)|ξ|p−1 for all ξ ∈ RN , a.e x ∈ Ω, where p θ ∈ L p−1 (Ω), σ ∈ L1loc (Ω), θ(x) 0, and σ(x) for a.e x ∈ Ω (A3) There exists k0 > such that A x, ξ+ψ 1 A(x, ξ) + A(x, ψ) − k0 σ(x)|ξ − ψ|p 2 for all ξ, ψ ∈ RN , a.e x ∈ Ω, that is, A is p-uniformly convex (A4) a(x, ξ) · ξ pA(x, ξ) holds for all ξ ∈ RN , a.e x ∈ Ω (A5) There exists k1 > such that A(x, ξ) k1 σ(x)|ξ|p for all ξ ∈ RN , a.e x ∈ Ω; There are many functions a : Ω × RN → RN satisfying all conditions (A1)–(A5), see for example [7,10] Let W01,p (Ω) be the usual Sobolev space under the norm u 1,p = ( Ω |∇u| p dx) p and define the subspace 1,p (Ω) = {u ∈ W01,p (Ω) : u(x1 , x2 ) = u x1 , |x2 | , ∀ x = (x1 , x2 ) ∈ Ω}, W0,s which is a closed subspace of W01,p (Ω) Now, we introduce the space 1,p (Ω) : X = u ∈ W0,s Acta Mathematica Hungarica 135, 2012 Ω σ(x)|∇u|p dx < ∞ SOLUTIONS OF ELLIPTIC PROBLEMS OF p-LAPLACIAN TYPE 45 With the method as those used in [10], we can show that X is a Banach space with respect to the norm u X p = p σ(x)|∇u| dx Ω 1,p Moreover, the continuous embedding X → W0,s (Ω) holds true since σ(x) for a.e x ∈ Ω Firstly, we consider problem (1.1) in the case when f (x, u) = h(x)|u|q−2 u, then the problem becomes − div a(x, ∇u) = h(x)|u|q−2 u in Ω, (1.4) u=0 on ∂Ω Throughout this paper, we always assume that h : [0, ) is a Hăolder continuous function satisfying: (H1) h(x1 , x2 ) = h x1 , |x2 | for all x = (x1 , x2 ) ∈ Ω = Ω1 × Ω2 , h(x1 , 0) = for all x1 ∈ Ω1 ; (H2) lh > 0, where lh := sup λ > : sup x∈Ω h(x) |x2 |λ such that for any g ∈ −1,p (Ω) with < g −1 < ε, problem (1.5) has at least two nontrivial weak W0,s solutions Since q ∈ (p, p + τ ) with the number τ defined by Theorem 1.2, problems (1.4) and (1.5) contain the subcritical, critical and supercritical cases The paper is organized as follows In Section 2, we give some preliminaries and in Section 3, we prove the main theorems Preliminaries In this section, we give some useful results which are used in the proofs of our main theorems Firstly, the following result concerning the function A may be found in [8,10] Lemma 2.1 Assume that all conditions (A1)–(A5) are satisfied Then: (i) The function A satisfies the growth condition A(x, ξ) c0 θ(x)|ξ| + σ(x)|ξ|p for all ξ ∈ RN and a.e x ∈ Ω; (ii) A(x, tξ) A(x, ξ)tp for all t 1, and for all ξ ∈ RN , a.e x ∈ Ω Define the functionals Λ, I and J : X → R by Λ(u) = Ω A(x, ∇u) dx, Acta Mathematica Hungarica 135, 2012 I(u) = q h(x)|u|q dx + Ω g(x)u dx, Ω SOLUTIONS OF ELLIPTIC PROBLEMS OF p-LAPLACIAN TYPE 47 and J(u) = Λ(u) − I(u) for all u ∈ X Due to the presence of σ ∈ L1loc (Ω), the functional Λ (and thus J ) may not belong to C (X, R) So, we cannot apply the mountain pass theorem by A Ambrosetti and P H Rabinowitz [1] To overcome this difficulty, we need to recall a useful concept in [9] (see also in [10,19]) as follows Definition 2.2 Let Y be a Banach space We say that a map F : Y → R is weakly continuously differentiable on Y if and only if (i) for any u ∈ Y , there exists a linear map DF(u) from Y into R such that F(u + tϕ) − F(u) = DF(u)(ϕ), for all ϕ ∈ Y ; lim t→0 t (ii) for any ϕ ∈ Y , the map u → DF(u)(ϕ) is continuous on Y Remark 2.3 If we suppose further that for every u ∈ Y , the map ϕ → DF(u)(ϕ) is continuous and linear on Y , then F is Gˆ ateaux differentiable on Y For information and connection on functionals that are not Gˆateaux differentiable but only differentiable along directions of a certain subspace of Y , we refer to [2] Denote by Cw1 (Y, R) the set of weakly continuously differentiable functionals on Y Then, it is clear that C (Y, R) ⊂ Cw1 (Y, R), where C (Y, R) is the set of all continuously Fr´echet differentiable functionals on Y Now let F ∈ Cw1 (Y, R) We put for any u ∈ Y , DF(u) Y = sup { DF (u)(h) , h ∈ Y with h Y = 1}, where DF(u) Y may be ∞ We say that F satisfies the Palais–Smale condition at level c ∈ R (denote by (PS)c ) if any sequence {um } ⊂ Y for which F(um ) → c and DF(um ) → in Y , possesses a convergent subsequence If this is true at every level c then we simply say that F satisfies the Palais–Smale condition (denote by (PS)) on Y Hence, our idea is to obtain the existence of solutions for problem (1.1) using the weak version of the mountain pass theorem for weakly continuously differentiable functionals by D M Duc [9] In our arguments, we need the following results whose proofs are standard and simple Lemma 2.4 Assume that all conditions (A1)–(A5) are satisfied Then: 1,p (i) The functional Λ is weakly lower semicontinuous in W0,s (Ω), i.e., if 1,p {um } converges weakly to u in W0,s (Ω) then we have Λ(u) lim inf Λ(um ) m→∞ Acta Mathematica Hungarica 135, 2012 48 N T CHUNG and H Q TOAN (ii) It holds that 1 u+v Λ(u) + Λ(v) − Λ 2 k0 u − v p X for all u, v ∈ X The following result is a natural generalization of W Wang [20, Theorem 2.7], whose proof is standard so we omit it Theorem 2.5 Assume that Ω = Ω1 × Ω2 , dim (Ω1 ) = m 1, dim (Ω2 ) older continuous function satisfying the = k 2, and h : Ω → [0, ∞) is a Hă conditions (H1), (H2) Then, there is a positive number τ = τ (h, p, m, k) depending on h, p, m, k (well-defined) such that the embedding 1,p W0,s (Ω) → Lr (h, Ω), r ∈ (1, p + τ ), is compact, where Lr (h, Ω) is the usual Lebesgue space with weight h Proof of the main results In our arguments, the proof of Theorem 1.4 contains the existence result which is stated as in Theorem 1.2 So, for the sake of brevity, we will deal only with problem (1.5) in details As we mentioned in Section 2, in order to prove our main results, we will use critical point theory Let us define the functional J : X → R by (3.1) J(u) = Ω A(x, ∇u) dx − q Ω h(x)|u|q dx − Ω gu dx, = Λ(u) − I(u), where (3.2) Λ(u) = Ω A(x, ∇u) dx, I(u) = q h(x)|u|q dx + Ω gu dx Ω for all u ∈ X By Theorem 2.5, for any q ∈ (p, p + τ ), similar arguments as in [10] ensure that the functional J is well-defined and weakly continuously differentiable on X Moreover, we have DJ(u)(ϕ) = Ω a(x, ∇u) · ∇ϕ dx − Ω h(x)|u|q−2 uϕ dx − gϕ dx Ω for all u, ϕ ∈ X Thus, weak solutions of problem (1.5) are exactly the critical points of the functional J Lemma 3.1 The functional J is weakly lower semiconituous on X Acta Mathematica Hungarica 135, 2012 SOLUTIONS OF ELLIPTIC PROBLEMS OF p-LAPLACIAN TYPE 49 Proof Let a sequence {um } ⊂ X be weakly convergent to u in X 1,p Since the embedding X → W0,s (Ω) is continuous, {um } converges weakly 1,p to u in W0,s (Ω) By Lemma 2.4, it follows that (3.3) lim inf Λ(um ) Λ(u) m→∞ On the other hand, Theorem 2.5 and the fact that {um } converges weakly 1,p to u in W0,s (Ω) imply that {um } converges strongly to u in Lq (h, Ω) for all q ∈ (p, p + τ ) Therefore, (3.4) lim m→∞ Ω h(x)|um |q dx = h(x)|u|q dx Ω 1,p We also deduce since {um } converges weakly to u in W0,s (Ω) that (3.5) lim m→∞ Ω gum dx = gu dx Ω Combining (3.4) and (3.5), we have (3.6) lim I(um ) = I(u) m→∞ Finally, we conclude from (3.3) and (3.6) that lim inf J(um ) = lim inf Λ(um ) − I(um ) m→∞ Λ(u) − I(u) = J(u) m→∞ and thus, the functional J is weakly lower semicontinuous on X The following lemma shows that the functional J has the geometry of the mountain pass theorem Lemma 3.2 (i) There exist ε > 0, η > 0, and α > such that J(u) α for all u ∈ X with u X = η , provided that < g −1 < ε (ii) For η > as in (i), there exists e ∈ X such that e X > η and J(e) < Proof (i) For any ε > 0, using Young’s inequality we deduce that gu dx g Ω ε u p p X −1 + u pε (ε 1,p p p g p p −1 , u X ) 1 εp g −1 1 + =1 p p for all u ∈ X Acta Mathematica Hungarica 135, 2012 50 N T CHUNG and H Q TOAN Hence, by Theorem 2.5, it follows that for any ε > and for all u ∈ X, (3.7) J(u) = Ω Ω A(x, ∇u) dx − k1 σ(x)|∇u|p dx − = k1 − q Ω u qSqq ε − u p qSqq h(x)|u|q dx − q X q−p X − ε u p p X u p X − pε pε − gu dx Ω p p g g p p p −1 p −1 1,p where Sq is the best constant in the embedding W0,s (Ω) → Lq (h, Ω) By fixing ε ∈ (0, k1 p), we can find η > 0, ε > and α > such that the conclusion of the lemma holds true For example, we can take η= M qSqq where M = q−p , ε= p p εp q M p (q−p) qSqq q p (q−p) , q α = M q−p qSqq (k1 − pε ) > 1,p (ii) Let ϕ0 ∈ C0∞ (Ω) ∩ W0,s (Ω) such that for any t > we obtain by Lemma 2.1 that J(tϕ0 ) = Ω Ω A(x, t∇ϕ0 ) dx − A(x, ∇ϕ0 ) dx − tq q tq q Ω Ω Ω h(x) ϕ0 (x) h(x)|ϕ0 |q dx − t h(x)|ϕ0 |q dx − t q q q−p , dx > Then gϕ0 dx Ω gϕ0 dx, Ω which approaches −∞ as t → +∞ since q > p > Lemma 3.3 The functional J satisfies the Palais–Smale condition in X Proof Let {um } be a Palais–Smale sequence for the functional J in X, i.e (3.8) J(um ) → c, DJ(um ) → in X as m → ∞, where X is the dual space of X In order to prove that J satisfies the Palais–Smale condition, we first prove that {um } has a bounded subsequence Indeed, assume by contradiction that um X → ∞ as m → ∞ Then, using relation (3.8) and (A4)–(A5), we deduce that for m large enough the following inequalities hold: (3.9) c + + um X Acta Mathematica Hungarica 135, 2012 J(um ) − DJ(um )(um ) q 51 SOLUTIONS OF ELLIPTIC PROBLEMS OF p-LAPLACIAN TYPE Ω 1 A(x, ∇um ) − a(x, ∇um ) · ∇um dx − − q q k1 − p q um p X q − 1− g um −1 gum dx Ω X Dividing the above inequality by um pX and letting m → ∞ we obtain a contradiction since < p < q This implies that {um } has a bounded subse1,p (Ω) is continuous, {um } is quence, still denoted by {um } Since X → W0,s 1,p 1,p bounded in W0,s (Ω) and thus there exists u ∈ W0,s (Ω) such that, passing to 1,p a subsequence, still denoted by {um }, it converges weakly to u in W0,s (Ω) By (3.8), it follows from Lemma 2.4(i), Theorem 2.5 that σ(x)|∇u|p dx = k1 (3.10) Ω Ω lim inf Λ(um ) = lim m→∞ m→∞ A(x, ∇u) dx = Λ(u) I(um ) + J(um ) = I(u) + c < ∞, um − u X is bounded which yields that u ∈ X and then the sequence Combining this with (3.8), we conclude that DJ(um )(um − u) → as m → ∞ On the other hand, using Hă olders inequality we have h(x) |un |q−2 u − |u|q−2 u (un − u) dx Ω Ω = Ω (h q (x)|un |) q−1 ( h(x) |un |q−1 + |u|q−1 |un − u| dx h q (x)|un − u| dx + un q−1 Lq (h,Ω) + u Ω q−1 Lq (h,Ω) (h ) q q−1 (x)|u|) un − u h q (x)|un − u| dx Lq (h,Ω) This implies using Theorem 2.5 that (3.11) lim n→∞ Ω h(x) |un |q−2 u − |u|q−2 u (un − u) dx = 1,p (Ω), that Moreover we deduce, since {um } converges weakly to u in W0,s (3.12) lim m→∞ Ω g(um − u) dx = Acta Mathematica Hungarica 135, 2012 52 N T CHUNG and H Q TOAN Relations (3.11) and (3.12) imply that (3.13) lim DI(um )(um − u) = lim m→∞ m→∞ Ω + lim m→∞ Ω h(x) |un |q−2 u − |u|q−2 u (un − u) dx g(um − u) dx = By Lemma 2.4(ii), the functional Λ is convex So, the relation lim DΛ(um )(um − u) = lim DI(um )(um − u) + lim DJ(um )(um − u) = m→∞ m→∞ m→∞ helps us to get Λ(u) − lim sup Λ(um ) = lim inf Λ(u) − Λ(um ) m→∞ m→∞ lim DΛ(um )(u − um ) = m→∞ and thus, (3.14) lim Λ(um ) = Λ(u) m→∞ Now, we will show that the sequence {um } converges strongly to u in X Indeed, assume by contradiction that {um } is not strongly convergent to u in X Then there exist a constant ε0 > and a subsequence of {um }, still denoted by {um } such that um − u X ε0 for any m = 1, 2, Hence, by Lemma 2.4(ii) we have (3.15) um + u Λ(um ) + Λ(u) − Λ 2 On the other hand, notice that the sequence in X, so using Lemma 2.4(i) again we have (3.16) lim inf Λ Λ(u) m→∞ k0 u m − u {u +u m Λ(u) − lim sup Λ um + u m→∞ which contradicts (3.16) Acta Mathematica Hungarica 135, 2012 um + u k0 εp0 } converges weakly to u It follows from (3.15) by letting m → ∞ that (3.17) p X k0 εp0 , SOLUTIONS OF ELLIPTIC PROBLEMS OF p-LAPLACIAN TYPE 53 Therefore, the sequence {um } converges strongly to u in X and thus the functional J satisfies the Palais–Smale condition in X Proof of Theorem 1.4 By Lemmas 3.2 and 3.3, all assumptions of the mountain pass theorem in [9] are satisfied Hence we deduce the existence of u1 ∈ X as a non-trivial weak solution of (1.5) and J(u1 ) = c > We now prove that there exists a second weak solution u2 ∈ X such that u2 ≡ u1 For η > given as in Lemma 3.2, we define the number c by (3.18) c := inf {u∈X: u X η} J(u) Then we have c J(0) = It is clear that if c = J(0) then is a minimum point of J and thus it implies from (A1) that DJ(0)(ϕ) = − ϕ ∈ X, gϕ dx = for all Ω which contradicts the fact that g ≡ Therefore, c < J(0) = Denote by B η (0) the closed ball of radius η centered at the origin in X, i.e B η (0) := u ∈ X : u X η , it follows that the set B η (0) is a complete metric space with respect to the distance dist (u, v) := u − v X for all u, v ∈ B η (0) By Lemma 3.1, the functional J is weakly lower semi-continuous and bounded from below in X since relation (3.7) holds true Let ε be such that < ε < inf ∂Bη (0) J − inf Bη (0) J Applying Ekeland’s variational principle [11] for the functional J : B η (0) → R, there exists a function uε ∈ B η (0) such that J(uε ) < inf J + ε, B η (0) J(uε ) < J(u) + ε u − uε X, u = uε Since J(uε ) inf J + ε B η (0) inf J + ε < inf J Bη (0) ∂Bη (0) it follows that uε ∈ Bη (0) We now define the functional K : B η (0) → R by K(u) = J(u) + ε u − uε X Acta Mathematica Hungarica 135, 2012 54 N T CHUNG and H Q TOAN It is clear that uε is a minimum point of K and thus, (3.19) K(uε + tϕ) − K(uε ) t for t > small enough and ϕ ∈ Bη (0) Relation (3.19) yields (3.20) J(uε + tϕ) − J(uε ) +ε ϕ t X It follows from (3.20) by letting t → 0+ that DJ(uε )(ϕ) + ε ϕ X It should be noticed that −ϕ also belongs to Bη (0), so replacing ϕ by −ϕ, we get DJ(uε )(−ϕ) + ε − ϕ X or DJ(uε )(ϕ) ε ϕ X , which helps us to deduce that DJ(uε ) −1 ε From the above information, there exists a sequence {um } ⊂ Bη (0) such that (3.21) J(um ) → c and DJ(um ) → in X as m → ∞ Using Lemma 3.3, we can show that {um } converges strongly to some u2 ∈ X Thus, u2 is a weak solution of (1.5) and u2 is non-trivial since J(u2 ) = c < Finally, 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W0,s Acta Mathematica Hungarica 135, 2012 Ω σ(x)|∇u|p dx < ∞ SOLUTIONS OF ELLIPTIC PROBLEMS OF p-LAPLACIAN TYPE 45 With the method as those used in [10], we can show that X is a Banach space