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Abstract and Applied Analysis On an asymptotically linear elliptic Dirichlet problem
On an asymptotically linear elliptic Dirichlet problem
Zhang, Zhitao, Li, Shujie, Liu, Shibo, Feng, Weijie이 책이 얼마나 마음에 드셨습니까?
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7
년:
2002
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english
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Abstract and Applied Analysis
DOI:
10.1155/s1085337502207046
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ON AN ASYMPTOTICALLY LINEAR ELLIPTIC DIRICHLET PROBLEM ZHITAO ZHANG, SHUJIE LI, SHIBO LIU, AND WEIJIE FENG Received 2 July 2002 Under very simple conditions, we prove the existence of one positive and one negative solution of an asymptotically linear elliptic boundary value problem. Even for the resonant case at infinity, we do not need to assume any more conditions to ensure the boundness of the (PS) sequence of the corresponding functional. Moreover, the proof is very simple. 1. Introduction In this paper, we consider the existence of onesigned solutions for the following Dirichlet problem: −∆u = f (x,u), u = 0, x ∈ Ω, x ∈ ∂Ω, (1.1) where Ω is a bounded domain in RN (N ≥ 1) with smooth boundary ∂Ω. The conditions imposed on f (x,t) are as follows: (f1 ) f ∈ C(Ω × R, R); f (x,0) = 0, for all x ∈ Ω. (f2 ) limt→0 ( f (x,t)/t) = µ, limt→∞ ( f (x,t)/t) = uniformly in x ∈ Ω. Since we assume (f2 ), problem (1.1) is called asymptotically linear at both zero and infinity. This kind of problems have captured great interest since the pioneer work of [1]. For more information, see [2, 3, 4, 5, 6, 7, 8, 11, 12] and the references therein. Obviously, the constant function u = 0 is a trivial solution of problem (1.1). Therefore, we are interested in finding nontrivial solutions. Let F(x,u) = u 0 f (x,s)ds. It follows from (f1 ) and (f2 ) that the functional 1 J(u) = 2 2 Ω ∇u dx − Copyright © 2002 Hindawi Publishing Corporation Abstract and Applied Analysis 7:10 (2002) 509–516 2000 Mathematics Subject Classification: 35J65 URL: http://dx.doi.org/10.1155/S1085337502207046 Ω F(x,u)dx (1.2) 510 On an asymptotically linear elliptic Dirichlet problem is of class C 1 on the Sobolev space H01 := H01 (Ω) with norm u := 1/2 Ω ∇u2 , (1.3) and the critical points of J are solutions of (1.1). Thus we will try to find critical points of J. In doing so, we have to prove that the functional J satisfies the (PS) condition. We denote by 0 < λ1 < λ2 ≤ λ3 ≤ · · · ≤ λi ≤ · · · the eigenvalu; es of (−∆,H01 ) with eigenfunctions φi . If is an eigenvalue of (−∆,H01 (Ω)), then the problem is resonant at infinity. This case is more delicate. To ensure that J satisfies the (PS) condition usually one needs to assume additional conditions, such as the wellknown LandesmanLazer condition, see, for example, [3, 4]; the angle condition at infinity, see [2]. Recently, in the case 0 ≤ µ < λ1 < , Zhou [12] obtained a positive solution of problem (1.1) under (f2 ) and the following conditions: (H1 ) f ∈ C(Ω × R, R); f (x,t) ≥ 0, for all t ≥ 0, x ∈ Ω and f (x,t) ≡ f (x,0) ≡ 0, for all t ≤ 0, x ∈ Ω. (H2 ) ( f (x,t)/t) is nondecreasing with respect to t ≥ 0, a.e. on x ∈ Ω. Note that our assumption (f1 ) is weaker than (H1 ). And condition (H2 ) is a strong assumption. In this paper, we prove that (f1 ) and (f2 ) are suﬃcient to obtain a positive solution and a negative solution of problem (1.1). Our main result is the following. Theorem 1.1. Assume that f satisfies (f1 ) and (f2 ). If µ < λ1 < , then problem (1.1) has at least two nontrivial solutions, one is positive, the other is negative. Note that in Theorem 1.1, even in the resonant case, we do not need to assume any additional conditions to ensure that J satisfies the (PS) condition. Thus Theorem 1.1 greatly improves previous results, such as Zhou’s [12]. This fact is interesting. The proof of Theorem 1.1 will be stated in Section 2. We can also consider the asymptotically linear Dirichlet problem for the pLaplacian −∆ p u = f (x,u), u = 0, p p x ∈ Ω, x ∈ ∂Ω, (1.4) p where 1 < p < +∞. Let 0 < λ1 < λ2 ≤ λ3 ≤ · · · be the sequence of variational eigenvalues of the eigenvalue problem −∆ p u = λu p−2 u, u = 0, x ∈ Ω, x ∈ ∂Ω. (1.5) Zhitao Zhang et al. 511 It is known that − p has a smallest eigenvalue (see [5]), that is, the princip ple eigenvalue, λ1 , which is simple and has an associated eigenfunction ϕ1 ∈ p 1,p p W0 (Ω) ∩ C 1 (Ω) that is strictly positive in Ω and Ω ϕ1 = 1. λ1 is defined as p λ1 = min Ω  u p 1,p : u ∈ W0 (Ω), p Ω u = 1 . (1.6) Assuming (f1 ) and the following condition: (f2 ) limt→0 ( f (x,t)/ t  p−2 t) = µ, limt→∞ ( f (x,t)/ t  p−2 t) = uniformly in x ∈ Ω, we obtain the following theorem. p Theorem 1.2. Assume that f satisfies (f1 ) and (f 2 ). If µ < λ1 < , then problem (1.4) has at least two nontrivial solutions, one is positive, the other is negative. Remark 1.3. (1) The existence of a positive solution of problem (1.4) was obtained by Li and Zhou [7, Theorem 1.1], under (H1 ), (f2 ) with µ = 0 and (H2 ) ( f (x,t)/t p−1 ) is nondecreasing in t > 0, for x ∈ Ω. Condition (H2 ) is a strong assumption. Moreover, if is an eigenvalue of (1.5), they need another condition (f F) limt→∞ { f (x,t)t − pF(x,t)} = +∞ uniformly a.e. x ∈ Ω to produce a positive solution. Thus Theorem 1.2 extends [7, Theorem 1.1] greatly. (2) Obviously, Theorem 1.1 is a special case of Theorem 1.2. But we would rather state the proof of Theorem 1.1 separately, because the proof is very simple and clear. 2. Proof of Theorem 1.1 In this section, we will always assume that (f1 ) and (f2 ) hold and give the proof of Theorem 1.1. Consider the following problem: −∆u = f+ (x,u), u = 0, where x ∈ Ω, x ∈ ∂Ω, f (x,t), f+ (x,t) = 0, Define a functional J+ : H01 → R, J+ (u) = where F+ (x,t) = t 1 2 0 f+ (x,s)ds. Ω (2.1) t ≥ 0, t ≤ 0. (2.2) F+ (x,u)dx, (2.3) ∇u2 dx − Ω We know J+ ∈ C 1 (H01 , R). 512 On an asymptotically linear elliptic Dirichlet problem Lemma 2.1. J+ satisfies the (PS) condition. Proof. Let {un } ⊂ H01 be a sequence such that J+ un J+ un −→ 0. ≤ c, (2.4) It is easy to see that f+ (x,u)u ≤ C 1 + u2 . (2.5) Now (2.4) implies that for all φ ∈ H01 Ω ∇un ∇φ − f+ x,un φ dx −→ 0. (2.6) Set φ = un , we have un 2 = ≤ Ω Ω f+ x,un un dx + J+ un ,un f+ x,un un dx + o(1) un ≤ C + C un 2 2 + o(1) (2.7) un , where · 2 is the standard norm in L2 := L2 (Ω). We claim that un 2 is bounded. For otherwise, we may assume that un 2 → +∞. Let vn = un / un 2 , then vn 2 = 1. Moreover, from (2.7) we have vn 2 ≤ o(1) + C + un o(1) · un 2 un = o(1) + C + o(1) vn . (2.8) 2 That is, vn is bounded. So, up to a subsequence, we have vn v in H01 , vn −→ v in L2 , for some v with v2 = 1. (2.9) From (2.6) it follows that Ω ∇v ∇φ − v + φ dx = 0, ∀φ ∈ H01 , (2.10) where v+ = max{0,v}. From this and the regularity theory we have −∆v = v + , v = 0, x ∈ Ω, x ∈ ∂Ω. (2.11) The maximum principle implies that v = v+ ≥ 0. But > λ1 and hence v ≡ 0 which contradicts with v2 = 1. Zhitao Zhang et al. 513 Since un 2 is bounded, from (2.7) we get the boundness of un . A standard argument shows that {un } has a convergent subsequence. Therefore, J+ satisfies the (PS) condition. Lemma 2.2. Let φ1 > 0 be a λ1 eigenfunction of (−∆,H01 ) with φ1 = 1, if µ < λ1 < , then (a) there exist ρ,β > 0 such that J+ (u) ≥ β for all u ∈ H01 with u = ρ; (b) J+ (tφ1 ) → −∞ as t → +∞. Proof. See the proof of [12, Lemma 2.5]. Now, we are in a position to state the proof of Theorem 1.1. Proof of Theorem 1.1. By Lemmas 2.1, 2.2, and the Mountain Pass Theorem [9, Theorem 2.2], the functional J+ has a critical point u+ with J+ (u+ ) ≥ β. But J+ (0) = 0, that is, u+ = 0. Then u+ is a nontrivial solution of (2.1). From the strong maximum principle, u+ > 0. Hence u+ is also a positive solution of (1.1). Similarly, we obtain a negative solution u− of (1.1). The proof is completed. Remark 2.3. If we assume further that f ∈ C 1 (Ω × R, R) and is not an eigenvalue of (−∆,H01 ), that is, ∈ (λi ,λi+1 ) for some i ≥ 2. Then the functional J defined in Section 1 satisfies the (PS) condition. Using Morse theory, we can prove that problem (1.1) has one more nontrivial solution u with Ci (J,u) = 0, where Ci (J,u) is the ith critical group of J at u. Remark 2.4. If we assume that µ = µ(x), = (x), and µ(x) < λ1 , (x) ∈ L∞ (Ω), 1.1 (x) ≥ λ1 , mes{x ∈ Ω : (x) > λ1 } > 0, then the conclusion of Theorem is valid too. Since under this assumption, by (2.11) we can get λ vφ 1 Ω 1 = ∇ v ∇ φ = (x)vφ , thus v ≡ 0. 1 1 Ω Ω 3. Proof of Theorem 1.2 and final remarks In this section, we sketch the proof of Theorem 1.2 and give some remarks. First, we recall the concept Fučik spectrum and a related result. The Fučik spectrum of pLaplacian with Dirichlet boundary condition is defined as the set Σ p of those (a,d) ∈ R2 such that −∆ p u = a u+ p −1 u = 0, − d u− p −1 x ∈ ∂Ω, , x ∈ Ω, (3.1) has a nontrivial solution, where u+ = max{u,0} and u− = max{−u,0}. By [5], p p p we know that if (a,d) ∈ Σ p and (a,d) ∈ R × λ1 , (a,d) ∈ λ1 × R, then a > λ1 , p d > λ1 . We will also need the following lemma, which is due to Zhang and Li [11, Lemma 3]. 514 On an asymptotically linear elliptic Dirichlet problem Lemma 3.1. Assume that h ∈ C(Ω × R, R), lim t →+∞ h(t) = a, t  p−2 t h(t) lim t →−∞ t  p−2 t = d. (3.2) 1,p If (a,d) ∈ / Σ p , then the functional ϕ : W0 (Ω) → R, ϕ(u) = 1 p Ω ∇u p dx − satisfies the (PS) condition, where H(u) = u 0 Ω H(u)dx (3.3) h(t)dt. Sketch of the proof of Theorem 1.2. As in Section 2, consider the trancated problem −∆ p u = f+ (x,u), u = 0, x ∈ Ω, (3.4) x ∈ ∂Ω, where f+ is defined as in (2.2). Due to the maximum principle (see [10]), solutions of (3.4) are positive, thus are solutions of (1.4). We have lim t →−∞ f+ (x,t) t  p−2 t = 0, f+ (x,t) lim t →+∞ t  p−2 t = . (3.5) p Since > λ1 , one deduces directly from the definition of Fučik spectrum that (,0) ∈ / Σ p , thus by Lemma 3.1, we deduce that the C 1 functional 1 J+ (u) = p p Ω ∇u dx − Ω F+ (x,u)dx (3.6) 1,p satisfies the (PS) condition on the Sobolev space W0 (Ω) with norm u1,p = t Ω ∇u p dx 1/ p , (3.7) where F+ (x,t) = 0 f+ (x,s)ds. As [7, Lemma 2.3], the functional J+ admits the “Mountain Pass Geometry.” Thus J+ has a nonzero critical point, which is a nontrivial solution of (3.4). From the strong maximum principle (see [10]), it is also a positive solution of (1.4). Similarly, we obtain a negative solution of (1.4). Zhitao Zhang et al. 515 Remark 3.2. Problems (1.1) and (1.4) can be resonant at infinity, this is the main diﬃculty in verifying the (PS) condition. But after trancating, the problems are not resonant with respect to the Fučik spectrum. Thus, from the Fučik spectrum point of view, the corresponding functionals of the trancated problems satisfies the (PS) condition naturally. And our limit conditions at zero allow us to use the trancation technique and apply the Mountain Pass Theorem. These are the main ingredient of this work. Remark 3.3. In fact, let P := {u ∈ H01 : u(x) ≥ 0, a.e.}, the functional J does not satisfies the (PS) condition on the whole space H01 whenever = λi , i > 1, but from our proof J satisfies the (PS) condition on P. That is, the unbounded (PS) sequences do not belong to P. This idea may be used to weaken the compact conditions for other problems. Acknowledgment This work was supported by the National Natural Science Foundation of China (NSFC). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] H. Amann and E. Zehnder, Nontrivial solutions for a class of nonresonance problems and applications to nonlinear diﬀerential equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 7 (1980), no. 4, 539–603. T. Bartsch and S. Li, Critical point theory for asymptotically quadratic functionals and applications to problems with resonance, Nonlinear Anal. 28 (1997), no. 3, 419– 441. K. C. Chang, Solutions of asymptotically linear operator equations via Morse theory, Comm. Pure Appl. Math. 34 (1981), no. 5, 693–712. , InfiniteDimensional Morse Theory and Multiple Solution Problems, Progress in Nonlinear Diﬀerential Equations and Their Applications, vol. 6, Birkhäuser Boston, Massachusetts, 1993. M. Cuesta, D. de Figueiredo, and J.P. Gossez, The beginning of the Fučik spectrum for the pLaplacian, J. Diﬀerential Equations 159 (1999), no. 1, 212–238. E. N. Dancer and Z. Zhang, Fučik spectrum, signchanging, and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity, J. Math. Anal. Appl. 250 (2000), no. 2, 449–464. G. Li and H.S. Zhou, Asymptotically linear Dirichlet problem for the pLaplacian, Nonlinear Anal. 43 (2001), no. 8, Ser. A: Theory Methods, 1043–1055. S. Li and Z. Zhang, Fucik spectrum, signchanging and multiple solutions for semilinear elliptic boundary value problems with jumping nonlinearities at zero and infinity, Sci. China Ser. A 44 (2001), no. 7, 856–866. P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Regional Conference Series in Mathematics, vol. 65, American Mathematical Society, Rhode Island, 1986. J. L. Vázquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984), no. 3, 191–202. 516 [11] [12] On an asymptotically linear elliptic Dirichlet problem Z. Zhang and S. Li, On signchanging and multiple solutions of pLaplacian, preprint, 2001. H.S. Zhou, Existence of asymptotically linear Dirichlet problem, Nonlinear Anal. 44 (2001), no. 7, Ser. A: Theory Methods, 909–918. Zhitao Zhang: Institute of Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, Beijing 100080, China Email address: zzt@math03.math.ac.cn Shujie Li: Institute of Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, Beijing 100080, China Email address: lisj@math03.math.ac.cn Shibo Liu: Institute of Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, Beijing 100080, China Email address: liusb@amss.ac.cn Weijie Feng: Institute of Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, Beijing 100080, China Advances in Operations Research Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Advances in Decision Sciences Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Applied Mathematics Algebra Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Probability and Statistics Volume 2014 The Scientific World Journal Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Differential Equations Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Submit your manuscripts at http://www.hindawi.com International Journal of Advances in Combinatorics Hindawi Publishing Corporation http://www.hindawi.com Mathematical Physics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Journal of Complex Analysis Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Mathematics and Mathematical Sciences Mathematical Problems in Engineering Journal of Mathematics Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Discrete Mathematics Journal of Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Discrete Dynamics in Nature and Society Journal of Function Spaces Hindawi Publishing Corporation http://www.hindawi.com Abstract and Applied Analysis Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 International Journal of Journal of Stochastic Analysis Optimization Hindawi Publishing Corporation http://www.hindawi.com Hindawi Publishing Corporation http://www.hindawi.com Volume 2014 Volume 2014