Multiplicity of positive solution of p-Laplacian problems with sign-changing weight functions

全文

(1)Int. Journal of Math. Analysis, Vol. 1, 2007, no. 12, 557 - 563. Multiplicity of Positive Solution of p-Laplacian Problems with Sign-Changing Weight Functions Tsung-fang Wu1 Department of Applied Mathematics National University of Kaohsiung Kaohsiung 811, Taiwan [email protected] Abstract In this paper, we study the multiplicity of positive solutions for the p-Laplacian problems with sign-changing weight functions. Using the decomposition of the Nehari manifold, we prove that an elliptic equation has at least two positive solutions.. Mathematics Subject Classification: 35J50, 35J65 Keywords: Palais–Smale, Nehari manifold, Multiple positive solutions. 1. Introduction. In this paper, we study the multiplicity of positive solutions for the following elliptic equation: −Δp u = λf (x) |u|q−2 u + g (x) |u|r−2 u in Ω, (Eλ ) u=0 on Ω, if N > p, p∗ = ∞ if N ≤ p), Ω ⊂ RN where 1 < q < p < r < p∗ (p∗ = NpN −p is a bounded domain, λ ∈ R\ {0} , and the weight functions f, g ∈ C Ω are satisfying f ± = max {±f, 0} ≡ 0 and g ± = max {±g, 0} ≡ 0. When p = 2. The fact that the number of positive solutions of equation (Eλ ) is affected by the nonlinearity terms has been the focus of a great deal of research in recent years. If the weight functions f ≡ g ≡ 1, the authors Ambrosetti-Brezis-Cerami [1] have investigated equation (Eλ ) . They found 1. Partially supported by the National Science Council of Republic of Taiwan (R.O.C.).

(2) Tsung-fang Wu. 558. that there exists λ0 > 0 such that equation (Eλ ) admits at least two positive solutions for λ ∈ (0, λ0) , has a positive solution for λ = λ0 and no positive solution exists for λ > λ0 . Wu [7] proved that equation (Eλ ) has at least two positive solutions under the assumptions the weight functions f change sign in Ω, g ≡ 1 and λ is sufficiently small. For more general results, were done by de Figueiredo-Grossez-Ubilla [5], Wu [8] and Brown-Wu [2]. In this paper, we give a very simple variational proof of the existence of at least two positive solutions of equation (Eλ ) for p ∈ (1, p∗ ). In fact, we use the decomposition of the Nehari manifold as λ varies to prove that the following result. Theorem 1.1 There exists λ0 > 0 such that for 0 < |λ| < λ0 , equation (Eλ ) has at least two positive solutions. This paper is organized as follows. In section 2, we give some notations and preliminaries. In section 3, we establish the existence of Palais–Smale sequences. In section 4, we prove the equation (Eλ ) has at least two positive solutions.. 2. Notations and Preliminaries. Throughout this paper, we denote by Sl the best Sobolev constant for the operators W01,p (Ω) → Ll (Ω) is given by |∇u|p Ω Sl = inf p/l > 0, l u∈W01,p (Ω)\{0} |u| Ω −l where 1 < l ≤ p∗ . In particular, Ω |u|l ≤ Sl p u l for all u ∈ W01,p (Ω) with 1 the standard norm u = Ω |∇u|p p . Equation (Eλ ) is posed in the framework of the Sobolev space W01,p (Ω) . Moreover, a function u ∈ W01,p (Ω) is said to be a weak solution of problem (Eλ ) if p−2 q−2 |∇u| ∇u∇ϕ − λ f |u| uϕ − g |u|r−2 uϕ = 0 for all ϕ ∈ W01,p (Ω) . Ω. Ω. Ω. Thus, the corresponding energy functional of problem (Eλ ) is defined by 1 λ 1 p q f |u| − f |u|p for u ∈ W01,p (Ω) . Jλ (u) = u − 2 q Ω r Ω.

(3) Multiplicity of positive solution. 559. As the energy functional Jλ is not bounded below on W01,p (Ω) , it is useful to consider the functional on the Nehari manifold. Mλ = u ∈ W01,p (Ω) \ {0} | Jλ (u) , u

(4) = 0 . Thus, u, ∈ Mλ if and only if p. u − λ. Ω. . q. f |u| −. Ω. Define ψλ (u) =. Jλ. ψλ. . p. (u) , u

(5) = u − λ. Then for u ∈ Mλ ,. . p. g |u|r = 0.. Ω. . q. f |u| −. q. (1). Ω. g |u|r .. . (u) , u

(6) = p u − λq f |u| − r g |u|r Ω Ω = (p − q) λ f |u|q + (p − r) g |u|r . Ω. (2). Ω. Now, we split Mλ into three parts: = {u ∈ Mλ | ψλ (u) , u

(7) > 0} ; M+ λ M0λ = {u ∈ Mλ | ψλ (u) , u

(8) = 0} ; = {u ∈ Mλ | ψλ (u) , u

(9) < 0} . M− λ Then, we have the following results. / M0λ , then Lemma 2.1 If u0 is a local minimizer for Jλ on Mλ and u0 ∈ Jλ (u0 ) = 0 in W −1,p (Ω) . Proof. Our proof is almost the same as that in Brown-Zhang [3, Theorem 2.3]. Lemma 2.2 The energy functional Jλ,μ is coercive and bounded below on Mλ . Proof. If u ∈ Mλ , then by the Sobolev trace imbedding theorem.

(10) r−q r−p p Jλ (u) =. u − λ f |u|q pr qr.

(11) Ω q r−q r−p. u p − Sqp |λ| f ∞ u q . ≥ pr qr Thus, Jλ is coercive and bounded below on Mλ .. .

(12) Tsung-fang Wu. 560 q Lemma 2.3 (i) For any u ∈ M+ λ , we have λ Ω f|u| > 0; (ii) for any u ∈ M0λ , we have λ Ω f |u|q > 0 and Ω g |u|r > 0; r (iii) For any u ∈ M− λ , we have Ω g |u| > 0. Proof. The proofs are immediate from (1) and (2) .. . Lemma 2.4 There exists λ0 > 0 such that for 0 < |λ| < λ0 we have M0λ = ∅. Proof. Suppose otherwise, that is M0λ = ∅ for all λ ∈ R\ {0}. Then by Lemma 2.3, p 0 = Jλ (u) , u

(13) = (p − q) u − (r − q) g |u|r Ω p q = (p − r) u − (q − r) λ f |u| Ω. for all u ∈ M0λ . By the Hölder inequality and the Sobolev imbedding theorem,. u ≥. r p−q. g ∞ Srp r−q.

(14) 1/(p−r). and. u ≤. r−q. f ∞ r−p. 1

(15) p−q. q. 1. Spp(p−q) |λ| p−q. If |λ| is sufficiently small, this is impossible. Thus, we can conclude that there exists λ0 > 0 such that if 0 < |λ| < λ0 , we have M0λ = ∅. + − By Lemma 2.4, for 0 < |λ| < λ0 we write Mλ = Mλ ∪ Mλ and define αλ+ = inf + Jλ (u) and αλ− = inf − Jλ (u) . u∈Mλ. u∈Mλ. Then we have the following results. ± Proposition 2.5 There exist minimizing sequences {u± n } in Mλ for Jλ such that. ± ± −1,p (Ω) . Jλ u ± n = αλ + o (1) and Jλ un = o (1) in W Proof. The proof is almost the same as that in Wu [7, Proposition 9] and is omit here. .

(16) Multiplicity of positive solution. 3. 561. Proof of Theorem 1. Throughout this section, we assume that the parameter λ satisfies 0 < |λ| < λ0 . Then we have the following results. + Theorem 3.1 Equation (Eλ ) has a positive solution u+ 0 ∈ Mλ such that + Jλ u0 = αλ+ < 0.. Proof. First, we show αλ+ < 0. For u ∈ M+ λ , we have.

(17)

(18) 1 1 1 1 p Jλ (u) = g |u|r −. u + − p q q r Ω (p − q) (r − p) < −. u p < 0. pqr + This implies αλ+ < 0. By Proposition 2.5, there exists {u+ n } ⊂ Mλ such that + + −1,p (Ω) . Jλ u + n = αλ + o (1) and Jλ un = o (1) in W. Then by Lemma 2.2 (ii) and the Rellich–Kondrachov theorem there exist a 1,p + subsequence {u+ n } and u0 ∈ W0 (Ω) is a solution of equation (Eλ ) such that 1,p + u+ n u0 weakly in W0 (Ω). and + l ∗ u+ n → u0 strongly in L (Ω) for all 1 ≤ l < p .. Then we have Ω. q = f u+ n. Ω. q + o (1) and λ f u+ 0. Ω. q ≥ 0. f u+ 0. q > 0. Otherwise, then Now we prove that λ Ω f u+ 0 r + p un = + o (1) g u+ n . Ω. and.

(19) p 1 1 u+ − n W 1,p p r + q 1 r p λ 1 + u − + o (1) = f un − g u+ n n 2 q Ω r Ω = αλ+ + o (1).

(20) Tsung-fang Wu. 562. q + +. this is contradicts αλ+ < 0. Thus, λ Ω f u+ 0 > 0. In particular, u0 ∈ Mλ is a + nontrivial solution of equation (Eλ ) and Jλ u+ 0 ≥ αλ . Moreover, αλ+.

(21)

(22) + q r 1 1 1 1. − − = ≤ Jλ g u+ λ f u0 + 0 p q p r Ω Ω + = lim Jλ u+ n = αλ . . u+ 0. . n→∞. + and u+. Consequently, Jλ u+ = αλ+ . Since Jλ u+ = Jλ u+ 0 0 0 0 ∈ Mλ . By Lemma 2.1 we may assume that u+ 0 ≥ 0. Moreover, by the Harnack inequality + due to Trudinger [6], we obtain u0 is a positive solution of equation (Eλ ) . − Theorem 3.2 Equation (Eλ ) has a positive solution u− 0 ∈ Mλ such that − − Jλ u0 = αλ .. Proof. By Proposition 2.5, there exists {un } ⊂ M− λ such that − − −1,p Jλ u − (Ω) . n = αλ + o (1) and Jλ un = o (1) in W By Lemma 2.2 and the Rellich–Kondrachov theorem, there exist a subsequence − − {u− n } and u0 ∈ Mλ is a nonzero solution of equation (Eλ ) such that 1,p − u− n u0 weakly in W0 (Ω). and − l u− n → u0 strongly in L (Ω) for all 1 ≤ l < 2∗ .. Moreover, αλ−.

(23)

(24) − q r 1 1 1 1. = ≤ Jλ g u− − λ f u0 + − 0 p q p r Ω Ω − = α = lim Jλ u− . n λ . u− 0. . n→∞. − and u−. = αλ− . Since Jλ u− = Jλ u− Consequently, Jλ u− 0 0 0 0 ∈ Mλ . By Lemma 2.1 we may assume that u− 0 ≥ 0. Moreover, by the Harnack inequality due to Trudinger [6], we obtain u− 0 is a positive solution of equation (Eλ ) . Now, we begin to show the proof of Theorem 1.1: By Theorems 3.1, 3.2, − + + equation (Eλ ) has two positive solutions u+ 0 and u0 such that u0 ∈ Mλ and − − + − + − u0 ∈ Mλ . Since Mλ ∩ Mλ = ∅, this implies that u0 and u0 are distinct..

(25) Multiplicity of positive solution. 563. References [1] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), 519-543. [2] K. J. Brown and T. F. Wu, A fibrering map approach to a semilinear elliptic boundary value problem (submitted for publication). [3] K. J. Brown and Y. Zhang, The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function, J. Diff. Equns 193 (2003), 481-499. [4] I. Ekeland, On the variational principle, J. Math. Anal. Appl. 17 (1974), 324-353. [5] D. G. de Figueiredo, J. P. Gossez and P. Ubilla, Local superlinearity and sublinearity for indefinite semilinear elliptic problems, J. Funct. Anal. 199 (2003), 452–467. [6] N. S. Trudinger, On Harnack type inequalities and their application to quasilinear elliptic equations, Comm. Pure Appl. Math. 20 (1967), 721– 747. [7] T. F. Wu, On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl. 318 (2006), 253–270. [8] T. F. Wu, Multiplicity results for a semilinear elliptic equation involving sign-changing weight function, Rocky Mountain Journal of Mathematics, in press. Received: January 11, 2007.

(26)