Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 524862, 21 pages doi:10.1155/2010/524862 Research Article Multiple Positive Solutions of Semilinear Elliptic Problems in Exterior Domains Tsing-San Hsu and Huei-Li Lin Department of Natural Sciences, Center for General Education, Chang Gung University, Tao-Yuan 333, Taiwan Correspondence should be addressed to Huei-Li Lin, hlin@mail.cgu.edu.tw Received 30 July 2010; Accepted 30 November 2010 Academic Editor: Wenming Z. Zou Copyright q 2010 T S. Hsu and H L. Lin. This is an open access article d istributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Assume that q is a positive continuous function in N and satisfies the suitable conditions. We prove that the Dirichlet problem −Δu u qz|u| p−2 u admits at least three positive solutions in an exterior domain. 1. Introduction For N ≥ 3and2<p<2 ∗ 2N/N − 2, we consider the semilinear elliptic equations − Δu u q z | u | p−2 u in Ω, u ∈ H 1 0 Ω , 1.1 − Δu u q ∞ | u | p−2 u in Ω, u ∈ H 1 0 Ω , 1.2 where Ω is an unbounded domain N .Letq be a positive continuous function in N and satisfy lim | z | →∞ q z q ∞ > 0,q z / ≡ q ∞ . q1 2 Boundary Value Problems Associated with 1.1 and 1.2,wedefinethefunctionala, b, b ∞ ,J,andJ ∞ ,foru ∈ H 1 0 Ω a u Ω | ∇u | 2 u 2 dz u 2 H 1 , b u Ω q z u p dz, b ∞ u Ω q ∞ u p dz, J u 1 2 a u − 1 p b u , J ∞ u 1 2 a u − 1 p b ∞ u , 1.3 where u max{u, 0}≥0. By Rabinowitz 1,Proposition B.10, the functionals a, b, b ∞ , J, and J ∞ are of C 2 . It is well known that 1.1 admits infinitely many solutions in a bounded domain. Because of the lack of compactness, it is difficult to deal with this problem in an unbounded domain. Lions 2, 3 proved that if qz ≥ q ∞ > 0, then 1.1 has a positive ground state solution in N . Bahri and Li 4 proved that there is at least one positive solution of 1.1 in N when lim |z|→∞ qzq ∞ > 0andqz ≥ q ∞ − C exp−δ|z| for δ>2. Zhu 5 has studied the multiplicity of solutions of 1.1 in N as follows. Assume N ≥ 5, lim |z|→∞ qz q ∞ , qz ≥ q ∞ > 0, and there exist positive constants C, γ, R 0 such that qz ≥ q ∞ C/|z| γ for |z|≥R 0 ,then1.1 has at least two nontrivial solutions oneispositiveandtheother changes sign.Esteban6, 7 and Cao 8 have studied the multiplicity of solutions of −Δu u qz|u| p−2 u with Neumann condition in an exterior domain N \ D,whereD is a C 1,1 bounded domain in N .Hirano9 proved that if q − q ∞ ∞ is sufficiently small and qz ≥ q ∞ 1 C exp−δ|z| for 0 <δ<1, then 1.1 admits at least three nontrivial solutions one is positive and the other changes sign in N . Recently, under the same conditions, Lin 10 showed that 1.1 admits at least two positive solutions and one nodal solution in an exterior domain. Let qzazμbz.Wu11 showed that for sufficiently small μ,ifa and b satisfy some hypotheses, then 1.1 has at least three positive solutions in N . In this paper, we consider the multiplicity of positive solutions of 1.1 in an exterior domain. If q satisfies the suitable conditions q − q ∞ ∞ is sufficiently small and qz ≥ q ∞ C exp−δ|z| for 0 <δ<2, then we can show that 1.1 admits at least three positive solutions in an exterior domain. First, in Section 3, we use the concentration-compactness argument of Lions 2, 3 to obtain the “ground-state solution” see Theorem 3.7.InSection 4,westudy the idea of category in Adachi-Tanaka 12 and Bahri-Li minimax method to get that there are at least three positive solutions of 1.1 in N \ D see Theorems 4.10 and 4.15. 2. Existence of (PS)—Sequences Let Ω be an unbounded domain in N . We define the Palais-Smale denoted by PS sequences, PS-values, and PS-conditions in H 1 0 Ω for J as follows. Boundary Value Problems 3 Definition 2.1. i For β ∈ ,asequence{u n } is a PS β -sequence in H 1 0 Ω for J if Ju n β o n 1 and J u n o n 1 strongly in H −1 Ω as n →∞. ii β ∈ is a PS-value in H 1 0 Ω for J if there is a PS β -sequence in H 1 0 Ω for J. iii J satisfies the PS β -condition in H 1 0 Ω if every PS β -sequence in H 1 0 Ω for J contains a convergent subsequence. Lemma 2.2. Let u ∈ H 1 0 Ω be a critical point of J,thenu is a nonnegative solution of 1.1. Moreover, if u / ≡ 0,thenu is positive in Ω. Proof. Suppose that u ∈ H 1 0 Ω satisfies J u,ϕ 0foranyϕ ∈ H 1 0 Ω,thatis, Ω ∇u∇ϕ uϕ Ω q z u p−1 ϕ for any ϕ ∈ H 1 0 Ω . 2.1 Thus, u is a weak solution of −Δu u qzu p−1 in Ω.Sinceq>0in N , by the maximum principle, u is nonnegative. If u / ≡ 0, we have that u is positive in Ω. Define α Ω inf u∈MΩ J u , 2.2 where MΩ {u ∈ H 1 0 Ω \{0}|aubu } and α ∞ Ω inf u∈M ∞ Ω J ∞ u , 2.3 where M ∞ Ω {u ∈ H 1 0 Ω \{0}|aub ∞ u }. Lemma 2.3. Let β ∈ and let {u n } be a PS β -sequence in H 1 0 Ω for J.Then, i {u n } is a bounded sequence in H 1 0 Ω, ii au n bu n o n 12p/p − 2β o n 1 as n →∞and β ≥ 0. By Chen et al. 13 and Chen and Wang 14, we have the following lemmas. Lemma 2.4. i For each u ∈ H 1 0 Ω \{0} with u / ≡ 0, there exists the unique number s u > 0 such that s u u ∈ MΩ and sup s≥0 JsuJs u u. ii Let β>0 and {u n } a sequence in H 1 0 Ω \{0} for J such that u n / ≡ 0,Ju n β o n 1 and au n bu n o n 1. Then, there is a sequence {s n } in such that s n 1 o n 1, {s n u n } in MΩ and Js n u n β o n 1 as n →∞. Lemma 2.5. There exists a positive constant c such that u H 1 ≥ c>0 for each u ∈ MΩ.Moreover, αΩ > 0. Lemma 2.6. Let Ω 1 Ω 2 .IfJ satisfies the PS αΩ 1 -condition or αΩ 1 is a critical value, then αΩ 2 <αΩ 1 . Proof. See Chen et al. 13 or Lin et al. 15. 4 Boundary Value Problems Remark 2.7. The above definitions and lemmas hold not only for J ∞ and M ∞ Ω but also for α ∞ Ω. Lemma 2.8. Every minimizing sequence {u n } in M ∞ Ω of α ∞ Ω is a PS α ∞ Ω -sequence in H 1 0 Ω for J.Moreover,α ∞ Ω is a (PS)-value. 3. Existence of Ground State Solution From now on, let Ω N \ D be an exterior domain, where D is a C 1,1 bounded domain in N .ByLions2, 3,Struwe16, and Lien et al. 17, we have the following decomposition lemmas. Lemma 3.1 Palais-Smale Decomposition Lemma for J. Assume that q is a positive continuous function in N and lim |z|→∞ qzq ∞ > 0.Let{u n } be a PS β -sequence in H 1 0 Ω for J.Then, there are a s ubsequence {u n }, a nonnegative integer l, sequences {z i n } ∞ n1 in N , functions u in H 1 0 Ω, and w i / 0 in H 1 N for 1 ≤ i ≤ l such that z i n − z j n −→ ∞ for 1 ≤ i, j ≤ l, i / j, −Δu u q z | u | p−2 u in Ω, −Δw i w i q ∞ w i p−2 w i in N , u n u l i1 w i ·−z i n o n 1 strongly in H 1 N , J u n J u l i1 J ∞ w i o n 1 . 3.1 Lemma 3.2 Palais-Smale Decomposition Lemma for J ∞ . Let {u n } be a PS β -sequence in H 1 0 Ω for J ∞ . Then, there are a subsequence {u n }, a nonnegative integer l, sequences {z i n } ∞ n1 in N , functions u in H 1 0 Ω,andw i / 0 in H 1 N for 1 ≤ i ≤ l such that z i n − z j n −→ ∞ for 1 ≤ i, j ≤ l, i / j, −Δu u q ∞ | u | p−2 u in Ω, −Δw i w i q ∞ w i p−2 w i in N , u n u l i1 w i ·−z i n o n 1 strongly in H 1 N , J ∞ u n J ∞ u l i1 J ∞ w i o n 1 . 3.2 Boundary Value Problems 5 Lemma 3.3. i α ∞ Ω α ∞ N (denoted by α ∞ ). ii Let {u n }⊂MΩ be a PS β -sequence in H 1 0 Ω for J with 0 <β<α ∞ . Then, there exist a subsequence {u n } and a nonzero u 0 ∈ H 1 0 Ω such that u n → u 0 strongly in H 1 0 Ω,thatis,J satisfies the PS β -condition in H 1 0 Ω.Moreover,u 0 is a positive solution of 1.1 such that Ju 0 β. Proof. i Since Ω is an exterior domain, by Lien et al. 17, Ω is a ball-up domain for any r>0, there exists z ∈ Ω such that B N z; r ⊂ Ω and α ∞ Ω α ∞ N . ii Since {u n }⊂MΩ is a PS β -sequence in H 1 0 Ω for J with 0 <β<α ∞ ,by Lemma 2.3, {u n } is bounded. Thus, there exist a subsequence {u n } and u 0 ∈ H 1 0 Ω such that u n u 0 weakly in H 1 0 Ω. It is easy to check that u 0 is a solution of 1.1. Applying Palais-Smale Decomposition Lemma 3.1,weget α ∞ >β J u n ≥ lα ∞ . 3.3 Then, l 0andu 0 / 0. Hence, u n → u 0 strongly in H 1 0 Ω and Ju 0 β. Moreover, by Lemma 2.2, u 0 is positive in Ω. It is well known that there is the unique up to translation, positive, smooth, and radially symmetric solution w of 1.2 in N such that J ∞ wα ∞ . See Bahri and Lions 18, Gidas et al. 19, 20 and Kwong 21.Recallthefacts i for any ε>0, there exist constants C 0 , C 0 > 0suchthatforallz ∈ N w z ≤ C 0 exp − | z | , | ∇w z | ≤ C 0 exp − 1 − ε | z | , 3.4 ii for any ε>0, there exists a constant C ε > 0suchthat w z ≥ C ε exp − 1 ε | z | ∀z ∈ N . 3.5 Suppose D ⊂ B N 0; R{z ∈ N ||z| <R} for some R>0. Let ψ R : N → 0, 1 be a C ∞ -function on N such that 0 ≤ ψ R ≤ 1, |∇ψ R |≤c and ψ R z ⎧ ⎨ ⎩ 1for | z | ≥ R 1, 0for | z | ≤ R. 3.6 We define w z z ψ R z w z − z for z ∈ N . 3.7 Clearly, w z z ∈ H 1 0 Ω. We need the following lemmas to prove that sup t≥0 Jtw z <α ∞ for sufficiently large | z|. 6 Boundary Value Problems Lemma 3.4. Let E be a domain in N .Iff : E → satisfies E f z e σ|z| dz < ∞ for some σ>0, 3.8 then E f z e −σ|z−z| dz e σ|z| E f z e σz,z/|z| dz o 1 as | z | −→ ∞ . 3.9 Proof. Since σ| z|≤σ|z| σ|z − z|,wehave f z e −σ|z−z| e σ|z| ≤ f z e σ|z| . 3.10 Since −σ|z − z| σ|z| σz, z/|z|o1 as |z|→∞, then the lemma follows from the Lebesque-dominated convergence theorem. Next, assume that q is a positive continuous function in N and satisfies q1 and q z ≥ q ∞ C exp −δ | z | for some C>0and0<δ<2. q2 Then, we have the following lemmas. Lemma 3.5. i There exists a number t 0 > 0 such that for 0 ≤ t<t 0 and each w z ∈ H 1 0 Ω,wehave J tw z <α ∞ . 3.11 There exists a number t 1 > 0 such that for any t>t 1 and |z|≥R 2,wehave J tw z < 0. 3.12 Proof. i Since α ∞ > 0 J0,Jis continuous in H 1 0 Ω and {w z } is bounded in H 1 0 Ω,then there exists t 0 > 0suchthatfor0≤ t<t 0 and each w z ∈ H 1 0 Ω J tw z <α ∞ . 3.13 Boundary Value Problems 7 For | z|≥R 2: since 0 ≤ ψ R ≤ 1, |∇ψ R |≤c and qz q ∞ , we have that J tw z t 2 2 Ω ∇ ψ R z w z − z 2 ψ R z w z − z 2 dz − t 2 p Ω q z ψ R z w z − z p dz ≤ t 2 2 N ∇ψ R w z − z ψ R ∇w z − z 2 w z − z 2 dz − t p p Bz;1 q ∞ w z − z p dz ψ R z 1forz ∈ B z;1 ≤ t 2 2 N cw z | ∇w z | 2 w z 2 dz − t p p B0;1 q ∞ w z p dz. 3.14 Hence, there exists t 1 > 0suchthat J tw z < 0foranyt>t 1 , | z | ≥ R 2. 3.15 Lemma 3.6. There exists a number R 1 >R 2 > 0 such that for any |z|≥R 1 ,weobtain sup t≥0 J tw z <α ∞ . 3.16 Proof. Applying the above lemma, we only need to show that there exists a number R 1 > R 2 > 0suchthatforany| z|≥R 1 , sup t 0 ≤t≤t 1 J tw z <α ∞ . 3.17 For t 0 ≤ t ≤ t 1 ,since ∇ ψ R w z − z 2 ∇ψ R 2 w z − z 2 ψ 2 R | ∇w z − z | 2 2ψ R w z − z ∇ψ R ∇w z − z , 3.18 8 Boundary Value Problems then we have J tw z t 2 2 N ∇ ψ R z w z − z 2 ψ R z w z − z 2 dz − t p p N q z ψ R z w z − z p dz the defination of ψ R ≤ t 2 2 N | ∇w z − z | 2 w z − z 2 dz − t p p N q ∞ w z − z p dz t 2 2 N ∇ψ R 2 w z − z 2 2ψ R w z − z ∇ψ R ∇w z − z dz − t p p N q z ψ p R w z − z p − q ∞ w z − z p dz 3.18 and 0 ≤ ψ R ≤ 1 ≤ α ∞ t 2 1 2 N ∇ψ R 2 w z − z 2 2 | w z − z | ∇ψ R | ∇w z − z | dz − t p 0 p {| z | ≥R1 } q z − q ∞ w z − z p dz t p 1 p {| z | ≤R1 } q ∞ w z − z p dz sup t≥0 J ∞ tw α ∞ and the defination of ψ R . 3.19 Since the support of ∇ψ R is bounded, then supp ∇ψ R ∇ψ R 2 w z − z 2 dz ≤ C 1 exp −2 | z | , supp ∇ψ R | w z − z | ∇ψ R | ∇w z − z | dz ≤ C 2 exp − 2 − ε | z | . 3.20 Similarly, we have {| z | ≤R1 } q ∞ w z − z p dz ≤ C 3 exp −p | z | . 3.21 Boundary Value Problems 9 Since qz ≥ q ∞ C exp−δ|z| for some 0 <δ<2, by Lemma 3.4,thereexistsR 1 >R 2 > 0 such that for any | z| >R 1 {| z | ≤R1 } q z − q ∞ w z − z p dz ≥ C ε exp − min δ, p 1 ε | z | ≥ C ε exp −δ | z | . 3.22 Choosing 0 <ε<2 − δ and using 3.20–3.22,thereexistsR 1 >R 1 such that for |z|≥R 1 ,we have sup t 0 ≤t≤t 1 J tw z <α ∞ , 3.23 that is, sup t≥0 Jtw z <α ∞ . Using the Ekeland variational principle or se e Stuart 22,thereisaPS αΩ - sequence {u n }⊂MΩ for J. Then, we apply Lemma 3.3ii to obtain the existence of positive ground state solution of 1.1 in Ω. Theorem 3.7. Assume that q is a positive continuous function in N and satisfies q1 and q2. Then, there exists at least one positive ground state solution u 0 of 1.1 in Ω. Proof. Since w z ∈ H 1 0 Ω,byLemma 2.4i,thereexistss z > 0suchthats z w z ∈ MΩ. Thus, by Lemma 3.6, αΩ ≤ Js z w z ≤ sup t≥0 Jtw z <α ∞ for |z|≥R 1 . Using the Ekeland variational principle, there is a PS αΩ -sequence {u n }⊂MΩ for J. Apply Lemma 3.3ii, there exists at least one positive solution u 0 of 1.1 in Ω such that Ju 0 αΩ. 4. Existence of Multiple Solutions In this section, we use two methods to obtain the existence of multiple positive solutions of 1.1 in an exterior domain. Part I : we study the idea of category to prove Theorem 4.10.Part II: we study the Bahri-Li minimax method to prove Theorem 4.15. Lemma 4.1. Assume that q is a positive continuous function in N .Ifq satisfies q1, q2 and m/2q ∞ qz where m>2, then there exists m 0 > 2 such that for m ≤ m 0 ,weobtainthat 2αΩ >α ∞ . Proof. Since qz q ∞ ,byLions2, 3,letw 0 ∈ H 1 N be a positive solution of −Δw 0 w 0 qz|w 0 | p−2 w 0 in N and Jw 0 α N .ByLemma 2.4i and Remark 2.7,thereexistss 0 > 0 such that s 0 w 0 ∈ M ∞ N and J ∞ s 0 w 0 ≥ α ∞ and N | ∇ s 0 w 0 | 2 s 0 w 0 2 dz N q ∞ s 0 w 0 p dz ≥ 2p p − 2 α ∞ . 4.1 10 Boundary Value Problems Moreover, we have 1 N | ∇w 0 | 2 w 2 0 N q z w p 0 < N | ∇w 0 | 2 w 2 0 N q ∞ w p 0 s p−2 0 < N m/2 q ∞ w p 0 N q ∞ w p 0 m 2 . 4.2 Hence, using the above inequalities, we get α N J w 0 sup s≥0 J sw 0 >J s 0 w 0 J ∞ s 0 w 0 − 1 p N q z − q ∞ s 0 w 0 p dz ≥ α ∞ − 1 p m 2 − 1 N q ∞ s 0 w 0 p dz α ∞ − s 2 0 p m 2 − 1 N | ∇w 0 | 2 w 2 0 dz >α ∞ − 1 p m 2 − 1 m 2 2/p−2 2p p − 2 α N , 4.3 that is, 1 m − 2/p − 2m/2 2/p−2 α N >α ∞ . Choose some m 0 > 2suchthatfor 2 <m≤ m 0 ,then2α N >α ∞ .ByLemma 2.6 and Theorem 3.7,2αΩ > 2α N >α ∞ . Lemma 4.2. There exists a number δ 0 > 0 such that if u ∈ M ∞ Ω and J ∞ u ≤ α ∞ δ 0 ,then N z | z | | ∇u | 2 u 2 dz / −→ 0 . 4.4 Proof. On the contrary, there exists a sequence {u n } in M ∞ Ω such that J ∞ u n α ∞ o n 1 as n →∞and N z | z | | ∇u n | 2 u 2 n dz −→ 0 ∀n. 4.5 By Lemma 2.8, {u n } is a PS α ∞ -sequence in H 1 0 Ω for J ∞ .Sinceα ∞ Ω α ∞ N ,Lienet al. 17 proved that 1.2 does not have any ground state solution in an exterior d omain, that is, inf v∈M ∞ Ω J ∞ vα ∞ Ω is not achieved. Applying the Palais-Smale Decomposition Lemma 3.2, we have that ther e exists a sequence {z n } in N such that |z n |→∞as n →∞ and u n z w z − z n o n 1 strongly in H 1 N , 4.6 [...]... 997–1020, 2001 Boundary Value Problems 21 10 H.-L Lin, “Multiple solutions of semilinear elliptic equations in exterior domains,” Proceedings of the Royal Society of Edinburgh A, vol 138, no 3, pp 531–549, 2008 11 T.-F Wu, “The existence of multiple positive solutions for a semilinear elliptic equation in ÊN ,” Nonlinear Analysis: Theory, Methods & Applications, vol 72, no 7-8, pp 3412–3 421, 2010 12 S... 4.34 by 2 2 ÊN z/|z| |∇u| |u| dz Gu 2 Ê z/|z| |∇u| N |u|2 dz 4.35 Lemma 4.9 For each n ≥ n0 , the map G ◦ Fn : SN−1 −→ SN−1 4.36 ζn θ, z : 0, 1 × SN−1 −→ SN−1 4.37 is homotopic to the identity Proof Define 16 Boundary Value Problems by ⎧ ⎪ ⎪G ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ζn θ, z ⎪G ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩z 1 − 2θ s n, z ψR w z − nz 2θψR w z − nz 1 − 2θ s n, z ψR w z − nz 2θψR w z − nz ψR w z − n/2 1 − θ z ψR w z − n/2 1... continuity of G, it is easy to check that lim − ζn θ, z G θ → 1/2 ψR w z − nz ψR w z − nz 4.41 H1 Thus, ζn θ, z ∈ C 0, 1 × SN−1 , SN−1 and ζn 0, z G Fn z ζn 1, z z provided n ≥ n0 This completes the proof ∀z ∈ SN−1 , ∀z ∈ SN−1 , 4.42 Boundary Value Problems 17 Theorem 4.10 Assume that q is a positive continuous function in ÊN and satisfies q1 and q2 Then, J u has at least two critical points in K < α∞ , 4.43... t∞ u dz 4.18 p Ω p q z − q∞ t∞ u dz Boundary Value Problems 13 From 4.17 and 4.18 , we have 1 p J ∞ t∞ u < α∞ p Ω q z − q∞ t∞ u dz m−2 q∞ t2 ∞ 2 ≤ α∞ 1 pq∞ < α∞ m − 2 2/ p−2 ∞ m α p−2 4.19 Hence, there exists m0 ≥ m1 > 2 such that if 2 < m < m1 , then J ∞ t∞ u ≤ α∞ δ0 , where t∞ u ∈ M∞ Ω 4.20 By Lemma 4.2, we obtain z |∇ t∞ u |2 |z| Ê N t∞ u 2 → − dz / 0 , 4 .21 or z |∇u|2 |z| Ê N → − u2 dz / 0 ... and 0 < δ < 2 2 q2 In this section, assume that q is a positive continuous function in ÊN and satisfies q1 , and 1 ψR z w z − nz ∈ H0 Ω for each n ∈ Æ By Lemma 2.4 i , q2 Let z ∈ SN−1 and wn z Boundary Value Problems 15 there exist unique numbers n, z > 0 such that s n, z wn ∈ M Ω We define a map Fn : 1 SN−1 → H0 Ω by s n, z wn z s n, z wn z Fn z z for z ∈ SN−1 4.30 H1 Then, we have the following... , ϕ J su u , su , ϕ u ∀ϕ ∈ Tu Σ 4.11 1 0 if and only if J su u , ϕ 0 for all ϕ ∈ Tu Σ Since H0 Ω is a ii By i , K u −1 0, so it is equivalent to J su u 0 in H Ω Hilbert space and J su u , u 12 Boundary Value Problems Lemma 4.4 Assume that q is a positive continuous function in ÊN and satisfies q1 and for m > 2 and 0 < δ < 2 m q∞ 2 ©q z ≥ q∞ C exp −δ|z| where 0 < C ≤ m−2 q∞ 2 4.12 We have that there.. .Boundary Value Problems 11 where w is the positive solution of 1.2 in ÊN Suppose the subsequence zn /|zn | → z0 as n → ∞, where z0 is a unit vector in ÊN Then, by the Lebesgue dominated convergence theorem,... and a unique s∞ > 0 such that su u ∈ M Ω and u ∈ M∞ Ω Then, we have the following results Lemma 4.11 For each u ∈ Σ, we have that p−m ∞ ∞ J su u ≤ J su u ≤ J ∞ s∞ u , u p−2 where m > 2 4.48 18 Boundary Value Problems Proof Since m/2 q∞ ©q © q∞ , where m > 2, we obtain that for each u ∈ Σ and z J su u ≤ J ∞ su u ≤ supJ ∞ su J ∞ s∞ u , u s≥0 J su u 1 ∞ s u 2 u supJ su ≥ J s∞ u u s≥0 1 ≥ 2 Ω q∞ s∞ u u... Rabinowitz 26 Lemma 4.13 Let V be a compact metric space, V0 ⊂ V a closed set, X a Banach space, χ ∈ C V0 , X and let us define the complete metric space M by M g ∈ C V, X | g s χ s if s ∈ V0 4.53 Boundary Value Problems 19 with the usual distance d Let ϕ ∈ C1 X, Ê and let us define c inf max ϕ g s , c1 g∈M s∈V max ϕ χ V0 4.54 If c > c1 , then for each ε > 0 and each g ∈ M such that max ϕ g s s∈V ≤c ε,... where 0 < C ≤ m−2 q∞ and 0 < δ < 2, 2 q2 then 1.1 admits at least three positive solutions in Ω Proof Applying Lemma 4.11 iii to obtain p−m ∞ α ≤ α Ω ≤ α∞ , p−2 p−m ∞ γ Ω ≤ γ Ω ≤ γ∞ Ω p−2 4.58 20 Boundary Value Problems Since α∞ < γ ∞ Ω < 2α∞ , given 0 < ε < 2α∞ − γ ∞ Ω /2, there is a number min{m1 , p} ≥ m2 > 2 such that for any 2 < m ≤ m2 , we have γ ∞ Ω < α∞ α Ω ≤ 2α∞ 4.59 Choosing some min{m2 , p} . Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 524862, 21 pages doi:10.1155/2010/524862 Research Article Multiple Positive Solutions of Semilinear Elliptic Problems. We define the Palais-Smale denoted by PS sequences, PS-values, and PS-conditions in H 1 0 Ω for J as follows. Boundary Value Problems 3 Definition 2.1. i For β ∈ ,asequence{u n } is a. PS αΩ 1 -condition or αΩ 1 is a critical value, then αΩ 2 <αΩ 1 . Proof. See Chen et al. 13 or Lin et al. 15. 4 Boundary Value Problems Remark 2.7. The above definitions and lemmas