Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 524862, 21 pages docx

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  qz|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 qz ≥ 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|→∞ qzq ∞ > 0andqz ≥ 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|→∞ qz q ∞ , qz ≥ q ∞ > 0, and there exist positive constants C, γ, R 0 such that qz ≥ q ∞  C/|z| γ for |z|≥R 0 ,then1.1 has at least two nontrivial solutions oneispositiveandtheother changes sign.Esteban6, 7 and Cao 8 have studied the multiplicity of solutions of −Δu  u  qz|u| p−2 u with Neumann condition in an exterior domain N \ D,whereD is a C 1,1 bounded domain in N .Hirano9 proved that if q − q ∞  ∞ is sufficiently small and qz ≥ 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 qzazμbz.Wu11 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 qz ≥ 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 Ju 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  qzu 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∈MΩ J  u  , 2.2 where MΩ  {u ∈ H 1 0 Ω \{0}|aubu  } and α ∞  Ω   inf u∈M ∞ Ω J ∞  u  , 2.3 where M ∞ Ω  {u ∈ H 1 0 Ω \{0}|aub ∞ 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 au n bu  n o n 12p/p − 2β  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 ∈ MΩ and sup s≥0 JsuJs u u. ii Let β>0 and {u n } a sequence in H 1 0 Ω \{0} for J such that u n / ≡ 0,Ju n β  o n 1 and au n bu  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 MΩ and Js 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 ∈ MΩ.Moreover, αΩ > 0. Lemma 2.6. Let Ω 1 Ω 2 .IfJ satisfies the PS αΩ 1  -condition or αΩ 1  is a critical value, then αΩ 2  <αΩ 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 .ByLions2, 3,Struwe16, 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|→∞ qzq ∞ > 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 } ∞ n1 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  i1 w i  ·−z i n   o n  1  strongly in H 1  N  , J  u n   J  u   l  i1 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 } ∞ n1 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  i1 w i  ·−z i n   o n  1  strongly in H 1  N  , J ∞  u n   J ∞  u   l  i1 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 }⊂MΩ 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 Ju 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 }⊂MΩ 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 Ju 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 ∞ wα ∞ . 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 Jtw 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 σz,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|  σz, z/|z|o1 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  J0,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 qz 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  Bz;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  B0;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 | ≥R1 }  q  z  − q ∞  w  z − z  p dz  t p 1 p  {| z | ≤R1 } 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 | ≤R1 } q ∞ w  z − z  p dz ≤ C 3 exp  −p | z |  . 3.21 Boundary Value Problems 9 Since qz ≥ 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 | ≤R1 }  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 Jtw z  <α ∞ . Using the Ekeland variational principle or se e Stuart 22,thereisaPS αΩ - sequence {u n }⊂MΩ for J. Then, we apply Lemma 3.3ii 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 2.4i,thereexistss z > 0suchthats z w z ∈ MΩ. Thus, by Lemma 3.6, αΩ ≤ Js z w z  ≤ sup t≥0 Jtw z  <α ∞ for |z|≥R 1 . Using the Ekeland variational principle, there is a PS αΩ -sequence {u n }⊂MΩ for J. Apply Lemma 3.3ii, there exists at least one positive solution u 0 of 1.1 in Ω such that Ju 0 αΩ. 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/2q ∞ qz where m>2, then there exists m 0 > 2 such that for m ≤ m 0 ,weobtainthat 2αΩ >α ∞ . Proof. Since qz q ∞ ,byLions2, 3,letw 0 ∈ H 1  N  be a positive solution of −Δw 0  w 0  qz|w 0 | p−2 w 0 in N and Jw 0 α N .ByLemma 2.4i 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 − 2m/2 2/p−2 α N  >α ∞ . Choose some m 0 > 2suchthatfor 2 <m≤ m 0 ,then2α N  >α ∞ .ByLemma 2.6 and Theorem 3.7,2αΩ > 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 ∞ vα ∞ 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 αΩ 1  -condition or αΩ 1  is a critical value, then αΩ 2  <αΩ 1 . Proof. See Chen et al. 13 or Lin et al. 15. 4 Boundary Value Problems Remark 2.7. The above definitions and lemmas