Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 325415, 16 pages doi:10.1155/2010/325415 Research Article Variational Approach to Impulsive Differential Equations with Dirichlet Boundary Conditions Huiwen Chen and Jianli Li Department of Mathematics, Hunan Normal University, Changsha, Hunan 410081, China Correspondence should be addressed to Jianli Li, ljianli@sina.com Received 18 September 2010; Accepted November 2010 Academic Editor: Zhitao Zhang Copyright q 2010 H Chen and J Li This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited We study the existence of n distinct pairs of nontrivial solutions for impulsive differential equations with Dirichlet boundary conditions by using variational methods and critical point theory Introduction Impulsive effects exist widely in many evolution processes in which their states are changed abruptly at certain moments of time Such processes are naturally seen in control theory 1, , population dynamics , and medicine 4, Due to its significance, a great deal of work has been done in the theory of impulsive differential equations In recent years, many researchers have used some fixed point theorems 6, , topological degree theory , and the method of lower and upper solutions with monotone iterative technique to study the existence of solutions for impulsive differential equations On the other hand, in the last few years, some researchers have used variational methods to study the existence of solutions for boundary value problems 10–16 , especially, in 14–16 , the authors have studied the existence of infinitely many solutions by using variational methods However, as far as we know, few researchers have studied the existence of n distinct pairs of nontrivial solutions for impulsive boundary value problems by using variational methods 2 Boundary Value Problems Motivated by the above facts, in this paper, our aim is to study the existence of n distinct pairs of nontrivial solutions to the Dirichlet boundary problem for the second-order impulsive differential equations u t λh t, u t −Δu tj 0, I j u tj u t / tj , a.e t ∈ 0, T , , j u T 1, 2, , p, 1.1 0, where t0 < t1 < · · · < < T , λ > 0, h ∈ C 0, T × R, R , Ij ∈ C R, R , j 1, 2, , p, u tj − u t− , u tj and u t− denote the right and the left limits, respectively, of Δu tj j j u tj at t tj , j 1, 2, , p Preliminaries Definition 2.1 Suppose that E is a Banach space and ϕ ∈ C1 E, R If any sequence {uk } ⊂ E for which ϕ uk is bounded and ϕ uk → as k → ∞ possesses a convergent subsequence in E, we say that ϕ satisfies the Palais-Smale condition Let E be a real Banach space Define the set Σ {A | A ⊂ E \ {θ} as symmetric closed set} Theorem 2.2 see 17, Theorem 3.5.3 Let E be a real Banach space, and let ϕ ∈ C1 E, R be an even functional which satisfies the Palais-Smale condition, ϕ is bounded from below and ϕ0 0; suppose that there exists a set K ⊂ Σ and an odd homeomorphism h : K → Sn−1 n − one-dimensional unit sphere and supx∈K ϕ x < 0, then ϕ has at least n distinct pairs of nontrivial critical points To begin with, we introduce some notation Denote by X the Sobolev space H0 0, T , and consider the inner product T u t v t dt u, v 2.1 and the norm 1/2 T u t u dt 2.2 Hence, X is reflexive We define the norm in C 0, T as x ∞ maxt∈ 0,T |x t | For u ∈ H 0, T , we have that u and u are absolutely continuous and u ∈ L2 0, T u t − u t− for every t ∈ 0, T If u ∈ H0 0, T , then u is absolutely Hence, Δu t continuous and u ∈ L 0, T In this case, the one-sided derivatives u t− , u t may not exist As a consequence, we need to introduce a different concept of solution Suppose that u| tj ,tj satisfies uj ∈ H tj , tj , and u ∈ C 0, T such that for every j 1, 2, , p, uj it satisfies the equation in problem 1.1 for t / tj , a.e t ∈ 0, T , the limits u tj , u t− , and j Boundary Value Problems j 1, 2, , p exist, and impulsive conditions and boundary conditions in problem 1.1 hold, we say it is a classical solution of problem 1.1 Consider the functional ϕ : X −→ R, 2.3 defined by u ϕu −λ T p H t, u t dt − u tj Ij s ds, j 2.4 u h t, s ds Clearly, ϕ is a Fr´ chet differentiable functional, whose Fr´ chet e e where H t, u derivative at the point u ∈ X is the functional ϕ u ∈ X ∗ given by T ϕ u v T u t v t dt − λ p h t, u t v t dt − I j u tj v tj , 2.5 j for any v ∈ X Obviously, ϕ is continuous Lemma 2.3 If u ∈ X is a critical point of the functional ϕ, then u is a classical solution of problem 1.1 Proof The proof is similar to the proof of 16, Lemma 2.4 , and we omit it here √ Lemma 2.4 Let u ∈ X, then u ∞ ≤ T u Proof For u ∈ X, then u t |u t | Hence, for t ∈ 0, T , by Holder’s inequality, we have ¨ u T T u s ds ≤ u s ds ≤ √ 1/2 T u s T ds √ T u , 2.6 which completes the proof Main Results Theorem 3.1 Suppose that the following conditions hold i There exist a, b > and γ ∈ 0, such that |h t, u | ≤ a b|u|γ for any t, u ∈ 0, T × R 3.1 ii h t, u is odd about u and H t, u > for every t, u ∈ 0, T × R \ {0} iii Ij u j 1, 2, , p are odd and u I j s ds ≤ for any u ∈ R j 1, 2, , p Boundary Value Problems Then for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct pairs of nontrivial classical solutions Proof By 2.4 , ii , and iii , ϕ ∈ C1 X, R is an even functional and ϕ Next, we will verify that ϕ is bounded from below In view of i , iii , and Lemma 2.4, we have u ϕu ≥ ≥ u u 2 T −λ p Ij s ds j T −λ u tj H t, u t dt − a|u t | b|u t |γ dt 3.2 − λaT 3/2 u − λbT γ /2 u γ > −∞, for any u ∈ X That is, ϕ is bounded from below In the following we will show that ϕ satisfies the Palais-Smale condition Let {uk } ⊂ X, Then, there exists M > such that {ϕ uk } is a bounded sequence and limk → ∞ ϕ uk such that ϕ uk ≤ M 3.3 In view of 3.2 , we have M≥ uk 2 − λaT 3/2 uk − λbT γ /2 uk γ 3.4 So {uk } is bounded in X From the reflexivity of X, we may extract a weakly convergent u in X Next, we will verify that {uk } subsequence that, for simplicity, we call {uk }, uk strongly converges to u in X By 2.5 , we have ϕ uk − ϕ u uk − u uk − u −λ T h t, uk t − h t, u t uk t − u t dt 3.5 p I j uk t j − I j u tj uk t j − u t j j By uk u in X, we see that {uk } uniformly converges to u in C 0, T So, T λ h t, uk t − h t, u t uk t − u t dt −→ 0, 3.6 p I j uk t j j − I j u tj uk t j − u t j −→ as k −→ ∞ Boundary Value Problems By limk → ∞ ϕ uk and uk u, we have uk − u −→ as k −→ ∞ ϕ uk − ϕ u 3.7 In view of 3.5 , 3.6 , and 3.7 , we obtain uk − u → as k → ∞ Then, ϕ satisfies the Palais-Smale condition √ 2T /mπ sin mπ/T t, m 1, 2, , n, then Let vm t vm T m2 π T2 |vm t |2 dt, m 1, 2, , n 3.8 r > 3.9 Define n Kn r cm vm | m n cm r2 , m √ Then, for any r > 0, there exists an odd homeomorphism f : Kn r → Sn−1 Let < r < 1/ T , √ √ T r < for any u ∈ Kn r By ii , we have then u ∞ ≤ T u ut h t, s ds > as u t / 0, H t, u t then T 3.10 H t, u t dt > for any u ∈ Kn r p T u tj Ij s ds, then αn > 0, βn ≤ Let αn infu∈Kn r H t, u t dt, βn infu∈Kn r j 1/2 r − βn α−1 > 0, then when λ > λn , for any u ∈ Kn r , we have Let λn n ϕ u ≤ < r − λαn − βn 2 r − λn αn − βn 3.11 By Theorem 2.2, ϕ possesses at least n distinct pairs of nontrivial critical points That is, problem 1.1 has at least n distinct pairs of nontrivial classical solutions Corollary 3.2 Let the following conditions hold: i h t, u is bounded, ii h t, u is odd about u and H t, u > for every t, u ∈ 0, T × R \ {0}, iii Ij u j 1, 2, , p are odd and u I j s ds ≤ for any u ∈ R j 1, 2, , p Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct pairs of nontrivial classical solutions 6 Boundary Value Problems Proof Let γ in Theorem 3.1, then Corollary 3.2 holds Theorem 3.3 Suppose that the following conditions hold i There exists a, b > and γ ∈ 0, such that b|u|γ |h t, u | ≤ a for any t, u ∈ 0, T × R ii There exists aj , bj > and γj ∈ 0, Ij u iii h t, u and Ij u 0, T × R \ {0} bj |u|γj ≤ aj 3.12 1, 2, , p such that j for any u ∈ R j 1, 2, , p 3.13 1, 2, , p are odd about u and H t, u > for every t, u ∈ j Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct pairs of nontrivial classical solutions Proof By 2.4 and iii , ϕ ∈ C1 X, R is an even functional and ϕ Next, we will verify that ϕ is bounded from below Let M1 max{a1 , a2 , , ap }, M2 max{b1 , b2 , , bp } In view of i , ii , and Lemma 2.4, we have u ϕu ≥ u 2 p T −λ u tj H t, u t dt T −λ Ij s ds j b|u t |γ a|u t | dt p − aj u tj bj u tj γj 3.14 j ≥ u 2 − λaT 3/2 u − λbT γ /2 u γ √ − pM1 T u p − M2 T γj /2 u γj j > −∞, for any u ∈ X That is, ϕ is bounded from below In the following, we will show that ϕ satisfies the Palais-Smale condition As in the proof of Theorem 3.1, by 3.3 and 3.14 , we have M≥ uk 2 − λaT 3/2 uk − λbT γ /2 uk γ √ − pM1 T uk − M2 p T γj /2 uk γj j 3.15 Boundary Value Problems It follows that {uk } is bounded in X In the following, the proof of the Palais-Smale condition is the same as that in Theorem 3.1, and we omit it here for Take the same Kn r as in Theorem 3.1, then √ any r > 0, there exists an odd √ √ Tr < homeomorphism f : Kn r → Sn−1 Let < r < 1/ T , then u ∞ ≤ T u for any u ∈ Kn r By iii , we have ut h t, s ds > as u t / H t, u t Then, λn T 3.16 H t, u t dt > for any u ∈ Kn r p T u tj Let αn infu∈Kn r H t, u t dt, βn infu∈Kn r Ij s ds, then αn > Let j −1 max{0, 1/2 r − βn αn }, then when λ > λn , for any u ∈ Kn r , we have ϕu ≤ r − λαn − βn < r − λn αn − βn ≤ 2 3.17 By Theorem 2.2, ϕ possesses at least n distinct pairs of nontrivial critical points That is, problem 1.1 has at least n distinct pairs of nontrivial classical solutions Corollary 3.4 Let the following conditions hold: i h t, u is bounded, ii Ij u j 1, 2, , p are bounded, iii h t, u and Ij u 0, T × R \ {0} j 1, 2, , p are odd about u and H t, u > for every t, u ∈ Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct pairs of nontrivial classical solutions Proof Let γ and γj j 1, 2, , p in Theorem 3.3, then Corollary 3.4 holds Theorem 3.5 Suppose that the following conditions hold i There exist constants σ > such that h t, σ 0, h t, u > for every u ∈ 0, σ ii h t, u is odd about u iii Ij u j 1, 2, , p are odd and u I j s ds ≤ for any u ∈ R j 1, 2, , p Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct pairs of nontrivial classical solutions Proof Let h1 t, u ⎧ ⎪h t, σ , ⎪ ⎪ ⎨ h t, u , ⎪ ⎪ ⎪ ⎩ h t, −σ , u > σ, |u| ≤ σ, u < −σ, 3.18 Boundary Value Problems then h1 t, u is continuous, bounded, and odd Consider boundary value problem u t λh1 t, u t −Δu tj t / tj , a.e t ∈ 0, T , 0, I j u tj u , j u T 1, 2, , p, 3.19 Next, we will verify that the solutions of problem 3.19 are solutions of problem 1.1 In fact, let u0 t be the solution of problem 3.19 If max0≤t≤T u0 t > σ, then there exists an interval a, b ⊂ 0, T such that u0 a u0 b σ, for any t ∈ a, b u0 t > σ 3.20 When t ∈ a, b , by i , we have u0 t −λh1 t, u −λh t, σ 3.21 c for any t ∈ a, b We consider the following Thus, there exist constants c such that u0 t two possible cases Case c ≥ 0, then u0 is nondecreasing in a, b By u0 a ≥ and u0 b ≤ 0, we have ≤ u0 a ≤ u0 t ≤ u0 b ≤ for every t ∈ a, b 3.22 That is, u0 t ≡ for any t ∈ a, b So, there exists a constant d such that u0 t ≡ d, which contradicts 3.20 Then, max0≤t≤T u0 t ≤ σ Similarly, we can prove that min0≤t≤T u0 t ≥ −σ Case c < 0, the arguments are analogous, then u0 t is solution of problem 1.1 For every u ∈ X, we consider the functional ϕ1 : X −→ R, 3.23 defined by u ϕ1 u −λ T p H1 t, u t dt − u tj Ij s ds, j 3.24 u h t, s ds where H1 t, u e It is clear that ϕ1 is Fr´ chet differentiable at any u ∈ X and T ϕ1 u v u t v t dt − λ T p h1 t, u t v t dt − I j u tj j v tj , 3.25 Boundary Value Problems for any v ∈ X Obviously, ϕ1 is continuous By Lemma 2.3, we have the critical points of ϕ1 as solutions of problem 3.19 By 3.24 , ii , and iii , ϕ1 ∈ C1 X, R is an even functional and ϕ1 0 for In the following, we will show that ϕ1 is bounded from below since h1 t, u |u| ≥ σ, thus T T ut 0 h1 t, s ds dt ≤ T σ H1 t, u t dt h1 t, s ds dt e > 3.26 By iii , we have ϕ1 u u ≥ u 2 −λ T p H1 t, u t dt − u tj Ij s ds j 3.27 − λe ≥ −λe, for any u ∈ X That is, ϕ1 is bounded from below In the following we will show that ϕ1 satisfies the Palais-Smale condition Let {uk } ⊂ X Then, there exists M3 > such that {ϕ1 uk } is a bounded sequence and limk → ∞ ϕ1 uk such that ϕ1 uk ≤ M3 3.28 By 3.27 , we have uk 2 ≤ M3 λe 3.29 It follows that {uk } is bounded in X In the following, the proof of the Palais-Smale condition is the same as that in Theorem 3.1, and we omit it here for Take the same Kn r as in Theorem 3.1, then,√ any r > 0, there exists an odd √ √ Tr < σ homeomorphism f : Kn r → Sn−1 Let < r < σ/ T , then u ∞ ≤ T u for any u ∈ Kn r By i and ii , we have ut H1 t, u t ut h1 t, s ds Then, T H1 t, u t dt > for any u ∈ Kn r h t, s dt > 0 as u t / 3.30 10 Boundary Value Problems p T u tj Ij s ds, then αn > 0, βn ≤ Let αn infu∈Kn r H1 t, u t dt, βn infu∈Kn r j −1 1/2 r − βn αn > 0, then when λ > λn , for any u ∈ Kn r , we have Let λn r − λαn − βn ϕ1 u ≤ r − λn αn − βn < 3.31 By Theorem 2.2, ϕ1 possesses at least n distinct pairs of nontrivial critical points Then, problem 3.19 has at least n distinct pairs of nontrivial classical solutions, that is, problem 1.1 has at least n distinct pairs of nontrivial classical solutions Theorem 3.6 Let the following conditions hold i There exist constants σ > such that h t, σ ii There exist aj , bj > 0, and γj ∈ 0, ≤ aj Ij u iii h t, u and Ij u j bj |u|γj j 0, h t, u > for every u ∈ 0, σ 1, 2, , p such that for any u ∈ R j 1, 2, , p 3.32 1, 2, , p are odd about u Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct pairs of nontrivial classical solutions Proof The proof is similar to the proof of Theorem 3.5, and we omit it here Theorem 3.7 Let the following conditions hold i There exist constants σ1 > such that h t, σ1 ≤ ii There exist aj , bj > 0, and γj ∈ 0, ≤ aj Ij u iii h t, u and Ij u t ∈ 0, T j bj |u|γj j 1, 2, , p such that for any u ∈ R j 1, 2, , p 1, 2, , p are odd about u and limu → h t, u /u 3.33 uniformly for Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct pairs of nontrivial classical solutions Proof Let h2 t, u ⎧ ⎪h t, σ1 , ⎪ ⎪ ⎨ h t, u , ⎪ ⎪ ⎪ ⎩ h t, −σ1 , u > σ1 , |u| ≤ σ1 , u < −σ1 , 3.34 Boundary Value Problems 11 then h2 t, u is continuous, bounded, and odd Consider boundary value problem u t λh2 t, u t −Δu tj t / tj , a.e t ∈ 0, T , 0, I j u tj u , j u T 1, 2, , p, 3.35 Next, we will verify that the solutions of problem 3.35 are solutions of problem 1.1 In fact, let u0 t be the solution of problem 3.35 If max0≤t≤T u0 t > σ1 , then there exists an interval a, b ⊂ 0, T such that u0 a u0 b for any t ∈ a, b σ1 , u0 t > σ1 3.36 When t ∈ a, b , by i , we have −λh2 t, u u0 t −λh t, σ1 ≥ 3.37 Thus, u0 t is nondecreasing in a, b By u0 a ≥ and u0 b ≤ 0, we have ≤ u0 a ≤ u0 t ≤ u0 b ≤ for every t ∈ a, b 3.38 That is, u0 t ≡ for any t ∈ a, b So, there exists a constant d such that u0 t ≡ d, which contradicts 3.36 Then max0≤t≤T u0 t ≤ σ1 Similarly, we can prove that min0≤t≤T u0 t ≥ −σ1 Then, u0 t is solution of problem 1.1 For every u ∈ X, we consider the functional ϕ2 : X −→ R, 3.39 defined by u ϕ2 u −λ T p H2 t, u t dt − u tj Ij s ds, j 3.40 u h t, s ds where H2 t, u e It is clear that ϕ2 is Fr´ chet differentiable at any u ∈ X and T ϕ2 u v u t v t dt − λ T p h2 t, u t v t dt − I j u tj v tj , 3.41 j for any v ∈ X Obviously, ϕ2 is continuous By Lemma 2.3, we have the critical points of ϕ2 as solutions of problem 3.35 By 3.40 and iii , ϕ2 ∈ C1 X, R is an even functional and ϕ2 12 M2 Boundary Value Problems Next, we will show that ϕ2 is bounded from below Let M1 max{b1 , b2 , , bp } since uh2 t, u ≤ for |u| ≥ σ1 , thus T T ut 0 T h2 t, s ds dt ≤ σ1 H2 t, u t dt max{a1 , a2 , , ap }, h2 t, s ds dt e 3.42 By ii and Lemma 2.4, we have u ϕ2 u ≥ u 2 −λ T p u tj H2 t, u t dt − Ij s ds j √ − λe − pM1 T u − M2 p T γj /2 u 3.43 γj j > −∞, for any u ∈ X That is, ϕ2 is bounded from below In the following we will show that ϕ2 satisfies the Palais-Smale condition Let {uk } ⊂ X Then, there exists M4 > such that {ϕ2 uk } is a bounded sequence and limk → ∞ ϕ2 uk such that ϕ2 uk ≤ M4 3.44 By 3.43 , we have uk 2 ≤ M4 λe √ pM1 T uk p T M2 γj /2 uk γj 3.45 j It follows that {uk } is bounded in X In the following, the proof of the Palais-Smale condition is the same as that in Theorem 3.1, and we omit it here Take the same Kn r as in Theorem 3.1, then for any r > 0, there exists an odd homeomorphism f : Kn r → Sn−1 By iii , for any < ε < 1, there exists δ > 0, when |u| ≤ δ, we have h2 t, u ≥ u − ε|u| 3.46 √ √ √ √ Let < r < min{σ1 / T , δ/ T }, then u ∞ ≤ T u T r < min{σ1 , δ} for any u ∈ Kn r T T ut T Then, H2 t, u t dt h2 t, s dt ≥ 1/2 − ε |u t |2 dt > for any u ∈ Kn r 0 Boundary Value Problems 13 λn u tj p T infu∈Kn r H2 t, u t dt, βn infu∈Kn r Ij s ds, then αn > Let Let αn j −1 max{ 1/2 r − βn αn , 0}, then when λ > λn , for any u ∈ Kn r , we have ϕ1 u ≤ < r − λαn − βn 2 r − λn αn − βn 3.47 ≤ By Theorem 2.2, ϕ2 possesses at least n distinct pairs of nontrivial critical points Then, problem 3.35 has at least n distinct pairs of nontrivial classical solutions, that is, problem 1.1 has at least n distinct pairs of nontrivial classical solutions Theorem 3.8 Let the following conditions hold i There exist constants σ1 > such that h t, σ1 ≤ uniformly for t ∈ 0, T ii limu → h t, u /u iii h t, u and Ij u 1, 2, , p j u I j 1, 2, , p are odd about u and s ds ≤ for any u ∈ R j Then, for any n ∈ N, there exists λn such that λ > λn , and problem 1.1 has at least n distinct pairs of nontrivial classical solutions Proof The proof is similar to the proof of Theorem 3.7, and we omit it here Some Examples Example 4.1 Consider boundary value problem u t λ t u t −Δu tj t / tj , a.e t ∈ 0, π , 0, −u tj , j uπ 4.1 u 1, 2, It is easy to see that conditions i , ii , and iii of Theorem 3.1 hold Let inf αn u∈Kn r βn then λn when λ > solutions π u∈Kn r inf − u∈Kn r t |u t |4/3 dt > inf u tj sds j inf − u∈Kn r j π |u t |2 dt > u tj 3r , 4n2 4.2 ≥ −πr , 1/2 r − βn α−1 < 4π /3 n2 Applying Theorem 3.1, then for any n ∈ N, n 4π /3 n2 , problem 4.1 has at least n distinct pairs of nontrivial classical 14 Boundary Value Problems Example 4.2 Consider boundary value problem u t λ t u t −Δu tj t / tj , a.e t ∈ 0, π , 0, − u tj , u uπ j 4.3 1, 2, It is easy to see that conditions i , ii , and iii of Theorem 3.3 hold Let r αn inf u∈Kn r inf − βn u∈Kn r π u∈Kn r u tj 1/3 s j t |u t |4/3 dt > inf inf − ds u∈Kn r j π |u t |2 dt > u tj 4/3 √ 1/2 π, 3r , 4n2 4.4 >− , then λn 1/2 r − βn α−1 < 24π /3 n2 Applying Theorem 3.3, then for any n n ∈ N, when λ > 24π /3 n2 , problem 4.3 has at least n distinct pairs of nontrivial classical solutions Example 4.3 Consider boundary value problem u t λ t2 −Δu tj j uπ 4.5 1, 2, 1, it is easy to see that conditions i , ii , and iii of Theorem 3.5 hold Let π αn t / tj , a.e t ∈ 0, π , 0, −u tj , u Let σ u t − u t inf u∈Kn r t2 βn inf − u∈Kn r π 1 |u t |2 − |u t |4 dt > inf u∈Kn r 4 u tj inf − sds j u∈Kn r j u tj |u t |2 dt > r2 , 4n2 4.6 ≥ −πr , then λn 1/2 r − βn α−1 < 4π n2 Applying Theorem 3.5, then for any n ∈ N, n when λ > 4π n , problem 4.5 has at least n distinct pairs of nontrivial classical solutions Example 4.4 Consider boundary value problem u t λ u t − t u t −Δu tj u − u tj , uπ t / tj , a.e t ∈ 0, π , 0, j 1, 2, 4.7 Boundary Value Problems r Let σ1 √ 1/4 π, 1, it is easy to see that conditions i , ii , and iii of Theorem 3.7 hold Let π αn 15 inf u∈Kn r βn then λn N, when λ > solutions 1 |u t |2 − inf − u∈Kn r u tj 1/3 s j t |u t |4 dt > inf u∈Kn r ds inf − u∈Kn r j u tj π |u t |2 dt > 4/3 >− , r2 , 4n2 4.8 1/2 r − βn α−1 < 24π 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