The existence of a positive solution to a Kirchhoff type problem on R-N is proved by using variational methods, and the new result does not require usual compactness conditions. A cut-off functional is utilized to obtain the bounded Palais-Smale sequences. (c) 2012 Elsevier Inc. All rights reserved

Li, Fuyi

Li, Yuhua

Shi, Junping

English

College of William & Mary: W&M Publish

W&M ScholarWorks Arts & Sciences Articles Arts and Sciences 2012 Existence of a positive solution to Kirchhoff type problems without compactness conditions Yuhua Li Fuyi Li Junping Shi Junping Shi William & Mary, jxshix@wm.edu Follow this and additional works at: https://scholarworks.wm.edu/aspubs Recommended Citation Li, Y., Li, F., & Shi, J. (2012). Existence of a positive solution to Kirchhoff type problems without compactness conditions. Journal of Differential Equations, 253(7), 2285-2294. This Article is brought to you for free and open access by the Arts and Sciences at W&M ScholarWorks. It has been accepted for inclusion in Arts & Sciences Articles by an authorized administrator of W&M ScholarWorks. For more information, please contact scholarworks@wm.edu. J. Differential Equations 253 (2012) 2285–2294Contents lists available at SciVerse ScienceDirectJournal of Differential Equationswww.elsevier.com/locate/jdeExistence of a positive solution to Kirchhoff type problemswithout compactness conditions✩Yuhua Li a, Fuyi Li a,∗, Junping Shi a,ba School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, PR Chinab Department of Mathematics, College of William and Mary, Williamsburg, VA 23187-8795, USAa r t i c l e i n f o a b s t r a c tArticle history:Received 20 February 2012Revised 18 May 2012Available online 30 May 2012Keywords:Kirchhoff type problemVariational methodsCut-off functionalThe existence of a positive solution to a Kirchhoff type problemon RN is proved by using variational methods, and the new resultdoes not require usual compactness conditions. A cut-off functionalis utilized to obtain the bounded Palais–Smale sequences.© 2012 Elsevier Inc. All rights reserved.1. IntroductionIn this paper, we consider the positive solutions to the following nonlinear Kirchhoff type problem(a + λ∫RN|∇u|2 + λb∫RNu2)[−u + bu] = f (u), in RN , (1.1)where N  3, and a, b are positive constants, λ  0 is a parameter. Kirchhoff type problem ona bounded domain Ω ⊂RN⎧⎪⎨⎪⎩−(a + b∫Ω|∇u|2)u = f (u), in Ω,u = 0, on ∂Ω(1.2)✩ Partially supported by National Natural Science Foundation of China (Grant No. 11071149), Science Council of ShanxiProvince (2010011001-1), and Shanxi 100 Talent program.* Corresponding author.E-mail address: fyli@sxu.edu.cn (F. Li).0022-0396/$ – see front matter © 2012 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.jde.2012.05.0172286 Y. Li et al. / J. Differential Equations 253 (2012) 2285–2294has been studied by many authors, for example [2,4–6,8,12,13,19,20]. Many solvability conditionswith f near zero and infinity for problem (1.2) have been considered, such as the superlinearcase [12]; and asymptotical linear case [16]. In addition, the following growth condition on f is oftenassumed:(f) f (t)t  4F (t) for |t| large, where F (t) = ∫ t0 f (s)ds,which assures the boundedness of any (PS) or Cerami sequence. Indeed the condition (f) may appearin different forms as follows:(f0) there exists θ  1 such that θG(t) G(st) for all t ∈ R and s ∈ [0,1], where G(t) = f (t) − 4F (t)(see [16]);(f1) lim|t|→∞[ f (t)t − 4F (t)] = ∞ (see [19]); or(f2) lim|t|→∞ G(t) = ∞ and there exists σ > max{1,N/2} such that | f (t)|σ  CG(t)|t|σ for |t| large(see [12]).In the above papers, each of the conditions (f0)–(f2) implies that condition (f) holds. On the otherhand, the condition (f) is sufficient to show the boundedness of any (PS) or Cerami sequence, whichhas been proved in [18].There are few papers considering Kirchhoff type problems on RN except [18]. In [18], the authorstudied the problem−(a + b∫RN|∇u|2)u + V (x)u = f (u), x ∈RN .The existence of nontrivial solutions was proved in [18] under the condition (f) and(V) V ∈ C(RN ,R), infx∈RN V (x) > 0 and for each M > 0, meas {x ∈RN : V (x) M} < ∞;(H1) f ∈ C(R+,R+) and | f (t)| C(|t| + |t|p−1) for all t ∈ R+ = [0,∞) and some p ∈ (2,2∗), where2∗ = 2N/(N − 2) for N  3;(H2) limt→0 f (t)t = 0;(f3) limt→∞ F (t)t4 = ∞.In this paper, we prove the existence of positive solutions of (1.1) without the condition (f)(or (f0)–(f2)), and we use a cut-off functional to obtain bounded (PS) sequences. We assume thefollowing weaker condition:(H3) limt→∞ f (t)t = ∞.Our main result is as follows:Theorem 1.1. Assume that N  3, and a, b are positive constants, λ 0 is a parameter. If the conditions (H1),(H2) and (H3) hold, then there exists λ0 > 0 such that for any λ ∈ [0, λ0), (1.1) has at least one positivesolution.Theorem 1.1 appears to be the first existence result for Eq. (1.1). We also remark that the condi-tion (H3) is weaker than the ones in the above mentioned papers, in which lim|t|→∞ f (t)/t3 = ∞ ora constant (which implies (H3)) was assumed. Since the result in Theorem 1.1 holds for λ = 0, thenwe have the following corollary regarding the well-known semilinear equation.Corollary 1.2. Assume that N  3, and b is a positive constant. If the conditions (H1), (H2) and (H3) hold,then the problemY. Li et al. / J. Differential Equations 253 (2012) 2285–2294 2287−u + bu = f (u), in RN (1.3)has at least one positive solution.Note that the existence result like the one in Corollary 1.2 has been obtained by many authors, forexample, [1,3,10,11,14]. Hence our result in Theorem 1.1 can be regarded as an extension of the clas-sical result for the semilinear equation (1.3) to the case of the nonlinear Kirchhoff type problem (1.1).On the other hand, it is not clear whether the result in Theorem 1.1 still holds for large λ > 0. Inour result, the choice of λ0 depends on the nonlinearity f , constants N , a and b, Sobolev embeddingconstant, several test functions and constants used in the proof.We recall some preliminaries and prove some lemmas in Section 2, and we give the proof ofTheorem 1.1 in Section 3.2. PreliminariesLet H1(RN ) be the usual Sobolev space equipped with the inner product and norm(u, v) =∫RN[∇u · ∇v + buv], ‖u‖ = (u,u)1/2.We denote by | · |q the usual Lq(RN ) norm. Then we have that H1(RN ) ↪→ Lq(RN ) continuously forq ∈ [2,2∗]. Let H = H1r (RN ) be the subspace of H1(RN ) containing only the radial functions. ThenH ↪→ Lq(RN ) compactly for q ∈ (2,2∗) [17, Corollary 1.26, p. 18]. In this paper, we consider positivesolutions to (1.1), then we assume that f (t) = 0 for t < 0.Define a functional Jλ on the space H byJλ(u) = 12a‖u‖2 + 14λ‖u‖4 −∫RNF (u), u ∈ H .Then we have from (H1) that Jλ is well defined on H and is of C1 for all λ 0, and(J ′λ(u), v)= a(u, v) + λ‖u‖2(u, v) −∫RNf (u)v, u, v ∈ H .It is standard to verify that the weak solutions of (1.1) correspond to the critical points of the func-tional Jλ .To overcome the difficulty of finding bounded Palais–Smale sequences for the associated func-tional Jλ , following [7,9], we use a cut-off function ψ ∈ C∞(R+,R) satisfying⎧⎪⎪⎪⎨⎪⎪⎪⎩ψ(t) = 1, t ∈ [0,1],0ψ(t) 1, t ∈ (1,2),ψ(t) = 0, t ∈ [2,∞),∥∥ψ ′∥∥∞  2,and study the following modified functional J Tλ : H →R defined byJ Tλ (u) =12a‖u‖2 + 14λhT (u)‖u‖4 −∫RNF (u), u ∈ H,2288 Y. Li et al. / J. Differential Equations 253 (2012) 2285–2294where for every T > 0,hT (u) = ψ(‖u‖2T 2).With this penalization, for T > 0 sufficiently large and for λ sufficiently small, we are able to find acritical point u of J Tλ such that ‖u‖ T and so u is also a critical point of Jλ . We recall the followingresult. The “monotonicity trick” at the core of this theorem was invented by Struwe (see [15]).Theorem 2.1. (See [6].) Let (X,‖ · ‖) be a Banach space and I ⊂ R+ an interval. Consider the family of C1functionals on XJμ(u) = A(u) − μB(u), μ ∈ I,with B nonnegative and either A(u) → ∞ or B(u) → ∞ as ‖u‖ → ∞ and such that Jμ(0) = 0.For any μ ∈ I we setΓμ ={γ ∈ C([0,1], X): γ (0) = 0, Jμ(γ (1))< 0}.If for every μ ∈ I the set Γμ is nonempty andcμ = infγ∈Γμmaxt∈[0,1] Jμ(γ (t))> 0,then for almost every μ ∈ I there is a sequence {un} ⊂ X such that(i) {un} is bounded;(ii) Jμ(un) → cμ;(iii) J ′μ(un) → 0 in the dual X−1 of X .In our case, X = H ,A(u) = 12a‖u‖2 + 14λhT (u)‖u‖4, B(u) =∫RNF (u).So the perturbed functional which we study isJ Tλ,μ(u) =12a‖u‖2 + 14λhT (u)‖u‖4 − μ∫RNF (u),and((J Tλ,μ)′(u), v)= a(u, v) + λhT (u)‖u‖2(u, v) + λ2T 2ψ ′(‖u‖2T 2)‖u‖4(u, v) − μ∫RNf (u)v. (2.1)The following Lemmas 2.2–2.4 imply that J Tλ,μ satisfies the conditions of Theorem 2.1.Lemma 2.2. Γμ = ∅ for all μ ∈ I = [δ,1], where δ ∈ (0,1) is a positive constant.Y. Li et al. / J. Differential Equations 253 (2012) 2285–2294 2289Proof. We choose φ ∈ C∞0 (RN ) with φ  0,‖φ‖ = 1 and supp(φ) ⊂ B(0, R) for some R > 0. By (H3),we have that for any C1 > 0 with C1δ∫B(0,R) φ2 > a/2, there exists C2 > 0 such thatF (t) C1|t|2 − C2, t ∈R+. (2.2)Then for t2 > 2T 2,J Tλ,μ(tφ) =12at2‖φ‖2 + 14λψ(t2‖φ‖2T 2)t4‖φ‖4 − μ∫RNF (tφ)= 12at2 − μ∫RNF (tφ) 12at2 − δC1t2∫B(0,R)φ2 + C3.Then we can choose t > 0 large such that J Tλ,μ(tφ) < 0. The proof is completed. Lemma 2.3. There exists a constant c > 0 such that cμ  c > 0 for all μ ∈ I .Proof. For any u ∈ H and μ ∈ I , using (H1) and (H2), for ε ∈ (0,a/2), we haveJ Tλ,μ(u)12a‖u‖2 + 14λhT (u)‖u‖4 −∫RN(12εbu2 + Cε|u|p) 14a‖u‖2 − Cε∫RN|u|p.By Sobolev’s embedding theorem, we conclude that there exists ρ > 0 such that J Tλ,μ(u) > 0 for anyμ ∈ I and u ∈ H with 0 < ‖u‖ ρ . In particular, for ‖u‖ = ρ , we have J Tλ,μ(u) c > 0. Fix μ ∈ I andγ ∈ Γμ . By the definition of Γμ,‖γ (1)‖ > ρ . By continuity, we deduce that there exists tγ ∈ (0,1)such that ‖γ (tγ )‖ = ρ . Therefore, for any μ ∈ I ,cμ  infγ∈ΓμJ Tλ,μ(γ (tγ )) c > 0.The proof is completed. Lemma 2.4. For any μ ∈ I and 8λT 2 < a, each bounded Palais–Smale sequence of the functional J Tλ,μ admitsa convergent subsequence.Proof. Let μ ∈ I and {un} be a bounded (PS) sequence of J Tλ,μ , namely{un} and{J Tλ,μ(un)}are bounded,(J Tλ,μ)′(un) → 0 in H ′,2290 Y. Li et al. / J. Differential Equations 253 (2012) 2285–2294where H ′ is the dual space of H . Subject to a subsequence, we can assume that there exists u ∈ Hsuch thatun ⇀ u in H,un → u in Lp(RN),un → u a.e. in RN .By (H1) and (H2), for any ε ∈ (0,a/2), there exists Cε > 0 such that∣∣ f (t)∣∣ bε|t| + Cε|t|p−1, t ∈R, (2.3)hence,∣∣∣∣∫RNf (un)(un − u)∣∣∣∣∫RN∣∣ f (un)∣∣|un − u| bε|un|2|un − u|2 + Cε∫RN|un|p−1|un − u| εC‖un‖‖un − u‖ + Cε|un|p−1p |un − u|p .It follows that∫RNf (un)(un − u) → 0.Thus,0 ← (( J Tλ,μ)′(un),un − u)= a(un,un − u) + λhT (un)‖un‖2(un,un − u)+ λ2T 2ψ ′(‖un‖2T 2)‖un‖4(un,un − u) − μ∫RNf (un)(un − u)=(a + λhT (un)‖un‖2 + λ2T 2ψ ′(‖un‖2T 2)‖un‖4)(un,un − u) + o(1),and then(a + λhT (un)‖un‖2 + λ2T 2ψ ′(‖un‖2T 2)‖un‖4)(un,un − u) → 0.Since |ψ ′( ‖un‖2T 2)‖un‖4|  8T 4 and 8λT 2 < a, ‖un‖ → ‖u‖. This together with un ⇀ u shows thatun → u in H . The proof is completed. Lemma 2.5. Let 8λT 2 < a. For almost every μ ∈ I , there exists uμ ∈ H \ {0} such that ( J Tλ,μ)′(uμ) = 0 andJ Tλ,μ(uμ) = cμ .Y. Li et al. / J. Differential Equations 253 (2012) 2285–2294 2291Proof. By Theorem 2.1, for almost every μ ∈ I , there exists a bounded sequence {uμn } ⊂ H such thatJ Tλ,μ(uμn)→ cμ,(J Tλ,μ)′(uμn)→ 0.By Lemma 2.4, we can suppose that there exists uμ ∈ H such that uμn → uμ in H , then the assertionfollows from Lemma 2.3. According to Lemma 2.5, there exist sequences {μn} ⊂ I with μn → 1− and {un} ⊂ H as n → ∞such thatJ Tλ,μn(un) = cμn ,(J Tλ,μn)′(un) = 0.The Pohozaev identity is important for many problems. In this paper, we also use this identity toobtain ‖un‖ T . In fact we have the next lemma.Lemma 2.6. Let 8λT 2 < a and N  3. If u ∈ H is a weak solution of(a + λhT (u)‖u‖2 + λ2T 2ψ ′(‖u‖2T 2)‖u‖4)(−u + bu) = μ f (u), in x ∈RN , (2.4)then the following Pohozaev type identity holds(N − 22∫RN|∇u|2 + Nb2∫RNu2)(a + λhT (u)‖u‖2 + λ2T 2ψ ′(‖u‖2T 2)‖u‖4)= μN∫RNF (u). (2.5)Proof. Since u ∈ H is a weak solution of (2.4), by standard regularity results, u ∈ H2loc(RN ) ∩ H1(RN ).Letg(u) = μ f (u)a + λhT (u)‖u‖2 + λ2T 2 ψ ′( ‖u‖2T 2)‖u‖4− bu.Then u ∈ H is also a solution of−u = g(u).By [17, Corollary B.4, p. 138],N − 22∫RN|∇u|2 = N∫RNG(u),where G(t) = ∫ t0 g(s)ds. Then the conclusion holds. The following lemma shows that ‖un‖ T which is the key for this paper.Lemma 2.7. Let un be a critical point of J Tλ,μn at level cμn . Then for T > 0 sufficiently large, there existsλ0 = λ0(T ) with 8λ0T 2 < a such that for any λ ∈ [0, λ0), subject to a subsequence, ‖un‖ T for all n ∈N.2292 Y. Li et al. / J. Differential Equations 253 (2012) 2285–2294Proof. We argue by contradiction. Firstly, since ( J Tλ,μn )′(un) = 0, by (2.5), un satisfies the followingPohozaev identity(N − 22∫RN|∇un|2 + Nb2∫RNu2n)(a + λhT (un)‖un‖2 + λ2T 2ψ ′(‖un‖2T 2)‖un‖4)= μnN∫RNF (un). (2.6)By using J Tλ,μn (un) = cμn , we have that12aN‖un‖2 + 14λNhT (un)‖un‖2 − μnN∫RNF (un) = cμn N. (2.7)Therefore, by (2.6) and (2.7), we can obtain that12a∫RN|∇un|2 (a + λhT (un)‖un‖2 + λ2T 2ψ ′(‖un‖2T 2)‖un‖4)∫RN|∇un|2= cμn N +14λNhT (un)‖un‖4 + λN4T 2ψ ′(‖un‖2T 2)‖un‖6. (2.8)We estimate the right hand side of (2.8). By the min–max definition of the mountain pass level,Lemma 2.2 and (2.2), we havecμn maxt JTλ,μn(tφ)maxt{12at2 − μn∫RNF (tφ)}+maxt14λψ(t2T 2)t4maxt{12at2 − δC1t2∫B(0,R)φ2 + C3}+maxt14λψ(t2T 2)t4= C3 + A1(T ).If t2  2T 2, then ψ( t2T 2) = 0. Thus, we have thatA1(T ) λT 4.We have also that14λNhT (un)‖un‖4  λNT 4,λN4T 2∣∣∣∣ψ ′(‖un‖2T 2)∣∣∣∣‖un‖6  4λNT 4.Y. Li et al. / J. Differential Equations 253 (2012) 2285–2294 2293Then we have12a∫RN|∇un|2  NC3 + 6λNT 4.On the other hand, by (2.1) and (2.3), we have thata‖un‖2 + λhT (un)‖un‖4 + λ2T 2ψ ′(‖un‖2T 2)‖un‖6 = μn∫RNf (un)un  bε|un|22 + Cε|un|2∗2∗ .So(a − ε)‖un‖2  Cε|un|2∗2∗ −λ2T 2ψ ′(‖un‖2T 2)‖un‖6 C4|∇un|2∗2 + 8λT 4 C5(NC3 + 6λNT 4)2∗/2 + 8λT 4.We suppose by contradiction that there exists no subsequence of {un} which is uniformly boundedby T . Then we can assume that ‖un‖ > T , n ∈N. ThenT 2 < ‖un‖2  C6(NC3 + 6λNT 4)2∗/2 + C7λT 4,which is not true for T large and 8λT 4 < a. So by setting λ0 < a/(8T 4), we obtain the conclusion. Remark 2.8. In Lemma 2.7, the choice of λ0 depends on the nonlinearity f , constants N , a and b,Sobolev embedding constant γ2∗ , several test functions and constants used in the proof. So it isdifficult to give explicitly the value of λ0. However, for the special case f (t) = at2, we can chooseφ and δ in Lemma 2.2 to satisfy δ∫B(0,5) φ2 > 1/4. Similarly we can choose C1 = 2a, C2 = 32a/3,C3 = 32a/3|B(0,5)| in Lemma 2.2, where |B(0,5)| is the volume of B(0,5) in RN . Moreover, wechoose ε = a/2 in Lemma 2.7, then Cε = a(2/b)1/(2∗−3) . Hence, we can compute a lower bound of λ0to beλ0 = a32[(2/b)1/(2∗−3)γ2∗(2/a)2∗/2(32aN/3|B(0,5)| + a)2∗/2 + 1]2 ,where γ2∗ is the embedding constant in the embedding inequality (∫RN|∇u|2)2∗/2  γ2∗∫RN|u|2∗ forall u ∈ D1,2(RN ). So the existence result in Theorem 1.1 holds for any λ ∈ [0, λ0) for this specialcase.3. Proof of Theorem 1.1Proof of Theorem 1.1. Let T , λ0 be defined as in Lemma 2.7, and let un be a critical point for J Tλ,μnat level cμn . Then from Lemma 2.7 we may assume that‖un‖ T .2294 Y. Li et al. / J. Differential Equations 253 (2012) 2285–2294SoJ Tλ,μn (un) =12a‖un‖2 + 14λ‖un‖4 − μn∫RNF (un).Since μn → 1, we can show that {un} is a (PS) sequence of Jλ . Indeed, the boundedness of {un}implies that { Jλ(un)} is bounded. Also(J ′λ(un), v)= (( J Tλ,μn)′(un), v)+ (μn − 1)∫RNf (un)v, v ∈ H .Thus J ′λ(un) → 0, and then {un} is a bounded (PS) sequence of Jλ . By Lemma 2.4, {un} has a conver-gent subsequence. We may assume that un → u. Consequently, J ′λ(u) = 0. According to Lemma 2.3,we have that Jλ(u) = limn→∞ Jλ(un) = limn→∞ J Tλ,μn (un) c > 0 and u is a positive solution by thecondition (H1). The proof is completed. AcknowledgmentThe authors thank an anonymous referee for a careful reading and some helpful comments, whichgreatly improve the manuscript.References[1] H. Berestycki, P.-L. Lions, Nonlinear scalar field equations. I. Existence of a ground state, Arch. Ration. Mech. Anal. 82 (1983)313–345.[2] C. Chen, Y. Kuo, T. Wu, The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions, J. Dif-ferential Equations 250 (2011) 1876–1908.[3] W.-Y. Ding, W.-M. Ni, On the existence of positive entire solutions of a semilinear elliptic equation, Arch. Ration. Mech. 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Existence of a positive solution to Kirchhoff type problems without compactness conditions

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Existence of a positive solution to Kirchhoff type problems without compactness conditions

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