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Electronic Journal of Differential Equations, Vol. 2001(2001), No. 66, pp. 1–6. ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu (login: ftp) Existence of solutions to a superlinear p-Laplacian equation ∗ Shibo Liu Abstract Using Morse theory, we establish the existence of solutions to the equation R s −∆p u = f (x, u) with Dirichlet boundary conditions. We assume that f (x, t) dt lies between the first two eigenvalues of the p-Laplacian. 0 1 Introduction Consider the Dirichlet problem for the p-Laplacian (p > 1), −∆p u = f (x, u), in Ω, u = 0, on ∂Ω. (1.1) Here Ω is a bounded domain in RN with smooth boundary ∂Ω, and −∆p u is the p-Laplacian: −∆p u := div(|∇u|p−2 ∇u). We assume that f : Ω × R → R is a Carathéodory function with subcritical growth; that is, F1) The inequality |f (x, u)| ≤ C(1 + |u|q−1 ) holds for all u ∈ R, x ∈ Ω, and p for some positive constant C, where 1 ≤ q < NN−p if N ≥ p + 1, and 1 ≤ q < ∞ if 1 ≤ N < p. It is well known that weak solutions u ∈ W01,p (Ω) of (1.1) are the critical points of the C 1 functional Z Z 1 Φ(u) = |∇u|p dx − F (x, u) dx , p Z s where F (x, s) = f (x, t) dt. 0 Let λ1 and λ2 be the first and the second eigenvalues of −∆p on W01p (Ω). It is known that λ1 > 0 is a simple eigenvalue, and that σ(−∆p ) ∩ (λ1 , λ2 ) = ∅, where σ(−∆p ) is the spectrum of −∆p , (cf. [2]). We shall assume the following conditions: ∗ Mathematics Subject Classifications: 49J35, 35J65, 35B34. Key words: p-Laplacian, critical group. c 2001 Southwest Texas State University. Submitted August 21, 2001. Published October 11, 2001. 1 2 Existence of solutions EJDE–2001/66 F2) There exist r > 0, λ̄ ∈ (λ1 , λ2 ) such that |u| ≤ r implies λ1 |u|p ≤ pF (x, u) ≤ λ̄|u|p , F3) There exist θ > p, M > 0 such that |u| ≥ M implies 0 < θF (x, u) ≤ uf (x, u). Now, we are ready to state our main result. Theorem 1.1 Assume (F1), (F2), and (F3). Then (1.1) has a nontrivial weak solution in W01,p (Ω). There are many papers devoted to the existence of solutions of (1.1); see for example [1, 4, 5]. In these papers, the main tool is the minmax argument. However, it seems difficult to use the minmax argument in our situation. Thus we will use a different approach: Morse theory [3]. To the best of our knowledge, [7] is the only work using Morse theory to obtain the solvability of p-Laplacian equations. Our work is motivated by [7]. 2 Proof of main theorem In this section we give the proof of Theorem 1.1. Let E denote the Sobolev space W01,p (Ω), and k.k denote the norm in E. For Φ a continuously Fréchet differentiable map from E to R, let Φ0 (u) denote its Fréchet derivative. As stated in Section 1, weak solutions u ∈ W01,p (Ω) of (1.1) are the critical points of the C 1 functional Z Z 1 p |∇u| dx − F (x, u) dx . Φ(u) = p We will try to find a nontrivial critical point of the functional Φ. First we state the following lemmas. Lemma 2.1 Under conditions (F1) and (F3), the functional Φ satisfies the Palais-Smale condition. Proof Assume (un ) ⊂ E, |Φ(un )| ≤ B for some B ∈ R, and Φ0 (un ) → 0. Let d := supn Φ(un ). Then by (F3) we have θd + kun k ≥ θΦ(un ) + hΦ0 (un ), un i Z θ = ( − 1)kun kp − [θF (x, un ) − f (x, un )un ] p |un |≥M Z − [θF (x, un ) − f (x, un )un ] |un |≤M Z θ p ≥ ( − 1)kun k − [θF (x, un ) − f (x, un )un ] p |un |≤M θ ≥ ( − 1)kun kp − D, for some D ∈ R. p EJDE–2001/66 Shibo Liu 3 Thus (un ) is bounded in E. Up to a subsequence, we may assume that un * u in E. Now because of condition (F1), a standard argument shows that un → u in E and the proof is complete. ♦ Let V = span φ1 be the one-dimensional eigenspace associated to λ1 , where φ1 > 0 in Ω and kφ1 k = 1. Taking a subspace W ⊂ E complementing V , that is E = V ⊕ W . Obviously the genus of W \0 satisfies γ(W \0) ≥ 2. Therefore, by the variational characterization of λ2 , for ∀u ∈ W , Z Z p |∇u| ≥ λ2 |u|p . Lemma 2.2 Under Assumption (F2), the functional Φ has a local linking at the origin with respect to E = V ⊕ W . That is, there exists ρ > 0, such that Φ(u) ≤ 0, u ∈ V, kuk ≤ ρ, Φ(u) > 0, u ∈ W, 0 < kuk ≤ ρ. The proof of this lemma can be found in [7, Lemma 3.3]. For a C 1 -functional Φ : E → R and u an isolate critical point of Φ, Φ(u) = c, we define the critical group of Φ at u as Cq (Φ, u) := Hq (Φc , Φc \{u}). Where Hq (X, Y ) is the q-th homology group of the topological pair (X, Y ) over the ring Z. Since dim V = 1 < +∞, from Lemma 2.2 and Theorem 2.1 in [6], we have Lemma 2.3 Under assumption (F2), 0 is a critical point of Φ and C1 (Φ, 0) 6= 0. To find a nontrivial critical point of Φ, we investigate the behavior of Φ near infinity. Lemma 2.4 Under Assumption (F3), there exists a constant A > 0 such that Φa ' S ∞ , for a < −A, where S ∞ is the unit sphere in E. Proof Integrating on the inequality of (F2), we obtain a constant C1 > 0 such that F (x, t) ≥ C1 |t|θ , for |t| ≥ M. Thus, for u ∈ S ∞ , we have Φ(tu) → −∞, as t → +∞. Set 1 M |Ω| max |f (x, u)| + 1 . A := 1 + p Ω̄×[−M,M ] 4 Existence of solutions EJDE–2001/66 Using (F3) we obtain Z Z 1 F (x, v) − vf (x, v) p Z Z Z Z 1 1 = F (x, v) + F (x, v) − vf (x, v) − vf (x, v) p |v|≥M p |v|≤M |v|≥M |v|≤M Z Z Z 1 1 1 ≤ − vf (x, v) + F (x, v) − vf (x, v) θ p p |v|≤M |v|≥M |v|≤M Z 1 1 1 ≤ − vf (x, v) + 1 + M |Ω| max |f (x, u)| θ p p Ω̄×[−M,M ] |v|≥M Z 1 1 ≤ − vf (x, v) + A − 1. θ p |v|≥M For a < −A and Φ(tu) = |t|p − p Z F (x, tu) ≤ a, (u ∈ S ∞ ), we have d Φ(tu) dt Z = hΦ0 (tu), ui = |t|p−2 t − uf (x, tu) Z Z o pn 1 ≤ F (x, tu) − tuf (x, tu) + a t p Z n o p 1 1 ≤ ( − ) tuf (x, tu) + A − 1 + a t θ p |tu|≥M Z o pn 1 1 ≤ ( − ) tuf (x, tu) − 1 t θ p |tu|≥M Z o pn 1 1 ≤ ( − )C1 θ |tu|θ − 1 < 0. t θ p |tu|≥M By the Implicit Function Theorem, there is a unique T ∈ C(S ∞ , R) such that Φ(T (u)u) = a, For u 6= 0, set T̃ (u) = 1 u kuk T ( kuk ). ∀u ∈ S ∞ . Then T̃ ∈ C(E\0, R) and for all u ∈ E\0, Φ(T̃ (u)u) = a. Moreover, if Φ(u) = a, then T̃ (u) = 1. We define a function T̂ : E\0 → R as ( T̃ (u), if Φ(u) ≥ a, T̂ (u) := 1, if Φ(u) ≤ a. Since Φ(u) = a implies T̃ (u) = 1, we conclude that T̂ ∈ C(E\{0}, R). Finally we set η : [0, 1] × (E\0) → E\0 as η(s, u) = (1 − s)u + sT̂ (u)u. EJDE–2001/66 Shibo Liu 5 It is easy to see that η is a strong deformation retract from E\0 to Φa . Thus Φa ' E\0 ' S ∞ and present proof is complete. ♦ We also use the following topological result,which was proved by Perera [8]. Lemma 2.5 Let Y ⊂ B ⊂ A ⊂ X be topological spaces and q ∈ Z. If Hq (A, B) 6= 0 and Hq (X, Y ) = 0 or Hq−1 (B, Y ) 6= 0 . then Hq+1 (X, A) 6= 0 Now we can prove the main theorem. Proof of Theorem 1.1 By Lemma 2.1, Φ satisfies the Palais-Smale condition. Note that Φ(0) = 0, from [3] Chapter I, Theorem 4.2, there is a ε > 0, such that H1 (Φε , Φ−ε ) = C1 (Φ, 0) 6= 0. By Lemma 2.4, for a < −A (A is as in the lemma) we have Φa ' S ∞ . Since dim E = +∞, H1 (E, Φa ) = H1 (E, S ∞ ) = 0. So that Lemma 2.5 yields H2 (E, Φε ) 6= 0 or H0 (Φ−ε , Φa ) 6= 0. It follows that Φ has a critical point u for which Φ(u) > ε or − ε > Φ(u) > a . Therefore, u is a nonzero critical point of Φ, and (1.1) has a nontrivial solution. Remark Result similar to Lemma 2.4 has been proved (for p = 2) in [9] and [3], under the additional conditions f ∈ C 1 (Ω × R, R), f (x, 0) = ∂f (x, t) = 0. ∂t t=0 From these two references, we have obtained the motivation for this paper. References [1] A. R. El Amrouss & M. Moussaoui, Minimax principles for critical-point theory in applications to quasilinear boundary-value problems, Electron. J. Diff. Eqns.,, 2000(2000), No. 18, 1–9. [2] A. Anane & N. Tsouli, On the second eigenvalue of the p-Laplacian, Nonlinear Partial Differential Equations, Pitman Research Notes 343(1996), 1–9. 6 Existence of solutions EJDE–2001/66 [3] K. C. Chang, Infinite dimensional Morse theory and multiple solution problems, Birkhäuser, Boston, 1993. [4] D. G. Costa & C. A. Magalhães, Existence results for perturbations of the p-Laplacian, Nonlinear Analysis, 24(1995), 409–418. [5] X. L. Fan & Z. C. Li, Linking and existence results for perturbations of the p-Laplacian, Nonlinear Analysis, 42(2000), 1413-1420. [6] J. Q. Liu, The Morse index of a saddle point, Syst. Sc. & Math. Sc., 2(1989), 32-39. [7] J. Q. Liu & J. B. Su, Remarks on multiple nontrivial solutions for quasilinear resonant problems, J. Math. Anal. Appl., 258(2001), 209-222. [8] K. Perera, Critical groups of critical points produced by local linking with applications, Abstract and Applied Analysis, 3(1998), 437-446. [9] Z. Q. Wang, On a superlinear elliptic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8(1991), 43-57. Shibo Liu Institute of Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, Beijing, 100080, P. R. China e-mail address: liusb@math08.math.ac.cn