Sign Changing Solutions of Kirchhoff Type Problems via Invariant Sets of Descent Flow Zhitao Zhang∗ Academy of Mathematics and Systems Science Institute of Mathematics Chinese Academy of Sciences Beijing 100080, P.R.China
[email protected] Kanishka Perera∗∗ Department of Mathematical Sciences Florida Institute of Technology Melbourne, FL 32901, U.S.A.
[email protected] http://my.fit.edu/˜kperera/
Abstract We obtain sign changing solutions of a class of nonlocal quasilinear elliptic boundary value problems using variational methods and invariant sets of descent flow. MSC2000: Primary 35J65, Secondary 35J20, 47J10 Key Words and Phrases: nonlocal problems, Kirchhoff’s equation, variational methods, invariant sets of descent flow ∗
Supported in part by the National Natural Science Foundation of China, Ky and Yu-Fen Fan Endowment of the AMS, Florida Institute of Technology, and the Humboldt Foundation. ∗∗ Supported in part by the National Science Foundation.
1
1
Introduction
In this paper we obtain sign changing solutions of the problem Z 2 − a + b |∇u| ∆u = f (x, u) in Ω Ω
u=0
(1.1)
on ∂Ω
where Ω is a smooth bounded domain in Rn , a, b > 0, and f (x, t) is locally Lipschitz continuous in t ∈ R, uniformly in x ∈ Ω, and subcritical: 2n , n ≥ 3 p−1 ∗ |f (x, t)| ≤ C |t| +1 for some 2 < p < 2 = n − 2 (1.2) ∞, n = 1, 2 where C denotes a generic positive constant. This problem is related to the stationary analogue of the equation Z 2 (1.3) utt − a + b |∇u| ∆u = g(x, t) Ω
proposed by Kirchhoff [11] as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. Some early classical studies of Kirchhoff equations were Bernstein [5] and Pohozaev [14]. However, equation (1.3) received much attention only after Lions [12] proposed an abstract framework to the problem. Some interesting results can be found, for example, in [4, 6, 10]. More recently Alves et al. [2] and Ma and Rivera [13] obtained positive solutions of such problems by variational methods. Similar nonlocal problems also model several physical and biological systems where u describes a process which depends on the average of itself, for example the population density, see [1, 3, 7, 8, 17]. We assume that tf (x, t) ≥ 0
(1.4)
and consider three cases:
2
(i) 4-sublinear case: p < 4, (ii) asymptotically 4-linear case: f (x, t) = µ uniformly in x, |t|→∞ b t3 lim
(1.5)
(iii) 4-superlinear case: ∃ν > 4 :
ν F (x, t) ≤ tf (x, t), |t| large Z t where F (x, t) = f (x, s) ds, which implies
(1.6)
0
F (x, t) ≥ C (|t|ν − 1) .
(1.7)
By (1.2) and (1.5) (resp. (1.7)), 4 ≤ p < 2∗ (resp. 4 < ν ≤ p < 2∗ ),
(1.8)
so n = 1, 2, or 3 in (ii) and (iii). Case (ii) leads us to the nonlinear eigenvalue problem ( − kuk2 ∆u = µ u3 in Ω (1.9) u=0
on ∂Ω,
whose eigenvalues are the critical values of the functional Z 4 1 4 I(u) = kuk , u ∈ S := u ∈ H = H0 (Ω) : u =1 .
(1.10)
Ω
We will see in the next section that I satisfies the Palais-Smale condition (PS) and that the first eigenvalue µ1 > 0 obtained by minimizing I has an eigenfunction ψ > 0. We define a second eigenvalue ≥ µ1 by µ2 := inf
max I(u)
(1.11)
γ∈Γ u∈γ([0,1])
where Γ is the class of paths γ ∈ C([0, 1], S) joining ±ψ such that γ ∪ (−γ) is non-self-intersecting. 3
We are now ready to state our main result. Denote by 0 < λ1 < λ2 ≤ · · · the Dirichlet eigenvalues of −∆ on Ω. Theorem 1.1. Problem (1.1) has a positive solution, a negative solution, and a sign changing solution in the following cases: (i) p < 4 and ∃λ > λ2 :
F (x, t) ≥
aλ 2 t, 2
|t| small,
(1.12)
(ii) (1.5) and (1.12) hold and µ < µ1 , (iii) (1.5) holds, µ > µ2 is not an eigenvalue of (1.9), and F (x, t) ≤
aλ1 2 t, 2
|t| small,
(1.13)
(iv) (1.6) and (1.13) hold.
2
Variational Setting
Lemma 2.1. I satisfies (PS), i.e., every sequence (uj ) in H such that I(uj ) is bounded and I 0 (uj ) → 0, called a (PS) sequence, has a convergent subsequence. Proof. Since kuj k is bounded, for a subsequence, uj converges to some u weakly in H and strongly in L4 (Ω). Denoting by Z 3 Pj v = v − u j v uj (2.1) Ω
the projection of v ∈ H onto the tangent space to S at uj , we have 2
Z
kuj k
Z ∇uj · ∇(uj − u) = I(uj )
Ω
u3j (uj − u)
Ω
+ so, passing to a subsequence, uj → 0 or u. 4
1 0 (I (uj ), Pj (uj − u)) → 0, (2.2) 4
Since I is bounded from below and satisfies (PS), µ1 := inf I(u)
(2.3)
u∈S
is achieved and hence positive. If ψ is a minimizer, then so is |ψ|, so we may assume that ψ ≥ 0. Since ψ is a nontrivial solution of (1.9), ψ > 0 in Ω ∂ψ and the interior normal derivative > 0 on ∂Ω by the strong maximum ∂ν principle. Recall that a function u ∈ H is called a weak solution of (1.1) if Z Z 2 a + b kuk ∇u · ∇v = f (x, u) v ∀v ∈ H. (2.4) Ω
Ω
Weak solutions are the critical points of the C 2−0 functional Z b a 2 4 Φ(u) = kuk + kuk − F (x, u), u ∈ H. 2 4 Ω
(2.5)
They are also classical solutions if f is locally Lipschitz on Ω × R. Lemma 2.2. Φ satisfies (PS) in the following cases: (i) p < 4, (ii) µ is not an eigenvalue of (1.9), (iii) (1.6) holds. Proof. As usual, it suffices to show that every (PS) sequence (uj ) of Φ is bounded (see, e.g., Alves et al. [2], Lemma 1). (i) We have Z b a 4 2 kuj k = Φ(uj ) − kuj k + F (x, uj ) ≤ C (kuj kp + 1) (2.6) 4 2 Ω and hence (uj ) is bounded. (ii) Suppose that ρj = kuj k → ∞ for a subsequence. Setting vj = uj /ρj and passing to a further subsequence, vj converges to some v weakly in H, strongly in L4 (Ω), and a.e. in Ω. Passing to the limit in Z Z vj3 f (x, uj ) (Φ0 (uj ), w) ∇vj · ∇w − w = (2.7) b u3j 1 + a/bρ2j a + bρ2j ρj Ω Ω 5
gives Z
∇v · ∇w − µ v 3 w = 0
(2.8)
Ω
for each w ∈ H, and passing to the limit with w = vj − v shows that kvk = 1, so µ is an eigenvalue of (1.9), contrary to assumption. (iii) The conclusion follows from ν ν − 1 b kuj k4 = ν Φ(uj ) − (Φ0 (uj ), uj ) − − 1 a kuj k2 4 2 Z (2.9) + ν F (x, uj ) − uf (x, u) ≤ C + o(kuj k). Ω
Let X = C01 (Ω), with the usual norm kukX = max sup |Dα u(x)|,
(2.10)
0≤|α|≤1 x∈Ω
which is densely imbedded in H. By the elliptic regularity theory, the critical 0 e = Φ|X . A standard point set K := u ∈ H : Φ (u) = 0 ⊂ X. Let Φ e has the retracting property (see, e.g., Dancer and argument shows that Φ Zhang [9], proof of Lemma 1). Consider the initial value problem du = −W (u), t > 0 dt (2.11) u(0) = u ∈ X 0 where W (u) =
e 0 (u) Φ = u − Au, a + b kuk2
(2.12)
f (x, u(x)) , and K = (−∆)−1 : L∞ (Ω) → X. a + b kuk2 Since f (x, t) is locally Lipschitz in t, uniformly in Ω, G is locally Lipschitz, and K is a bounded linear operator, so W is locally Lipschitz on X. So a unique solution u(t, u0 ) of (2.11) satisfying Z t −t s u(t, u0 ) = e u0 + e Au(s, u0 ) ds (2.13) A = KG, G : X → L∞ (Ω), u 7→
0
6
exists in some maximal existence interval [0, T (u0 )), T (u0 ) ≤ ∞. Since
0 2
e Φ (u) d e du 0 e (u), =− ≤ 0, (2.14) Φ(u(t, u0 )) = Φ dt dt a + b kuk2 e is nonincreasing along the orbits. Φ Definition 2.3 (Sun [15]). We say that M ⊂ X is an invariant set of e if descent flow of Φ u(t, u0 ) : u0 ∈ M, t ∈ [0, T (u0 )) ⊂ M. (2.15) Denote by C(M ) = u0 ∈ X : u(t, u0 ) ∈ M for some t ∈ [0, T (u0 )) ⊃ M
(2.16)
the maximal subset of X retracted by an invariant set M . If M = C(M ), we say that M is complete. Lemma 2.4 (Sun [15]). Let M be an invariant set. e e achieves the infimum at a (i) If inf Φ(M ) > −∞ and M is closed, then Φ critical point in M . (ii) If M is open, then C(M ) is an open complete invariant set, ∂C(M ) is e e a closed complete invariant set, and inf Φ(∂C(M )) ≥ inf Φ(∂M ). Let P = u ∈ X : u ≥ 0 be the positive cone in X, which has nonempty ◦
interior P . Then P is a closed convex set and A(P ) ⊂ P by (1.4), so P is an invariant set (see Sun [16]). Combining this with (2.13) gives u0 ∈ P =⇒ u(t, u0 ) ≥ e−t u0 ,
(2.17)
◦
◦
so P is also invariant. Similarly, the negative cone −P and its interior − P are invariant sets. e Lemma 2.5. Let u be a nontrivial critical point of Φ. (i) If u ∈ P (resp. −P ), then u is a positive (resp. negative) solution of (1.1). ◦
◦
(ii) If u ∈ ∂C(P ) ∪ ∂C(− P ), then u is a sign changing solution of (1.1). 7
Proof. (i) By the strong maximum principle, if u ∈ P (resp. −P ), then, in ◦
◦
fact, u ∈P (resp. − P ).
◦
◦
(ii) We suppose that u ∈ ∂C(P ); the other case is similar. Since P is an ◦ ◦ ◦ ◦ / P . Since − P is open and does not intersect C(P ), open subset of C(P ), u ∈ ◦
/ P ∪ (−P ) as in (i). u∈ / − P . So u ∈
3
Proof of Theorem 1.1 ◦
e has nontrivial critical points in ±P and ∂C(P ) by It suffices to show that Φ Lemma 2.5. (i) By (1.2), b e Φ(u) ≥ kuk4 − C (kukp + 1) , 4
(3.1)
e is bounded from below. Let Y be the subspace of X spanned by the so Φ eigenfunctions of λ1 and λ2 . By (1.2) and (1.12), F (x, t) ≥
aλ 2 t − C |t|p , 2
(3.2)
so a λ b e Φ(u) ≤ − − 1 kuk2 + kuk4 + C kukp < 0, u ∈ Y ∩ ∂Bδ (3.3) 2 λ2 4 where Bδ = u ∈ X : kuk < δ with δ > 0 sufficiently small. Since Y ∩ Bδ ◦
◦
intersects ± P and hence also ∂C(P ) by connectedness, it follows that the ◦ e on ±P and ∂C(P ) are negative and hence achieved at nontrivial infima of Φ critical points by (i) of Lemma 2.4. (ii) By (1.5), F (x, t) ≤
b µ1 4 t + C, 4
(3.4)
so b e Φ(u) ≥ 4
4
Z
kuk − µ1
u
4
− C ≥ −C,
Ω
and the conclusion follows as in (i). 8
(3.5)
(iii) By (1.2) and (1.13), F (x, t) ≤
aλ1 2 t + C |t|p , 2
(3.6)
so b b e Φ(u) ≥ kuk4 − C kukp ≥ kuk4 , u ∈ B δ , (3.7) 4 8 δ > 0 sufficiently small, since p > 4 by (1.8). So 0 is a strict local minimum e and of Φ 4 b δ e U = u ∈ Bδ : Φ(u) < (3.8) 8 is an invariant neighborhood of 0. Then C(U ) is a complete invariant neighb δ4 e )) ≥ borhood of 0, ∂C(U ) is a complete invariant set, and inf Φ(∂C(U 8 by (ii) of Lemma 2.4. Fix ε ∈ (0, µ − µ2 ). There is a γ ∈ Γ such that max I(γ([0, 1])) < µ2 + ε/2 by (1.11) and F (x, t) ≥
b (µ − ε/2) 4 t −C 4
(3.9)
by (1.5). Let Z = tu : u ∈ γ([0, 1]) ∪ (−γ([0, 1])), t ≥ 0 ,
(3.10)
which is a surface in H homeomorphic to R2 . For u ∈ Z, b (µ − µ2 − ε) a e Φ(u) ≤− kuk4 + kuk2 + C → −∞ as kuk → ∞, (3.11) 4 (µ2 + ε/2) 2 so C(U )∩Z is a bounded neighborhood of 0 in Z. Denote by V its connected component containing 0. By Lemma 2 of Dancer and Zhang [9], ∂V has a connected component Σ that intersects each one sided ray in Z through 0 and ◦ ◦ hence contains some positive multiples of ±ψ ∈ ± P and intersects ∂C(P ). ◦
Since Σ ⊂ ∂C(U ), it follows that ∂C(U ) ∩ (±P ) and ∂C(U ) ∩ ∂C(P ) are e are positive and nonempty closed invariant sets on which the infima of Φ hence contain nontrivial critical points. e (iv) Since (1.7) holds and Y is a finite-dimensional subspace, Φ(u) → −∞ as kuk → ∞, uniformly in Y , so the conclusion follows as in (iii) with Y in place of Z. 9
References [1] C. O. Alves and F. J. S. A. Correa. On existence of solutions for a class of problem involving a nonlinear operator. Comm. Appl. Nonlinear Anal., 8(2):43–56, 2001. [2] C. O. Alves, F. J. S. A. Correa, and T. F. Ma. Positive solutions for a quasilinear ellitic equation of kirchhoff type. preprint. [3] D. Andrade and T. F. Ma. An operator equation suggested by a class of nonlinear stationary problems. Comm. Appl. Nonlinear Anal., 4(4):65– 71, 1997. [4] A. Arosio and S. Panizzi. On the well-posedness of the Kirchhoff string. Trans. Amer. Math. Soc., 348(1):305–330, 1996. [5] S. Bernstein. Sur une classe d’´equations fonctionnelles aux d´eriv´ees partielles. Bull. Acad. Sci. URSS. S´er. Math. [Izvestia Akad. Nauk SSSR], 4:17–26, 1940. [6] M. M. Cavalcanti, V. N. Domingos Cavalcanti, and J. A. Soriano. Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation. Adv. Differential Equations, 6(6):701–730, 2001. [7] M. Chipot and B. Lovat. Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Anal., 30(7):4619–4627, 1997. [8] M. Chipot and J.-F. Rodrigues. On a class of nonlocal nonlinear elliptic problems. RAIRO Mod´el. Math. Anal. Num´er., 26(3):447–467, 1992. [9] E. N. Dancer and Zhitao Zhang. Fucik spectrum, sign-changing, and multiple solutions for semilinear elliptic boundary value problems with resonance at infinity. J. Math. Anal. Appl., 250(2):449–464, 2000. [10] P. D’Ancona and S. Spagnolo. Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math., 108(2):247– 262, 1992. [11] G. Kirchhoff. Mechanik. Teubner, Leipzig, 1883.
10
[12] J.-L. Lions. On some questions in boundary value problems of mathematical physics. In Contemporary developments in continuum mechanics and partial differential equations (Proc. Internat. Sympos., Inst. Mat., Univ. Fed. Rio de Janeiro, Rio de Janeiro, 1977), volume 30 of NorthHolland Math. Stud., pages 284–346. North-Holland, Amsterdam, 1978. [13] T. F. Ma and J. E. Mu˜ noz Rivera. Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl. Math. Lett., 16(2):243– 248, 2003. [14] S. I. Pohoˇzaev. A certain class of quasilinear hyperbolic equations. Mat. Sb. (N.S.), 96(138):152–166, 168, 1975. [15] Jing Xian Sun. On Some Problems about Nonlinear Operators. PhD thesis, Shandong University, Jinan, 1984. [16] Jing Xian Sun. The Schauder condition in the critical point theory. Kexue Tongbao (English Ed.), 31(17):1157–1162, 1986. [17] Carlos Frederico Vasconcellos. On a nonlinear stationary problem in unbounded domains. Rev. Mat. Univ. Complut. Madrid, 5(2-3):309–318, 1992.
11
