Calc. Var. 12, 359–369 (2001) Digital Object Identifier (DOI) 10.1007/s005260000058
Solution of nonlinear equations having asymptotic limits at zero and infinity Kanishka Perera1 , Martin Schechter2 1 2
Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901-6997 (e-mail: kperera@winnie.fit.edu, http://www.winnie.fit.edu/∼kperera/) Department of Mathematics, University of California, Irvine, CA 92697-3875 (e-mail:
[email protected], http://www.math.uci.edu/mschecht.html)
Received September 28, 1999 / Accepted May 9, 2000 / c Springer-Verlag 2000 Published online: December 8, 2000 –
Abstract. We obtain nontrivial solutions for semilinear elliptic boundary value problems having asymptotic limits both at zero and at infinity. Mathematics Subject Classification (1991): 35J65, 58E05, 49B27 1 Introduction Let Ω be a smooth, bounded domain in Rn , and let A be a selfadjoint operator on L2 (Ω). We assume that C0∞ (Ω) ⊂ D := D(|A|1/2 ) ⊂ H m,2 (Ω)
(1.1)
holds for some m > 0, with A bounded from below. Let f (x, t) be a Carath´eodory function on Ω × R satisfying |f (x, t)| ≤ C(|t| + 1), x ∈ Ω, t ∈ R, f (x, t) = a0 t + p0 (x, t), p0 (x, t) = o(t) as t → 0,
(1.2)
f (x, t) = at + p(x, t), p(x, t) = o(t) as |t| → ∞.
(1.3)
and
If a0 ∈ σ(A), we assume that there is a σ ∈ (2, 2∗ ), 2∗ = 2n/(n − 2), such that tp0 (x, t)/|t|σ → α± as t → ±0,
(1.4)
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and 0 (y) := − 1 G σ
σ
y>0
α+ |y| +
σ
y 0, G 0 (y) < 0, or negative, or indefinite (with respect to f ) if G G0 (y) changes sign on E0 \{0}, respectively, where E0 = E(a0 ). Note that 0 (y) cannot change sign on E(a0 )\{0} unless a0 is a simple eigenvalue G If a ∈ σ(A), assume also that there is a τ ∈ (1, 2) such that tp(x, t)/|t|τ → β± as t → ±∞, and 1 G(y) := − τ
y>0
τ
β+ |y| +
y 0, G(y) < 0, or G(y) changes sign on E\{0}, respectively, where E = E(a). As in the case of a0 , G(y) cannot change sign on E\{0} unless a is a simple eigenvalue The object of the current paper is to prove Theorem 1.1. Assume, in addition, that there is a λ ∈ σ(A) such that either a0 ≤ λ ≤ a,
(1.8)
a ≤ λ ≤ a0 .
(1.9)
or
We also assume that a, a0 do not satisfy any of the following: 1. They are both indefinite. 2. They are equal and have the same designation. 3. There are no eigenvalues of A in the interval between them, and either (a) the smaller equals λ and is positive, while the larger is negative or in ρ(A). (b) the larger equals λ and is negative, while the smaller is positive or in ρ(A). Then Au = f (x, u) has a nontrivial solution.
(1.10)
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The proof of Theorem 1.1 will be accomplished by means of a series of lemmas given in the next section Our method consists of analyzing the functional G whose critical points coincide with the solutions of (1.10). We show, under the given hypotheses, that 0 |E , 0) Cq (G, 0) ∼ = Cq−m0 (G 0
∀q,
(1.11)
where m0 is the total of all the dimensions of eigenspaces of A corresponding to eigenvalues < a0 . Then we find a functional, GR , which coincides with G on a bounded set, has the same critical points and satisfies E , 0) Cq (GR , ∞) ∼ = Cq−m (G|
∀q,
(1.12)
where m is the sum of the dimensions of the eigenspaces of A corresponding to eigenvalues < a. In combining these results, we obtain Theorem 1.1 Many authors have studied special cases of problem (1.10) under hypotheses (1.2) and (1.3) beginning with the work of Amann-Zehnder [1], who considered the Dirichlet problem −∆u = f (u) in Ω, u = 0 on ∂Ω. They assumed that f (t) ∈ C 1 (R) and that either f (0) < λ < f (∞) or
f (∞) < λ < f (0).
They did not allow f (∞) to be in σ(A). Since then many authors have weakened some of these requirements (cf., e.g., [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], and the references mentioned there). In most cases, f (x, t) is required to be continuously differentiable with respect to t, and a and a0 are not both allowed to be in σ(A). In Theorem 1.1, we only require the continuity of f (x, t) with respect to t, allow either or both a0 and a to be in σ(A) and permit a = a0 = λ. 2 Some lemmas Theorem 1.1 will be established via a series of lemmas. In describing them, we let Ω be a smooth, bounded domain in Rn , and we let A be a selfadjoint operator on L2 (Ω). We assume that C0∞ (Ω) ⊂ D := D(|A|1/2 ) ⊂ H m,2 (Ω)
(2.1)
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holds for some m > 0, and use the notation a(u, v) = (Au, v), a(u) = a(u, u), u, v ∈ D. D becomes a Hilbert space if we use the scalar product (u, v)D = (|A|u, v) + (P0 u, v), u, v ∈ D,
(2.2)
and its corresponding norm, where P0 is the projection onto N (A). Let f (x, t) be a Carath´eodory function on Ω × R satisfying |f (x, t)| ≤ V (x)q (|t|q−1 + 1), x ∈ Ω, t ∈ R, and f (x, t)/V (x)q = o(|t|q−1 ) as |t| → ∞, unif ormly, where q > 2 satisfies q ≤ 2n/(n − 2m), 2m < n, q < ∞, n ≤ 2m, and V (x) > 0 is a function in Lq (Ω) such that V uq ≤ CuD , u ∈ D. (The norm on the left in (2.3) is that of Lq (Ω).) We let 2G(u) = a(u) − 2 F (x, u), Ω
(2.3)
(2.4)
where F (x, t) =
t
0
f (x, s)ds.
(2.5)
As is well known, G is in C 1 in D, and (G (u), h) = a(u, h) − (f (u), h), u, h ∈ D,
(2.6)
where we write f (u) in place of f (x, u(x)). Moreover, u is a solution of Au = f (x, u)
(2.7)
G (u) = 0.
(2.8)
iff it satisfies
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Let λ(b) (resp. λ(b)) denote the largest (resp. smallest) point in the negative (resp. positive) spectrum of A − bI, Lb (u) = (A − bI) u, u , (2.9) ⊥ V (b) = E(b), W (b) = V (b) ⊕ E(b) , (2.10) µ0 u 0 such that 0 is the only critical point of Gt with uD ≤ ρ, for all t ∈ [0, 1]. Assuming this for the moment it follows from the homotopy invariance of the critical groups that Cq (G, 0) = Cq (G0 , 0) ∼ = Cq (G1 , 0).
(2.16)
Since G1 has the special form G1 (u) =
1 0 (y), La (u) + G 2 0
(2.17)
it follows from standard arguments that 0 |E , 0) Cq (G1 , 0) ∼ = Cq−m0 (G 0
(2.18)
(see, e.g., Sect. I.5 of Chang [3]). Combining (2.16) and (2.18), we get (2.14)
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If the above claim is false, then there are sequences {uk } ⊂ D, {tk } ⊂ uk ˜k = , and write [0, 1] such that Gtk (uk ) = 0 and ρk = uk D → 0. Let u ρk u ˜k = v˜k + y˜k + w ˜k ∈ V (a0 ) ⊕ E0 ⊕ W (a0 ). Then ˜k − v˜k ) (Gtk (uk ), w , ≥ λ(a0 ) w ˜k 2 − λ(a0 ) ˜ vk 2 + o ρσ−2 0= k ρk (2.19) ˜k → 0, and hence a renamed subsequence y˜k → y˜ strongly in D so v˜k , w with ˜ y D = 1. But then (Gtk (uk ), y˜k ) p0 (x, uk ) y˜k 0 (˜ = −(1 − tk ) + σ tk G yk ) 0= σ−1 ρk ρσ−1 (2.20) Ω k 0 (˜ y ) = 0 → σG
for a subsequence, providing a contradiction. Lemma 2.2. Assume f (x, t)/t → 0 as |t| → ∞.
(2.21)
If N (A) = {0}, assume also that there is a constant τ ∈ (1, 2) such that f (x, t)t/|t|τ → β± (x) as t → ±∞, unif ormly in x, where
y>0
β+ |y|τ +
y0
Ω
365
[uk f (x, uk )/|uk |τ ][|˜ uk |τ /˜ uk ]˜ yk
β+ |˜ y |τ +
y˜ 0 such that inf
t∈[0,1], u>R
Gt (u) > 0,
inf
t∈[0,1], u≤R
Gt (u) > −∞,
(2.31)
then Cq (G0 , ∞) ∼ = Cq (G1 , ∞)
∀q.
(2.32)
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Proof. Take α
R by (2.33), and hence this flow exists by (2.31). It can also be reversed by replacing Gt with G1−t in (2.34). Hence η(1) is a homeomorphism of Gα0 onto Gα1 , and Cq (G0 , ∞) = Hq (X, Gα0 ) ∼ = Hq (X, Gα1 ) = Cq (G1 , ∞).
(2.37)
We have F (x, t) =
1 2 at − β(x, t) + Q(x, t), 2
(2.38)
where 1 β+ (x) (t+ )τ + β− (x) (t− )τ , τ Q(x, t) = o(|t|σ ) as |t| → ∞, uniformly in x. β(x, t) = −
(2.39) (2.40)
Take a smooth function ψ : R → [0, 1] such that ψ(t) = 1 for t ≤ 0 and ψ(t) = 0 for t ≥ 1, and let 1 Q(x, u), R > 0, GR (u) = La (u) + G(u) − ψ(uD − R) 2 Ω (2.41) where G(u) =
Ω
β(x, u).
(2.42)
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Arguments similar to those used in the proof of Lemma 2.2 show that the set of critical points of GR is bounded independently of R, so the critical points of G and GR are the same when R is sufficiently large Let E = E(a), m = dim V (a), m∗ = m + dim E.
(2.43)
Lemma 2.5. If G(y)
= 0
∀y ∈ E \{0} ,
(2.44)
then for any R > 0, E , 0) Cq (GR , ∞) ∼ = Cq−m (G|
∀q.
(2.45)
In particular, (i). if a is positive, then Cq (GR , 0) = δqm Z, (ii). if a is negative, then Cq (GR , 0) = δqm∗ Z, (iii). if a is indefinite, then Cq (GR , 0) = 0 for all q. Proof. Take a smooth function ψ0 : R → [0, 1] such that ψ0 (t) = 0 for |t| ≤ 1 and ψ0 (t) = 1 for |t| ≥ 2, and let 1 ψ0 (u) β(x, u). (2.46) G1 (u) = La (u) + 2 Ω Clearly, GR − G1 is bounded, and hence Cq (GR , ∞) ∼ = Cq (G1 , ∞) by Lemma 2.3. Let 1 G2 (u) = La (u) + 2
(2.47)
Ω
ψ0 (y) β(x, y)
(2.48)
where y = ProjE u, and set Gt (u) = (1 − t) G1 (u) + t G2 (u), t ∈ [0, 1].
(2.49)
Arguments similar to those used in the proof of Lemma 2.2 show that (2.31) of Lemma 2.4 hold, so Cq (G1 , ∞) ∼ = Cq (G2 , ∞).
(2.50)
1 La (u) + G(y). 2
(2.51)
Let G3 (u) =
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Then G2 − G3 is bounded, so Cq (G2 , ∞) ∼ = Cq (G3 , ∞)
(2.52)
by Lemma 2.3. Since G(y)
= 0 for y ∈ E \{0}, 0 is the only critical point of G3 , so E , 0). Cq (G3 , ∞) ∼ = Cq (G3 , 0) ∼ = Cq−m (G|
(2.53)
Combining (2.47), (2.50), (2.52), and (2.53), we get (2.45).
We can now give the proof of Theorem 1.1. Proof. Assume that 0 is the only critical point of G. Since GR (u) = G(u), uD ≤ R, and the critical points of G and GR are the same for R sufficiently large, we must have Cq (GR , 0) = Cq (G, 0) ∀q. Thus, by Lemma 2.1, Cq (GR , 0) = δqm0 Z, Cq (GR , 0) = δqm∗0 Z, Cq (GR , 0) = 0 ∀q, depending on whether a0 is in ρ(A) or negative, positive or indefinite, respectively. On the other hand, we have by Lemma 2.5, Cq (GR , 0) = δqm Z, Cq (GR , 0) = δqm∗ Z, Cq (GR , 0) = 0 ∀q, depending on whether a is in ρ(A) or negative, positive or indefinite, respectively. Thus, 0 can be the sole critical point of GR only if one of the possibilities in the first list coincides with one of the possibilities in the second list. These are excluded by the theorem. In all other cases, 0 cannot be the only critical point of GR . References 1. H. Amann and E. Zehnder. Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 7(4): 539–603, 1980 2. T. Bartsch and S. J. Li. Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. 28(3): 419–441, 1997 3. K. C. Chang. Infinite-dimensional Morse theory and multiple solution problems. Birkh¨auser Boston, Inc., Boston, MA, 1993 4. D. G. Costa and E. A. Silva. On a class of resonant problems at higher eigenvalues. Differential Integral Equations, 8(3): 663-671, 1995 5. N. Hirano and T. Nishimura. Multiplicity results for semilinear elliptic problems at resonance and with jumping nonlinearities. J. Math. Anal. Appl. 180(2): 566–586, 1993 6. A. C. Lazer and S. Solimini. Nontrivial solutions of operator equations and Morse indices of critical points of min-max type. Nonlinear Analysis TMA 12(8): 761–775, 1988
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7. S. J. Li and J. Q. Liu. Nontrivial critical points for asymptotically quadratic function. J. Math. Anal. Appl. 165(2): 333–345, 1992 8. S. J. Li and J. Q. Liu. Computations of critical groups at degenerate critical point and applications to nonlinear differential equations with resonance. Houston J. Math. 25(3):563–582, 1999 9. S. Li and M. Willem. Multiple solutions for asymptotically linear boundary value problems in which the nonlinearity crosses at least one eigenvalue. NoDEA 5:479– 490, 1998 10. S. J. Li and Z. Zhang. Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity. Discrete Contin. Dynam. Systems 5(3):489–493, 1999 11. S. J. Li and W. Zou. The computations of the critical groups with an application to elliptic resonant problems at higher eigenvalue. J. Math. Anal. Appl. 235(1):237–259, 1999 12. J. Mawhin and M. Willem. Critical point theory and Hamiltonian systems. SpringerVerlag, New York-Berlin 1989 13. K. Perera. Applications of local linking to asymptotically linear elliptic problems at resonance. NoDEA. Nonlinear Differential Equations Appl. 6(1): 55–62 1999 14. K. Perera and M. Schechter. The Fuˇc´ık spectrum and critical groups. Proc. Amer. Math. Soc., in press 15. K. Perera and M. Schechter. A generalization of the Amann-Zehnder theorem to nonresonance problems with jumping nonlinearities. to appear in NoDEA Nonlinear Differential Equations Appl. 7(3) 2000 16. K. Perera and M. Schechter. Type (II) regions between curves of the Fuˇc´ık spectrum and critical groups Topol. Methods Nonlinear Anal. 12(2): 227–243, 1998 17. K. Perera and M. Schechter. Nontrivial solutions of elliptic semilinear equations at resonance. Manuscripta Math. 101(3): 301–311, 2000 18. M. Schechter. Bounded resonance problems for semilinear elliptic equations. Nonlinear Anal. TMA 24(10): 1471–1482, 1995 19. M. Schechter Linking Methods in Critical Point Theory. Birkh¨auser Boston, Inc., Boston, MA, 1999 20. E. A. Silva. Linking theorems and applications to semilinear elliptic problems at resonance. Nonlinear Anal. 16(5): 455–477, 1991 21. J. Su and C. Tang. Multiplicity results for semilinear elliptic equations with resonance at higher eigenvalues. preprint 22. W. Zou and J.Q. Liu. Multiple solutions for resonant elliptic equations via local linking theory and Morse theory. preprint
