manuscripta math. 101, 301 – 311 (2000)
© Springer-Verlag 2000
Kanishka Perera · Martin Schechter
Nontrivial solutions of elliptic semilinear equations at resonance Received: 14 January 1999 / Revised version: 17 May 1999 Abstract. We find nontrivial solutions for semilinear boundary value problems having resonance both at zero and at infinity.
1. Introduction In a recent paper [7], Li and Zou considered the problem −4u = λm u + f (x, u) in ,
u = 0 on ∂,
(1.1)
where ⊂ Rn is a bounded domain with smooth boundary ∂, λm is an eigenvalue of the linear problem −4u = λu in ,
u = 0 on ∂,
(1.2)
and f ∈ C( × R, R) need not be bounded. They assume that f (x, t) is continuously differentiable with respect to t and consider the case when (1.1) is resonant both at ∞ and 0, i.e., when lim f (x, t)/t = lim f (x, t)/t = 0
t→0
t→∞
(1.3)
uniformly in x. (They assume that λm is not the first eigenvalue.) They assume that there is a σ ∈ (1, 2) such that F1 (x) ≤ lim inf f (x, t)t/|t|σ ≤ lim sup f (x, t)t/|t|σ ≤ F2 (x), |t|→∞
|t|→∞
(1.4)
uniformly in x, where F1 , F2 ∈ C(, R). In their first result, they assume, in addition to (1.3), that there is an r ∈ (2, 2∗ ), 2∗ = 2n/(n − 2), such that f1 (x) ≤ lim inf f (x, t)t/|t|r ≤ lim sup f (x, t)t/|t|r ≤ f2 (x) t→0
t→0
uniformly in x, where f1 , f2 ∈ C(, R), and either K. Perera, M. Schechter: Department of Mathematics, University of California, Irvine, CA 92697-3875, USA. e-mail:
[email protected];
[email protected] Mathematics Subject Classification (1991): Primary 35J65, 58E05, 49B27
302
K. Perera, M. Schechter
(a) f1 (x) ≥ 0 ≥ F2 (x) for a.e. x and Z Z f1 (x) > 0 > F2 (x)
or (b) f2 (x) ≤ 0 ≤ F1 (x) for a.e. x and Z Z f2 (x) < 0 < F1 (x).
They then show that (1.1) has a nontrivial solution. In their second result, they assume (1.4), f (x, t)/t ∈ [r1 , r2 ] ⊂ (λm−1 − λm , λm+1 − λm ), t ∈ R \ {0},
(1.5)
and either (c) There is a t0 > 0 such that F1 (x) ≥ 0 ≥ f (x, t)/t for a.e. x ∈ , |t| ≤ t0 and Z F1 (x) > 0, or (d) There is a t0 such that F2 (x) ≤ 0 ≤ f (x, t)/t for a.e. x ∈ , |t| ≤ t0 and Z F2 (x) < 0.
They then show that their second set of hypotheses produces a nontrivial solution. Many authors have studied resonance problems for (1.1), but few have allowed f (x, t) to be unbounded (a partial list is given in the bibliography). To the best of our knowledge, [7] and [15] are the only works to consider the case when (1.3) holds. In the present paper, the authors consider problem (1.1) and allow (1.3). In our first result, we assume (A) There is a constant σ ∈ (1, 2) such that tf (x, t)/|t|σ → α± (x) as t → ±∞ uniformly in x, where Z Z α− |y|σ + y0
α+ |y|σ < 0, y ∈ E(λm ) \ {0}
(E(λm ) is the eigenspace of λm ), and
(1.6)
(1.7)
Nontrivial solutions of elliptic semilinear equations at resonance
303
(B) There are a constant δ > 0 and a function r(x) such that 0 ≤ f (x, t)/t ≤ r(x)2 , |t| < δ,
(1.8)
lim sup krwk/kwkD < 1,
(1.9)
and kwkD →0
where kwk2D = k∇wk2 − λm kwk2 , w ⊥
M
N (A − λk ).
(1.10)
k≤m
We show that (A) and (B) imply that (1.1) has a nontrivial solution. We obtain the same result if we replace (1.7), (1.8) and (1.9) by Z Z σ α− |y| + α+ |y|σ > 0, y ∈ E(λm ) \ {0}, (1.11) y0
0 ≤ −f (x, t)/t ≤ r(x)2 , |t| < δ,
(1.12)
lim sup krvk/kvkD < 1,
(1.13)
and kvkD →0
where kvk2D = λm kvk2 − k∇vk2 , v ∈
M
N (A − λk ).
(1.14)
k 0, and σe (A) ⊂ (0, ∞). We 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). We assume that L ∞ λ≤0 N (A − λ) ⊂ L (). Let f (x, t) be a Carathéodory function on × R satisfying |f (x, t) − f (x, s)| ≤ V (x)q (|t|q−2 + |s|q−2 + 1)|t − s|, x ∈ , s, t ∈ R, and
f (x, t)/V (x)q = o(|t|q−1 ) as |t| → ∞, uniformly,
where q > 2 satisfies q ≤ 2n/(n − 2m), 2m < n, q < ∞, n ≤ 2m, and V (x) > 0 is a function in Lq () such that kV ukq ≤ CkukD , u ∈ D.
(2.3)
(The norm on the left in (2.3) is that of Lq ().) In our first result we make use of the following assumptions. (A) There is a constant σ ∈ (1, 2) such that f (x, t)t/|t|σ → α± (x) as t → ±∞, uniformly inx, where
Z
σ
y>0
α+ |y| +
Z y 0 and a function r(x) such that 0 ≤ f (x, t)/t ≤ r 2 (x), |t| < δ, and lim sup krwk/kwkD < 1, w ⊥
kwkD →0
M λ≤0
N (A − λ).
(2.6)
(2.7)
Nontrivial solutions of elliptic semilinear equations at resonance
305
We shall prove Theorem 2.1. Under hypotheses (A) and (B), there is at least one nontrivial solution of Au = f (x, u).
(2.8)
Theorem 2.2. The same conclusion holds if we replace (2.5) in (A) by Z Z α+ |y|σ + α− |y|σ > 0, y ∈ N (A)\{0} y>0
yδ
|f (x, u)u| ˆ ≤
|u|>δ
V (x)q [|u|q−1 + 1]|u| ˆ
Z
q
≤C 3|w|>δ
V q |w|q ≤ CkwkD .
(3.16)
Since uuˆ = w 2 − (v + y)2
(3.17)
f (x, u)uˆ = (f (x, u)/u)[w 2 − (v + y)2 ],
(3.18)
and
we see that (2.6) implies f (x, u)uˆ ≤ r 2 w 2 when |u| ≤ δ and uuˆ ≥ 0 and f (x, u)uˆ ≤ 0 when |u| ≤ δ and uuˆ < 0.
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K. Perera, M. Schechter
Hence,
Z |u| 0, A(P− + P0 ) ≤ 0.
(3.24)
By hypothesis,
Consequently, the Morse index of G1 is p. By the homotopy invariance of critical groups, we have Ck (G, 0) ∼ = Ck (G1 , 0) ∼ = δpk Z. This gives the desired conclusion. u t Next, we have Lemma 3.3. If 0 is the only solution of (2.8), and (A) holds, then Ck (G, 0) ∼ = δp1 k Z ∀k, where p1 = dim V . Proof. In this case we now define 2J (u) = kP+ uk2 − kP− uk2 + kP0 uk2 . Then
G001 (0) = A + P+ − P− + P0 .
Thus A(P+ + P0 ) ≥ 0, AP− < 0.
(3.25)
Nontrivial solutions of elliptic semilinear equations at resonance
Consequently,
309
G001 (0)(P+ + P− ) > 0, G001 (0)P− < 0.
Thus the Morse index of G1 at 0 is p1 . Next, we note that there is an R > 0 such that (Ht0 (u), J 0 (u)) > 0, kukD ≥ R.
(3.26)
For if (3.26) did not hold, there would be a sequence {uk } ⊂ D such that (Ht0 (uk ), J 0 (uk )) ≤ 0,
(3.27)
and ρk = kuk kD → ∞. Let u˜ k = uk /ρk , and write u˜ k = v˜k + y˜k + w˜ k , v˜k ∈ V , y˜k ∈ N (A), w˜ k ∈ W. In particular, we have (G0 (uk ), h)/2 = a(uk , h) − (f (uk ), h).
(3.28)
Thus, (Ht0 (uk ), J 0 (uk ))/ρk2 = kw˜ k k2D + kv˜k k2D − (1 − t)(f (uk ), uˆ k )/ρk2 . (Here we take uˆ = w − v + y.) From this we conclude that (3.27) implies kv˜k kD + kw˜ k kD → 0.
(3.29)
Since ku˜ k kD = 1, we must have a renamed subsequence such that y˜k → y˜ strongly in D with kyk ˜ D = 1. Consequently, (Ht0 (uk ), J 0 (uk ))/ρkσ ≥ −(1 − t)(f (uk ), uˆ k )/ρkσ . But
Z
f (x, uk )y˜k /ρkσ −1 =
Z
[uk f (x, uk )/|uk |σ ][|u˜ k |σ −2 u˜ k y˜k ] Z Z α+ |y| ˜σ+ α− |y| ˜σ 0 ˜
y 0
Z
and
|u| 0, we know that p 6 = p1 . But Lemma 3.2 tells us that (3.11) holds, while Lemma 3.3 tells us that (3.25) holds. This contradiction proves the theorem. u t Similarly, we prove Theorem 2.2. Proof. The same reasoning applies to the contradiction produced by Lemmas 3.4 and 3.5. u t References [1] Amann, H. and Zehnder, E.: Nontrivial solutions for a class of nonresonance problems and applications to nonlinear differential equations. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 7 (4), 539–603 (1980) [2] Bartsch, T. and Li, S.J.: Critical point theory for asymptotically quadratic functionals and applications to problems with resonance. Nonlinear Anal. 28 (3), 419–441 (1997) [3] Chang, K.C.: Infinite-dimensional Morse theory and multiple solution problems. Boston, MA: Birkhäuser, 1993 [4] Costa, D.G. and Silva, E.A.: On a class of resonant problems at higher eigenvalues. Differential Integral Equations 8 (3), 663–671 (1995) [5] Hirano, N. and Nishimura, T.: Multiplicity results for semilinear elliptic problems at resonance and with jumping nonlinearities. J. Math. Anal. Appl. 180 (2), 566–586 (1993) [6] Li, S.J. and Liu, J.Q.: Nontrivial critical points for asymptotically quadratic function. J. Math. Anal. Appl. 165 (2), 333–345 (1992) [7] Li, S.J. and Zou, W.: The computations of the critical groups with an application to elliptic resonant problems at higher eigenvalue. Preprint [8] Mawhin, J. and Willem, M.: Critical point theory and Hamiltonian systems. NewYork– Berlin: Springer-Verlag, 1989 [9] Perera, K.: Applications of local linking to asymptotically linear elliptic problems at resonance. NoDEA. Nonlinear Differential Equations Appl. 6 1, 55–62 (1999) [10] Perera, K. and Schechter, M.: The Fucik spectrum and critical groups. Preprint [11] Perera, K. and Schechter, M.: A generalization of the Amann-Zehnder theorem to nonresonance problems with jumping nonlinearities. To appear in NoDEA. Nonlinear Differential Equations and Applications [12] Perera, K. and Schechter, M.: Type (II) regions between curves of the Fucik spectrum and critical groups. Topol. Methods Nonlinear Anal. 12(2), 227–243 (1995) [13] Schechter, M.: Bounded resonance problems for semilinear elliptic equations. Nonlinear Anal. 24 (10), 1471–1482 (1995) [14] Silva, E.A.: Linking theorems and applications to semilinear elliptic problems at resonance. Nonlinear Anal. 16 (5), 455–477 (1991) [15] Su, J. and Tang, C.: Multiplicity results for semilinear elliptic equations with resonance at higher eigenvalues. Preprint
