− a + b Ω |∇u|2 Δu = f (x, u) in Ω,
u=0
on ∂Ω,
(1.1)
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:
(1.9)
u=0
on ∂Ω,
whose eigenvalues are the critical values of the functional
I (u) = u 4, u ∈ S := u ∈ H = H01(Ω): u4 = 1 .
Ω
(1.10)
458
Z. Zhang, K. Perera / J. Math. Anal. Appl. 317 (2006) 456–463
∃λ > λ2: F (x, t)
aλ t 2, 2
|t| small,
(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 t2, |t| small, 2
We assume that
tf (x, t) 0
(1.4)
and consider three cases:
(i) 4-sublinear case: p < 4, (ii) asymptotically 4-linear case:
lim
|t |→∞
f
(x, t) bt 3
=
μ
uniformly in x,
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. © 2005 Elsevier Inc. All rights reserved.
(1.5)
(iii) 4-superlinear case:
∃ν > 4: νF (x, t) tf (x, t), |t| large,
(1.6)
where F (x, t) =
t 0
f
(x
,
s
)
d
s
,
which
implies
F (x, t) C |t|ν − 1 .
(1.7)
By (1.2) and (1.5) (respectively (1.7)),
b Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA Received 11 September 2004
Available online 8 September 2005 Submitted by P. Smith
J. Math. Anal. Appl. 317 (2006) 456–463
/locate/jmaa
Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow
Keywords: Nonlocal problems; Kirchhoff’s equation; Variational methods; Invariant sets of descent flow
1. Introduction
In this paper we obtain sign changing solutions of the problem
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
Zhitao Zhang a,1, Kanishka Perera b,∗,2
a Academy of Mathematics and Systems Science, Institute of Mathematics, Chinese Academy of Sciences, Beijing 100080, PR China
μ2
:=
inf
γ ∈Γ
max
u∈γ ([0,1])
I
(u),
(1.11)
where Γ is the class of paths γ ∈ C([0, 1], S) joining ±ψ such that γ ∪ (−γ ) is non-self-
intersecting. We are now ready to state our main result. Denote by 0 < λ1 < λ2 · · · the Dirichlet eigen-
Proof. Since uj is bounded, for a subsequence, uj converges to some u weakly in H and strongly in L4(Ω). Denoting by
Pj v = v −
u3j v uj
(2.1)
Ω
the projection of v ∈ H onto the tangent space to S at uj , we have
>
0
in
Ω
and
the
interior
normal
derivative
∂ψ ∂ν
>
0
on ∂Ω by the strong maximum principle.
Z. Zhang, K. Perera / J. Math. Anal. Appl. 317 (2006) 456–463
459
Recall that a function u ∈ H is called a weak solution of (1.1) if
utt − a + b |∇u|2 Δu = g(x, t)
(1.3)
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 Pohožaev [14]. However, Eq. (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].
(iv) (1.6) and (1.13) hold.
(1.12) (1.13)
2. Variational setting
Lemma 2.1. I satisfies (PS), i.e., every sequence (uj ) in H such that I (uj ) is bounded and I (uj ) → 0, called a (PS) sequence, has a convergent subsequence.
μ1
:=
inf
u∈S
I
(u)
(2.3)
is achieved and hence positive. If ψ is a minimizer, then so is |ψ|, so we may assume that ψ 0.
Snontrivial
solution
of
(1.9),
ψ
Z. Zhang, K. Perera / J. Math. Anal. Appl. 317 (2006) 456–463