Infinitely many solutions for a fourth-order differential equation on a nonlinear elastic foundation
Boundary Value Problems
Infinitely many solutions for a fourth-order differential equation on a nonlinear elastic foundation
Xiaodan Wang
In this paper, existence results of infinitely many solutions for a fourth-order differential equation with nonlinear boundary conditions are established. The proof is based on variational methods. Some recent results are improved and extended.
⎨⎧⎪ u() = f (x; u); < x < ; u() = u () = ; ⎪⎩ u () = ; u () = g(u())
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1 Introduction
In this paper, we consider a beam equation with nonlinear boundary conditions of the
type
where f ∈ C([, ], R) and g ∈ C(R) are real functions. This kind of problem arises in the
study of deflections of elastic beams on nonlinear elastic foundations. The problem has
the following physical description: a thin flexible elastic beam of length is clamped at its
left end x = and resting on an elastic device at its right end x = , which is given by g.
Then, the problem models the static equilibrium of the beam under a load, along its length,
characterized by f . The derivation of the model can be found in [, ].
Owing to the importance of fourth-order two-point boundary value problems in
describing a large class of elastic deflection, there is a wide literature that deals with the
existence and multiplicity results for such a problem with different boundary conditions
(see, for instance, [–] and the references therein).
Motivated by the above works, in the present paper we study the existence of infinitely
many solutions for problem (.) when the nonlinear term f (x, u) satisfies the superlinear
condition and sublinear condition at the infinity on u, respectively. As far as we know, this
case has never before been considered.
Now we state our main results.
1.1 The superlinear case
We give the following assumptions.
(H) g is odd and satisfies
s
g(t) dt – g(s)s ≥ ,
g(t) dt ≥ for all s ∈ R.
s
(H) There exist constants a, b ≥ and γ ∈ [, ) such that
g(s) ≤ a + b|s|γ
for s ∈ R.
(H) lim|u|→+∞ F(ux,u) = +∞ uniformly for x ∈ [, ], where F(x, u) = u f (x, t) dt.
(H) F(x, ) ≡ , ≤ F(x, u) = o(|u|) as |u| → uniformly for x ∈ [, ].
(H) There exist constants α > , < β < + αα– , c, c > and L > such that for every
x ∈ [, ] and u ∈ R with |u| ≥ L,
f (x, u)u – F(x, u) ≥ c|u|α,
f (x, u) ≤ c|u|β .
Theorem . Assume that (H)-(H) hold and F is even in u. Then problem (.) has
infinitely many solutions.
Remark . There exist some functions satisfying (H)-(H), but not satisfying the
wellknown (AR)-condition,
< θ F(x, u) ≤ f (x, u)u,
∀u > , x ∈ [, ],
for some θ > .
For example, take f (x, u) = u ln( + u) + +uu . Then F(x, u) = |u| ln( + |u|).
Obviously, (H)-(H) are satisfied. Note that
f (x, u)u – F(x, u) = |u| ln + |u|
|u|
+ |u| ≥ |u| ln ,
∀|u| ≥ ,
and
f (x, u) ≤ ln + |u| |u| + +|u| u|| ln + |u| |u| ≤ |u| ,
∀|u| ≥ L,
for L being large enough, which implies (H). However, it is easy to see that f does not
satisfy (AR)-condition.
1.2 The sublinear case
We make the following assumptions.
(S) g is odd and satisfies g(s)s ≥ for any s ∈ R.
(S) There exist constants b > and γ ∈ [, ) such that
g(s) ≤ b|s|γ
for s ∈ R.
(S) F(x, ) ≡ for any x ∈ [, ].
(S) There are constants k > and ζ ∈ [, ) with ζ < γ + such that
F(x, u) ≥ k|u|ζ for any (x, u) ∈ [, ] × R.
(S) There exist constants k > and ζ ∈ [, ) such that
f (x, u) ≤ k|u|ζ– for any (x, u) ∈ [, ] × R.
Theorem . Assume that (S)-(S) hold and F is even in u. Then problem (.) has
infinitely many solutions.
Remark . The condition (S) implies that s g(t) dt ≥ .
The remainder of this paper is organized as follows. In Section , some preliminary
results are presented. In Section , we give the proofs of our main results.
2 Variational setting and preliminaries
In this section, the following two theorems will be needed in our argument. Let E be a
Banach space with the norm · and E = j∈N Xj with dim Xj < ∞ for any j ∈ N. Set Yk =
jk= Xj, Zk = j∞=k Xj and Bk = {u ∈ Yk : u ≤ ρk}, Nk = {u ∈ Zk : u = rk} for ρk > rk > .
Consider the C-functional λ : E → R defined by
λ(u) = A(u) – λB(u),
λ ∈ [, ].
Assume that:
(C) λ maps bounded sets to bounded sets uniformly for λ ∈ [, ]. Furthermore,
λ(–u) = λ(u) for all (λ, u) ∈ [, ] × E.
(C) B(u) ≥ for all u ∈ E; A(u) → ∞ or B(u) → ∞ as u → ∞; or
(C) B(u) ≤ for all u ∈ E; B(u) → –∞ as u → ∞.
For k ≥ , define k := {γ ∈ C(Bk, E) : γ is odd; γ |∂Bk = id},
ck(λ) := infγ ∈ k maxu∈Bk λ(γ (u)),
bk(λ) := infu∈Zk, u =rk λ(u),
ak(λ) := maxu∈Yk, u =ρk λ(u).
Theorem . ([, Theorem .]) Assume that (C) and (C) (or (C) ) hold. If bk(λ) > ak(λ)
for all λ ∈ [, ], then ck(λ) ≥ bk(λ) for all λ ∈ [, ]. Moreover, for a.e. λ ∈ [, ], there exists
a sequence {ukn(λ)}n∞= such that supn ukn(λ) < ∞, λ(ukn(λ)) → and λ(ukn(λ)) → ck(λ)
as n → ∞.
Theorem . ([, Theorem .]) Suppose that (C) holds. Furth (...truncated)