The study of a fourth-order differential equation: existence, uniqueness and a dynamical system approach

This paper is concerned with the problem of existence and uniqueness of solutions for the semilinear fourth-order differential equation uiv−ku′′+ a(x)u+c(x) f (u) = 0. Existence and uniqueness is proved using variational methods and maximum principles. We also give a dynamical system approach to the equation. We study the bifurcation of the system and show that the behaviour of the stationary points S (α, 0, 0, 0) depend on the relation between the parameter k and β = f ′(α).

We also note that the extension of (1), to the higher dimensional case (n ≥ 2) reads and also serves as a model of some applications: • the case φ ≡ 0 and f (u) = k 1 u 3 + k 2 u, k 1 , k 2 > 0 arises in bending of cylindrical shells [14] • the case φ ≡ 0 and f (u) = −ku, k > 0, arises in thin plate theory [16] • the case φ ≡ 0, f (u) = e u represents a natural higher order extension of the celebrated equation −∆u = e u , which describes the problem of self -ignition [12].
The purpose of this work is to study via variational methods and using maximum principles the existence and uniqueness of (1). We obtain, in some sense, generalizations of results obtained in Peletier et al. ([19]) , where (1) is considered with a(x) = −k and f (u) = ku 3 or generalizations of results presented in the paper of Tersian and Chaparova ( [22] We are able to treat the a general case when where We also give a dynamical system approach of equation (1). We show that if there exists α such that f (α) = 0 then the point S (α, 0, 0, 0) is an equilibrium point for the associated dynamical system. The behaviour of S (α, 0, 0, 0) depends on the relation between the parameter k and β = f ′ (α).
Throughout the paper C denotes a universal positive constant unless otherwise speci-fied.

Variational Settings and Preliminaries
A classical solution of (1) is a function u ∈ C 4 (Ω) that satisfies (1).
Note that a weak solution of (1) is also a classical solution of (1) (for a proof see [22]).
The problem (1) has a variational structure and the weak solutions in the space H(Ω) can be found as critical points of the functional which is Fréchet -differentiable and its Fréchet derivative is given by The following lemma will be useful.

Lemma 3.1 Suppose that F satisfies (3), k ≥ 0. Then there exists a minimizerũ ∈ H(Ω) of J.
If in addition we admit that f ′ ≥ 0 in IR, then the minimizer is unique.
We now show that J(u) is weakly lower semicontinous on the reflexive space W 2,2 (Ω).
Since k ≥ 0 we get that ) dx is (sequentially) weakly continuous.
Therefore, J(u) is weakly lower semicontinous by Criterion 6.1.3 in [2], p.301. We obtain thus the existence of the minimizer as required.
If f ′ ≥ 0 then J(u) is convex and the uniqueness of the minimizer follows.
Case r = 2 According to Lemma 2.1 the space H(Ω) endowed with the scalar product becomes a Hilbert space (k ∈ IR). This scalar product generates a norm denoted || · || 2 Hence J becomes Lemma 3.2 Suppose that F satisfies (3). If in addition we admit that then the boundary value problem (1) has at least a weak solution.
The proof uses the following inequality (which is a consequence of Lemma 2.1) ∫ and is similar to the proof of Lemma 2.1 The following result ensures that the solution we have found is nontrivial. where α ≥ 2, c 0 and Then there exists e ∈ H(Ω) such that J(e) < 0.
We are now able to prove our main result. Proof. The existence part of the theorem follows from Lemma 3.1, since the minimizer is the solution to (1). From Lemma 3.3 follows that the solution in nontrivial. f ′ ≥ 0 and F ≥ 0 implies f (0) = 0 and hence u ≡ 0 is the unique solution.
The proof (for a proof see [10]) is long and is based on the following four lemmas and on a version of the celebrated Mountain-Pass Theorem due to Brezis and Nirenberg ( [3], [11]). We just state the results without proof. [3], [11]) Let E be a real Banach space with its dual E * and suppose that J ∈ C 1 (E, R) satisfies

Theorem 3.2 (Mountain-Pass Theorem
for some constants µ < η, ρ > 0 and e ∈ E with ||e|| > ρ. Let δ ≥ η be characterized by The following two lemmas show that the functional J has the mountain pass geometry. Lemma 3.4 Let F satisfy (3). Then there exist two positive constants ρ and η such that where Then there exists e ∈ H(Ω) with ||e|| 2 > ρ such that J(e) < 0.
then the sequence {u n } defined by (8)

is bounded in H(Ω).
Remark. Lemma 3.6 still holds when γ = 2, if we impose some restrictions. An immediate consequence of Lemma 3.7 is the existence result:

Theorem 3.3 The problem (1) admits a nontrivial solution in H(Ω).
The following theorem collects several results and shows that uniqueness results may hold even if the coefficient of u is unbounded, nonsmooth or has arbitrary sign.
in Ω and a f ′ is of arbitrary sign in Ω, then there exists at most one classical solution of Proof.
We prove the uniqueness result under the assumption (12). It suffices to show that u ≡ 0 the unique solution of the homogenous, linear boundary value problem where u is a solution of (15). By computing, we see that for a constant γ > 0 we get By the generalized maximum principle, Theorem 10, p.73, [21], there exists a real constant such that or P w does not attain a nonnegative maximum in Ω.
The function P/w is smooth and hence (16) holds in Ω.
By the boundary conditions we have P = 0 on ∂Ω, i.e., the constant=0. It follows that P = 0 in Ω, which means u ≡ 0 in Ω.
We are left to check the condition (17), i.e., max By the boundary conditions we have i.e., u ≡ 0 in Ω. Note that the uniqueness results 10 and 11 hold in n dimensions and extend (in some sense) a classical result of Chow and Dunninger [5].
For further uniqueness results (in higher dimensions) the reader is referred to the paper [7] and [8].

A Dynamical System Approach
To give a dynamical approach to (1), we consider the following version of (1) in IR and rewrite (18) as a dynamical system (a fourth order system of four equations) Let us define F : Then the system (19) may be rewritten as where a solution to (19) has the form X(t) = (x 1 (t), x 2 (t), x 3 (t), x 4 (t)).
Suppose that f has a zero at α. Then the dynamical system (19) admits a stationary point which is S = (α, 0, 0, 0). Note that this point corresponds to the solution u ≡ α to (18).
We can prove: • if k > 2 √ β then the eigenvalues of A are purely complex and distinct and hence S is a center.
• if k = 2 √ β then eigenvalues of A are purely complex but not distinct (double).
• if k = −2 √ β then eigenvalues of A are real but not distinct (double).