Nonlinear Differential Equations, Electron. J. Diff. Eqns., Conf. 05, 2000, pp. 325–333 http://ejde.math.swt.edu or http://ejde.math.unt.edu ftp ejde.math.swt.edu or ejde.math.unt.edu (login: ftp)

A condition on the potential for the existence of doubly periodic solutions of a semi-linear fourth-order partial differential equation ^{∗}

Chen Chang

Abstract

We study the existence of solutions to the fourth order semi-linear equation

Δ^{2}u = g(u) + h(x) .

We show that there is a positive constant C_{∗}, such that if g(ξ)ξ ≥ 0 for |ξ| ≥ ξ_{0} and lim sup_{|ξ|→∞} 2G(ξ)/ξ^{2} < C_{∗}, then for all h ∈ L^{2}(Q) with ∫_{Q} hdx = 0, the above equation has a weak solution in H_{2π}^{2}.

1 Introduction

This paper is motivated by the study of the differential equation

u′′ + g(u) = h(t) = h(t + 2π) ,  (1.1)

where g and h are continuous functions. It is assumed that

∫_{0}^{2π} h(t) dt = 0 .  (1.2)

Indeed, if ĥ = \frac{1}{2π} ∫_{0}^{2π} h(t) dt, we may replace g(u) by g(u) − ĥ and h by h − ĥ in (1.1). We write g ∈ Σ if there exists a constant ξ_{0} ≥ 0 such that

g(ξ)ξ ≥ 0 for |ξ| ≥ |ξ_{0} .  (1.3)

Given g ∈ Σ, let G′(ξ) = g(ξ), G(0) = 0.

Recently Fernandes and Zanolin [2] proved the existence of 2π-periodic solutions of (1.1). Their work shows that if g ∈ Σ, (1.2) holds and either lim inf_{ξ→∞} 2G(ξ)/ξ^{2} < 1/4 or lim inf_{ξ→−∞} 2G(ξ)/ξ^{2} < 1/4, then there exists a 2π-periodic solution of (1.1). Earlier work of Mawhin and Ward showed that if either lim sup_{ξ→∞} g(ξ)/ξ < 1/4 or lim sup_{ξ→−∞} g(ξ)/ξ < 1/4, then (1.1) has a solution.

∗Mathematics Subject Classifications: 35J30, 35B10.

Key words: periodic solutions, elliptic, fourth-order PDE.

\begin{matrix}c \\ ◯ \end{matrix}2000 Southwest Texas State University.

Published October 31, 2000.

These results led us to consider a more modest question for the partial differential equation

Δu + g(u) = h(x) ,  (1.4)

where x = (x_{1}, x_{2}) and h(x_{1} + 2π, x_{2}) = h(x_{1}, x_{2} + 2π) = h(x_{1}, x_{2}). Namely if Q = [0, 2π] × [0, 2π], g ∈ Σ, h ∈ L^{2}(Q), and

∫_{Q} hdx = 0 ;  (1.5)

does there exist a constant C_{∗} such that the condition

lim sup_{|ξ|→∞} \frac{2G(ξ)}{ξ^{2}} < C_{∗}  (1.6)

implies the existence of a weak solution to (1.4) with the “boundary condition” u(x_{1} + 2π, x_{2}) = u(x_{1}, x_{2} + 2π)?

Thus we define a solution to be a member of the function space H_{2π}^{1} such that

∫_{Q}[u_{x_{1}}v_{x_{1}} + u_{x_{2}}v_{x_{2}} − g(u)v − h(x)v]dx = 0

for all v ∈ C_{2π}^{∞}, the space of C^{∞} functions defined on ℝ^{2} which are 2π-periodic in each variable. The space H_{2π}^{1} is the completion of this space with respect to the norm

∥u∥ = [ ∫_{Q}(u_{x_{1}}^{2} + u_{x_{2}}^{2} + u^{2})dx]^{1/2}.

The difficulty with this problem is that if g is only assumed to be continuous and u ∈ H_{2π}^{1}, it is not generally true that the function g(u(x)) is locally integrable. Also, unless g satisfies a suitable growth condition, the functional, f : H_{2π}^{1} → ℝ,

f(u) = ∫_{Q} \frac{|∇u|^{2}}{2} − G(u) + h(x)u dx

is not of class C^{1}. Thus we abandon this problem and considered the analogous fourth order semi-linear problem

Δ^{2}u = g(u) + h(x)  (1.7)

with u ∈ H_{2π}^{2}, where h is in L^{2}(Q) and H_{2π}^{2} denotes the completion of C_{2π}^{∞} with respect to the norm

{ ∫_{Q} [ ∑_{i=1}^{2} ∑_{j=1}^{2} u_{x_{i}x_{j}}^{2} + ∑_{i=1}^{2} u_{x_{i}}^{2} + u^{2}]dx}^{1/2} .

By a weak solution of (1.7) we mean a u ∈ H_{2π}^{2} such that _{∫Q}[ΔuΔv−g(u)v− h(x)v]dx = 0 for all v in C_{2π}^{∞}.

Since it can be shown that H_{2π}^{2} ⊂ C_{2π} (this is essentially the Sobolev embedding theorem), u ∈ H_{2π}^{2} implies that g(u(x)) is continuous. Moreover the compactness of H_{2π}^{2} in C_{2π} ensures that the functional f : H_{2π}^{2} → ℝ defined by

f(u) = _{∫Ω} [\frac{(Δu)^{2}}{2} − G(u) − h(x)u] dx

is of class C^{1}. We show that there exists C_{∗} > 0 such that if g ∈ Σ and (1.6) holds, then for all h satisfying (1.5), h ∈ L^{2}(Q), (1.7) has a weak solution.

We have shown that if

C_{∗} = \frac{1}{4π^{2}a_{∗}^{2} + 1}, where a_{∗}^{2} = \frac{1}{π^{2}} ∑_{i=1}^{∞}∑_{j=1}^{∞} \frac{1}{(i^{2} + j^{2})^{2}}

then this statement will be true. However, we feel that this is far from the optimal value of C_{∗}.

It is clear that the optimal value must be less than 1, since it can be shown that if g(ξ) = ξ, h(x_{1}, x_{2}) = sin x_{1}, then (1.7) does not have a weak solution, because of resonance.

2 Definitions and preliminary lemmas

In this section we state some preliminary lemmas. These results follow more or less from known results (see for example [1]). Full details will be given elsewhere.

Let Q = {(x_{1}, x_{2})|0 ≤ x_{1} ≤ 2π, 0 ≤ x_{2} ≤ 2π}. Let L_{2π}^{2}(ℝ^{2}) denote the set of real-valued measurable functions defined in ℝ^{2} such that if u ∈ L_{2π}^{2}(ℝ^{2}), then u(x_{1} + 2π, x_{2}) = u(x_{1}, x_{2} + 2π) = u(x_{1}, x_{2}) and such that u restricted to Q is in L^{2}(Q).

We denote C_{2π} and C_{2π}^{∞} the real-valued functions defined on ℝ^{2} which are 2π-periodic in each variable, which are continuous and of class C^{∞} respectively.

We denote by H_{2π}^{2}(ℝ^{2}) the set of u ∈ L_{2π}^{2}(ℝ^{2}) such that for p = 1, 2 there exists v_{p} ∈ L_{2π}^{2}(ℝ^{2}) such that for all ϕ ∈ C_{2π}^{∞},

−_{∫Q}(D_{p}ϕ)udx = _{∫Q}v_{p}ϕdx

and for 1 ≤ p, q ≤ 2 there exists v_{pq} ∈ L_{2π}^{2}(ℝ^{2}) such that for all ϕ ∈ C_{2π}^{∞},

∫Q(D_{p}D_{q}ϕ)udx = _{∫Q}ϕv_{pq}dx.

Here D_{p} = ∂/∂x_{p}, p = 1, 2. It is clear that v_{p}, p = 1, 2, and v_{pq}, p, q = 1, 2, are determined uniquely and we write v_{p} = D_{p}u, p = 1, 2, and v_{pq} = D_{p}D_{q}u, p, q = 1, 2.

The space H_{2π}^{2}(ℝ^{2}) is a real Hilbert space with inner product given by

⟨u, v⟩ = _{∫Q} [uv + ∑_{p=1}^{2}(D_{p}u)(D_{p}v) + ∑_{p,q=1}^{2}(D_{p}D_{q}u)(D_{p}D_{q}v)] dx

In the following we denote the Hilbert space H_{2π}^{2} by 𝔼 and ∥ · ∥_{𝔼} will denote the norm given by the inner product defined above.

Lemma 2.1 If u ∈ 𝔼 then u is equal almost everywhere to a unique function in C_{2π}. If this function is again denoted by u, then there exists a constant a_{0} such that for all u ∈ 𝔼, ∥u∥_{C_{2π}} = max_{x∈ℝ^{2}}|u(x)| ≤ a_{0}∥u∥_{𝔼}. (see [1, p 167]).

We denote by 𝔼̂ the set of u ∈ 𝔼 such that _{∫Q}udx = 0.

The following result can be proved using multiple Fourier series.

Lemma 2.2 An inner product on 𝔼̂ which is equivalent to the 𝔼-inner product is given by

⟨u, v⟩𝔼̂ = _{∫Q}(Δu)(Δv)dx

where, as usual Δu = D_{1}^{2}u + D_{2}^{2}u.

Lemma 2.3 The best possible constant a_{∗} such that for all u ∈ 𝔼̂,

∥u∥_{c_{2π}} = max_{x∈ℝ^{2}} |u(x)| ≤ a_{∗}∥u∥𝔼̂ ,

where ∥u∥𝔼̂ = ∥Δu∥_{L^{2}(Q)}, is

a_{∗} = \frac{1}{2π} ( ∑_{k≠(0,0)^{k∈Z^{2}}} \frac{1}{|k|^{4}} )^{1/2}  (2.1)

it where Z^{2} = Z × Z, Z = {0, ±1, ±2, ±3, …}, and if k = (k_{1}, k_{2}) ∈ Z^{2}, |k| = \sqrt{k_{1}^{2} + k_{2}^{2}}.

This lemma and the next are proved using multiple Fourier series.

Lemma 2.4 If u ∈ 𝔼̂, then _{∫Q}u^{2}dx ≤ _{∫Q}(Δu)^{2}dx.

The following result is proved using the idea of the proof given in [5, p. 216] except Fourier series are used instead of Fourier transform.

Lemma 2.5 Let 0 < α < 1. There exists M(α) such that if u ∈ 𝔼, then for x ∈ ℝ^{2} and y ∈ ℝ^{2}

|u(x) − u(y)| ≤ M_{(α)}∥u∥_{𝔼}|x − y|^{α} .

Here, for x = (x_{1}, x_{2}) and y = (y_{1}, y_{2}),

|x − y| = \sqrt{(x_{1} − y_{1})^{2} + (x_{2} − y_{2})^{2}} .

The final preliminary lemma follows from Lemma 2.1, Lemma 2.5 and Ascoli’s Lemma.

Lemma 2.6 The injection from 𝔼 to C_{2π} is compact, that is, if {u_{n}}_{n=1}^{∞} is a bounded sequence in 𝔼, then there exists a subsequence {u_{n_{i}}}_{i=1}^{∞} such that {u_{n_{i}}}_{i=1}^{∞} converges uniformly on ℝ^{2}.

3 Periodic solutions of a semi-linear elliptic fourthorder partial differential equation

In this section g will always denote a real-valued function defined and continuous on ℝ, and G will denote the function such that G′(ξ) = g(ξ) for ξ ∈ ℝ with G(0) = 0. L̂_{2π}^{2} will denote the closed subspace of L_{2π}^{2}(ℝ^{2}) such that for all h ∈ L̂_{2π}^{2}, ∫_{Q} h(x)dx = 0 .

We consider the question of existence of weak solution of the problem

Δ^{2}u = g(u) + h(x) (3.1)

u ∈ H_{2π}^{2}(ℝ^{2})

where h ∈ L̂_{2π}^{2}. This is defined to be a function u ∈ H_{2π}^{2}(ℝ^{2}) such that for all v ∈ 𝔼 (= H_{2π}^{2}(ℝ^{2})),

∫_{Q}[(Δu)(Δv) − g(u)v − h(x)v]dx = 0 .  (3.2)

If u is a function of class C^{4} which is 2π-periodic in each variable, then (3.1) holds if and only if (3.2) holds.

Let f : 𝔼 → ℝ be the function

f(u) = _{∫Q} [\frac{|Δu|^{2}}{2} − G(u) − h(x)u] dx .

Since 𝔼 ⊂ C_{2π}, standard arguments (see, for example, [4]) show that f ∈ C^{1}. For v ∈ 𝔼,

f′(u)(v) = _{∫Q}[(Δu)(Δv) − g(u)v − h(x)v]dx .

Therefore, weak solutions of (3.1) coincide with critical points of f.

Let Σ denote the set of continuous g : ℝ → ℝ such that there exists some ξ_{0}, depending on g, such that

g(ξ)ξ ≥ 0 for |ξ| ≥ ξ_{0} .  (3.3)

Theorem 3.1 Let a_{∗} be as in (2.1) and let

C_{∗} = \frac{1}{4π^{2}a_{∗}^{2} + 1}  (3.4)

If g ∈ Σ and

lim sup_{|ξ|→∞} \frac{2G(ξ)}{ξ^{2}} < C^{∗}  (3.5)

then, for all h ∈ L̂_{2π}^{2}, there exists a weak solution of (3.1).

Sketch of Proof: The proof is an application of Rabinowitz’s Saddle-Point Theorem [4]. Assume first that g satisfies the stronger condition: There exist δ > 0 and ξ_{0} ≥ 0 such that

|ξ| ≥ ξ_{0} implies sgn(ξ)g(ξ) ≥ δ.  (3.6)

Assuming that (3.5) holds there exist constants C_{2} ≥ 0 and C_{1} with

C_{1} < C_{∗}  (3.6)

such that for all ξ ∈ ℝ,

G(ξ) ≤ C_{1} (\frac{ξ^{2}}{2} ) + C_{2}.  (3.7)

We claim that the functional f defined above satisfies the Palais-Smale condition. To see this let {u_{n}}_{n=1}^{∞} be a sequence in 𝔼 such that {f(u_{n})}_{n=1}^{∞} is a bounded sequence in ℝ and f′(u_{n}) → 0 in 𝔼^{∗}, the topological dual space of 𝔼.

We first show that the sequence {u_{n}}_{n=1}^{∞} is bounded in L^{2}(Q). Assuming the contrary, we may assume, by considering a subsequence, that ∥u_{n}∥_{L^{2}} ≠ 0 for all n and that ∥u_{n}∥_{L^{2}} → ∞ as n → ∞.

By assumption, there exists a constant C_{3} such that f(u_{n}) ≤ C_{3} for all n ≥ 1 or

∫_{Q} [\frac{|Δu_{n}|^{2}}{2} − G(u_{n}) − h(x)u_{n}] dx ≤ C_{3}

for all n. From (3.7) we have that for n ≥ 1

∫Q|Δu_{n}|^{2}dx ≤ C_{1}∥u_{n}∥_{L^{2}}^{2} + 8π^{2}C_{2} + 2∥h∥_{L^{2}}∥u_{n}∥_{L^{2}} + 2C_{3}

Setting w_{n} = u_{n}/∥u_{n}∥_{L^{2}} for n = 1, 2, .. we obtain

∫Q(Δw_{n})^{2}dx ≤ C_{1} + \frac{2∥h∥_{L^{2}}}{∥u_{n}∥_{L^{2}}} + \frac{8π^{2}C_{2} + 2C_{3}}{∥u_{n}∥_{L^{2}}^{2}}  (3.8)

for all n ≥ 1.

If 𝔼̂ is defined as in the previous section and if we identify the constant functions with the real numbers ℝ, then

𝔼 = 𝔼̂ ⊕ ℝ  (3.9)

For n ≥ 1, let

w_{n} = z_{n} + τ_{n} ,  (3.10)

where z_{n} ∈ 𝔼̂ and τ_{n} ∈ ℝ. Since ∥Δz_{n}∥_{L^{2}} = ∥Δw_{n}∥_{L^{2}}, it follows from (3.8) and Lemma 2.2 that the sequence {z_{n}}_{n=1}^{∞} is bounded in 𝔼̂. Therefore, since for all n ≥ 1, 4π^{2}τ_{n}^{2} ≤ ∥w_{n}∥_{L^{2}}^{2} = 1, we infer the existence of a constant C_{4} such that ∥w_{n}∥_{𝔼} < C_{4} for all n.

It follows that there exists a subsequence of {w_{n}}_{n=1}^{∞} which converges weakly to w in 𝔼. By considering a subsequence, we may assume, without loss of generality, that the sequence {w_{n}}_{n=1}^{∞} itself converges weakly to w.

If w = z + τ where z ∈ 𝔼̂ and τ ∈ ℝ, then z_{n} converges weakly to z and τ_{n} converges to τ as n → ∞. From Lemma 2.6, it follows that the sequence {w_{n}}_{n=1}^{∞} converges uniformly to w on ℝ^{2}, and since lim_{n→∞}τ_{n} = τ, we see that {z_{n}(x)}_{n=1}^{∞} converges uniformly to z(x) on ℝ^{2}.

The uniform convergence implies that ∥w∥_{L^{2}} = lim_{n→∞}∥w_{n}∥_{L^{2}} = 1. From the lower semi-continuity of a norm with respect to weak convergence, it follows from (3.8) that

∥Δz∥_{L^{2}}^{2} = ∥Δw∥_{L^{2}}^{2} ≤ lim inf_{n→∞}∥Δw_{n}∥_{L^{2}}^{2} ≤ C_{1} .

Therefore, ∥Δz∥_{L^{2}}^{2} ≤ C_{1}∥w∥_{L^{2}}^{2} = C_{1}(∥z∥_{L^{2}}^{2} + 4π^{2}τ^{2}) and since, according to Lemma 2.4, ∥z∥_{L^{2}} ≤ ∥Δz∥_{L^{2}}, it follows that

∥Δz∥_{L^{2}}^{2} ≤ \frac{C_{1}4π^{2}τ^{2}}{1 − C_{1}} .  (3.11)

(That C_{1} < 1 follows from (3.4) and (3.6)). Since 1 = ∥w∥_{L^{2}}^{2} = ∥z∥_{L^{2}}^{2} + 4π^{2}τ^{2}, we see that τ ≠ 0.

According to lemma 2.3

max_{x∈ℝ^{2}} |z(x)|^{2} ≤ (\frac{a_{∗}^{2}C_{1}4π^{2}}{1 − C_{1}} ) τ^{2},

and from (3.4) and (3.6)

\frac{a_{∗}^{2}C_{1}4π^{2}}{1 − C_{1}} < \frac{a_{∗}^{2}C_{∗}4π^{2}}{1 − C_{∗}} = 1.

Therefore,

max_{x∈ℝ^{2}} |z(x)| < |τ|.

Since τ ≠ 0 it follows that either w(x) = z(x)+τ > 0 for all x ∈ ℝ^{2} or w(x) < 0 for all x ∈ ℝ^{2}. Since u_{n}(x) = ∥u_{n}∥_{L^{2}}w_{n} either u_{n}(x) → ∞ uniformly with respect to x ∈ ℝ^{2} or u_{n}(x) → −∞ uniformly with respect to x ∈ ℝ^{2}. From (3.6) it follows that in the first case

∫_{Q} g(u_{n}(x))dx ≥ 4π^{2}δ

for n sufficiently large, and in the second case

∫_{Q} g(u_{n}(x))dx ≤ −4π^{2}δ

for n sufficiently large. But since h ∈ L̂_{2π}^{2}, f′(u_{n})(1) = _{∫Q} − [g(u_{n}(x)) + h(x)]dx = −_{∫Q}g(u_{n}(x))dx. Since f′(u_{n})(1) → 0 as n → ∞, therefore we have a contradiction. This contradiction proves the sequence {u_{n}}_{n=1}^{∞} is bounded in L^{2}(Q).

From the condition f(u_{n}) ≤ C_{3} for all n and the condition (3.7), it follows from Lemma 2.2, that {u_{n}}_{n=1}^{∞} is bounded in 𝔼. Therefore, from the form of f′ and Lemma 2.6, standard arguments (see for example [4]) shows that f′ satisfies the Palais-Smale condition.

The existence of a critical point of f follows from Rabinowitz’s Saddle Point Theorem [4] corresponding to the direct sum decomposition 𝔼 = 𝔼̂ ⊕ ℝ. Since, according to Lemma 2.4, for all z ∈ 𝔼̂, ∥z∥_{L^{2}} ≤ ∥Δz∥_{L^{2}}, it follows that for all z ∈ 𝔼̂,

∫_{Q} [\frac{(Δz)^{2}}{2} − G(z) − h(x)z] dx

≥ _{∫Q} [\frac{(Δz)^{2}}{2} − \frac{C_{1}}{2} z^{2} − C_{2}] dx − ∥h∥_{L^{2}}∥z∥_{L^{2}}
≥ (\frac{1 − C_{1}}{2} )_{∫Q} \frac{(Δz)^{2}}{2} dx − C_{2}4π^{2} − ∥h∥_{L^{2}}∥Δz∥_{L^{2}} .

Since, as shown above, C_{1} < 1 it follows that

inf_{z∈𝔼̂} f(z) > −∞.

The condition g(ξ)sgn ξ ≥ δ for |ξ| ≥ ξ_{0} implies that G(ξ) → ∞ as |ξ| → ∞. Therefore, since h ∈ L̂_{2π}^{2}, it follows that for ξ ∈ ℝ,

f(ξ) = ∫_{Q}[−G(ξ) − ξh(x)]dx ≤ −4π^{2}G(ξ) → −∞

as |ξ| → ∞. Thus there exists b > 0 such that

max{f(b), f(−b)} < inf_{z∈𝔼̂} f(z).

Since f satisfies the Palais-Smale condition, it follows that if Γ denotes the set of all continuous mappings γ : [−b, b] → 𝔼 with γ(± b) = ± b,

C_{0} = inf_{γ∈Γ} max_{ξ∈[−b,b]}f(γ(ξ)),

then there exits u_{0} ∈ 𝔼 such that f(u_{0}) = C_{0} and f′(u_{0}) = 0. This u_{0} is a solution of problem (3.1).

To prove that (3.1) has a solution when it is only assumed that g(ξ) sgn ξ ≥ 0 for |ξ| ≥ ξ_{0}. We can use a perturbation argument. We define

r(ξ) = \{ \begin{matrix}−1 & if ξ ≤ −ξ_{0} , \\ −1 + \frac{2(ξ+ξ_{0})}{2ξ_{0}} & if |ξ| ≤ ξ_{0} , \\ 1 & if ξ ≥ ξ_{0} , \end{matrix}

For m = 1, 2, 3, …, set g_{m}(ξ) = g(ξ) + \frac{r(ξ)}{m}. Then g_{m}(ξ)ξ ≥ \frac{1}{m} for |ξ| ≥ ξ_{0} and we still have

lim sup_{|ξ|→∞} \frac{2G_{m}(ξ)}{ξ^{2}} < C^{∗}.

By what has been shown, (3.1) has a solution when g = g_{m}. The conditions of the theorem imply that there is a priori bound on this solution (the one characterized by the Saddle Point Theorem) in 𝔼, which is independent of m. Using a compactness argument this implies the existence of a solution of (3.1). The computational details of this proof will be published somewhere else.

Acknowledgment The author wants to express his gratitude to Professor A. C. Lazer for his guidance.

References

[1] L. Bers, F. John, and M. Schechter, Partial Differential Equations, John Wiley and Sons Inc., 1964.

[2] M.L.C. Fernandes and F. Zanolin, Periodic solutions of a second order differential equation with one-side growth restrictions on the restoring term, Arch. Math 59 (1992), 245-259.

[3] J. Mawhin and J.R. Ward Jr., Periodic solutions of some forced Lienard differential equations at resonance, Arch. Math. 41 (1983) 337-351.

[4] P.H. Rabinowitz, Minimax methods in critical point theory with applications to differential equations, CBMS Regional conference in Math., Vol 65. Amer. Math. Soc. Providence, R.I., 1986.

[5] M. Renardy and R.C. Rogers, An introduction to partial differential equations, Texts in Applied. Math. 13, Springer-Verlag, 1993.

Chen Chang Division of Mathematics and Statistics the University of Texas at San Antonio San Antonio, TX 78249 USA e-mail: chang@math.utsa.edu