Differential and Integral Equations Volume xx, Number xxx, , Pages xx–xx

A NOTE ON AN EXPONENTIAL SEMILINEAR EQUATION OF THE FOURTH ORDER

Dimitri Mugnai Dipartimento di Matematica e Informatica, Università di Perugia Via Vanvitelli, 1, 06123 Perugia, Italy

(Submitted by: Reza Aftabizadeh)

Abstract. In this note we prove some existence and nonexistence results for a semilinear equation of the fourth order involving an exponential function via variational techniques. Such equations may occur while studying traveling waves in suspension bridges.

1. Introduction

Recently Chen and McKenna ([2]) have suggested investigating the following equation:

u_{xxxx} + cu_{xx} = e^{u} − 1  (1.1)

as a generalization of the model for traveling waves in suspension bridges proposed by Lazer and McKenna in [4],

(LM)  \{ \begin{matrix}Δ^{2}u + cΔu − b[u + (u + 1)^{−}] = 0 & in Ω, \\ u = Δu = 0 & on ∂Ω, \end{matrix}

where b is a real parameter, Δ^{2} stands for the biharmonic operator and v^{−} = max{0, −v}. Moreover, in [4] the authors suggested studying problems similar to (LM), where the nonlinearity h(u) = u + (u + 1)^{−} is replaced by some function which behaves like h(u) (see Problem 11 in [4]).

For example, the functions h(u) = u + (u + 1)^{−} = (u + 1)^{+} − 1 and g_{2}(u) = e^{u} − 1 have similar behavior, at least if u is bounded from above.

Moreover, one can also consider nonlinear problems of the form

(P_{k})  \{ \begin{matrix}Δ^{2}u + cΔu − bu + a((u + 1)^{−})^{k−1} = 0 & in Ω, \\ u = Δu = 0 & on ∂Ω, \end{matrix}

where a, b ∈ ℝ and k is greater than 2 and subcritical (see [10]).

Accepted for publication: May 2003.

AMS Subject Classifications: 35J35, 35J65, 49J99.

Supported by M.I.U.R.

In this paper we are interested in a problem related to (1.1). More precisely we will consider the problem

(P)  \{ \begin{matrix}Δ^{2}u + cΔu − b(e^{u} − 1) = 0 & in Ω, \\ u = Δu = 0 & on ∂Ω, \end{matrix}

where Ω is a bounded and smooth domain of ℝ^{N} and b ∈ ℝ. By a (weak) solution we mean a function u in H_{Ω} = H_{0}^{1}(Ω) ∩ H^{2}(Ω) such that

∫_{Ω} ΔuΔv dx − c ∫_{Ω} Du · Dv dx − b ∫_{Ω}(e^{u}v − v) dx = 0  (1.2)

for every v ∈ H_{Ω}. Let us remark that all integrals appearing in (1.2) are well defined when N ≤ 3 and Ω is bounded.

We remark that u ≡ 0 is a solution of all the problems considered above. Moreover, one could expect that solutions of problems (P), (LM), and (P_{k}) behave similarly, since the nonlinearities are similar, but it is not the case, as we will see. In fact, it is not difficult to prove that, for example, if Ω is a bounded and smooth domain, c < λ_{1} (the first eigenvalue of −Δ in H_{0}^{1}(Ω)), b = 0, and a ≤ 0, then any solution u of (P_{k}) is less than or equal to 0 (and on the other hand, there are nontrivial solutions of such a problem). In fact, a maximum principle for equations of the fourth order show that, if u ≢ 0, then u < 0 in Ω (see Section 2). But in [11] it is proved that, if c < λ_{1}, then (LM) has a nontrivial solution if and only if b ≥ λ_{1}(λ_{1} − c) and that in such a case there exists a solution u < 0 in Ω. (Concerning existence and multiplicity results for problem (LM) there is a large literature. For example, see [3], [5], [6], [7], [9], and the references therein.)

Moreover, in the same spirit of [11], we are going to prove in a very simple way that, if b ≤ 0 and c < λ_{1}, then problem (P) does not have nontrivial solutions. In particular we will prove the following results.

Proposition 1. Suppose b ≤ 0, Ω is a bounded and smooth domain of ℝ^{N}, and c < λ_{1}. Then (P) admits only the trivial solution.

On the other hand, we will show a result for (P) which is different in nature from the one found in [11] for (LM). In fact, we will prove the existence of nontrivial solutions for positive values of b which are below λ_{1}(λ_{1} − c). More precisely, the following theorem holds.

Theorem 2. Suppose N ≤ 3, Ω is a bounded and smooth domain of ℝ^{N}, c < λ_{1} and 0 < b < λ_{1}(λ_{1} − c). Then (P) has a nontrivial solution.

We remark that such results cover the “physical case” (N ≤ 3).

Acknowledgments. Part of this work was done while the author was visiting Professor Paul H. Rabinowitz at the University of Wisconsin-Madison, under a grant of the Italian C.N.R. The author takes this opportunity to thank Professor Rabinowitz for his warm hospitality and for his kindness.

2. Notations, a remark on (P_{k}) and proof of Proposition 1

Let Ω be a bounded and smooth domain of ℝ^{N}, N ≥ 1, b, c ∈ ℝ, and b ≤ 0. Let us consider the following problem:

(P)  \{ \begin{matrix}Δ^{2}u + cΔu − b(e^{u} − 1) = 0 & in Ω, \\ u = Δu = 0 & on ∂Ω. \end{matrix}

Denote by λ_{i} the i-th eigenvalue of −Δ in H_{0}^{1}(Ω) with eigenvalue e_{i}. It will also be useful to introduce the couple of eigenvalues and eigenfunctions (Λ_{i}, E_{i})_{i} of the problem

\{ \begin{matrix}Δ^{2}E + cΔE = ΛE & in Ω, \\ E = ΔE = 0 & on ∂Ω. \end{matrix}

We remark that, if c < λ_{1}, it turns out that Λ_{i} = λ_{i}^{2} − cλ_{i} and E_{i} = e_{i} ∀ i. Recall that one can choose e_{1} > 0 in Ω.

We will use the following version of the maximum principle for fourthorder equations (see [8]).

Theorem 3 (Maximum Principle). Suppose c < λ_{1} and u satisfies

\{ \begin{matrix}Δ^{2}u + cΔu ≤ 0 & in Ω, \\ u = Δu = 0 & on ∂Ω. \end{matrix}  (2.1)

Then either u ≡ 0, or for every ball B contained in Ω, sup_{B} u < 0.

Of course (2.1) is considered in the sense of distributions; that is, ∀ v in H_{Ω}, v ≥ 0, one has

∫_{Ω} ΔuΔv dx − c ∫_{Ω} Du · Dv dx ≤ 0.

A straightforward application of this theorem implies the following proposition.

Proposition 4. Suppose b = 0, a ≤ 0, and c < λ_{1}. Then any nontrivial solution u of problem (P_{k}) is such that u < 0 in Ω.

On the other hand, one can prove that there is always a nontrivial solution of problem (P_{k}) (see [10], where also multiplicity results are shown). On the other hand, there are also solutions of (P_{k}) which are positive in Ω; for example, if b = λ_{1}^{2} − cλ_{1}, all functions of the form u = te_{1}, t > 0, solve (P_{k}).

On the contrary, we are now going to prove Proposition 1, which moves in an opposite direction. Proof of Proposition 1. Assume for the sake of contradiction that there exists u ≢ 0 which is a solution of (1.2). Then

∫_{Ω} |Δu|^{2} dx − c ∫_{Ω} |Du|^{2} dx = b ∫_{Ω}(e^{u} − 1)u dx ≤ 0,

since (e^{s} − 1)s ≥ 0 ∀ s ∈ ℝ.

But the left-hand side of such an inequality is greater than or equal to

(λ_{1}^{2} − cλ_{1}) ∫_{Ω} u^{2} dx,

which is positive, a contradiction. □

3. Proof of Theorem 2

Problem (P) has a natural variational structure. In fact, its solutions are the critical points of the functional defined on the Hilbert space H_{Ω} = H_{0}^{1}(Ω) ∩ H^{2}(Ω) as

f(u) = \frac{1}{2} ∫_{Ω} |Δu|^{2} dx − \frac{c}{2} ∫_{Ω} |Du|^{2} − b ∫_{Ω}(e^{u} − u − 1) dx.

It is clear, in fact, that f is a smooth functional on H_{Ω}, since N ≤ 3.

Let us remark that the space H_{Ω} can be endowed with the scalar product (u, v) = ∫ ΔuΔv, which induces the norm ∥u∥ = (∫_{Ω} |Δu|^{2})^{1/2}, equivalent to the standard norm of H^{2} (see [10]).

We first prove the usual compactness condition, known as the Palais-Smale condition at level a, (P S)_{a} for short: if (u_{n})_{n} in H_{Ω} is such that f(u_{n}) → a and f′(u_{n}) → 0, then there exists a subsequence (u_{n_{k}})_{k} which converges.

Proposition 5 (Palais-Smale). Assume N ≤ 3, c ≠ λ_{i} ∀ i, and b ∈ ℝ. Then f satisfies (P S)_{a} ∀ a in ℝ.

Proof. Let (u_{n})_{n} be such that f(u_{n}) → a and f′(u_{n}) → 0. In order to prove (P S)_{a} it is enough to prove that (u_{n})_{n} is bounded. Indeed, one has

grad f(u) = u + i^{∗}(cΔu − be^{u} − 1),

where i^{∗} is the adjoint (compact) operator to the immersion of H_{Ω} into L^{2}(Ω).

a note on an exponential semilinear equation 5

First of all observe that

\frac{f′(u_{n})(u_{n})}{∥u_{n}∥} = \frac{1}{∥u_{n}∥} \{∫_{Ω} |Δu_{n}|^{2} dx − c ∫_{Ω} |Du_{n}|^{2} dx − b ∫_{Ω}[e^{u_{n}}u_{n} − u_{n}] dx\}

= \frac{1}{∥u_{n}∥} \{2f(u_{n}) + b ∫_{Ω}[2e^{u_{n}} − e^{u_{n}}u_{n} − u_{n} − 2] dx\}

so that

\frac{∫_{Ω}[2e^{u_{n}} − e^{u_{n}}u_{n}] dx}{∥u_{n}∥} → ∫_{Ω} u dx.  (3.1)

Moreover, \frac{f(u_{n})}{∥u_{n}∥} → 0, which implies

lim_{n→∞} \{∥u_{n}∥ − \frac{c ∫_{Ω} |Du_{n}|^{2} dx}{∥u_{n}∥} − \frac{b ∫_{Ω} e^{u_{n}} dx}{∥u_{n}∥}\} = −b ∫_{Ω} u dx,  (3.2)

and \frac{f′(u_{n})(u_{n})}{∥u_{n}∥} → 0, so that

lim_{n→∞} \{∥u_{n}∥ − \frac{c ∫_{Ω} |Du_{n}|^{2} dx}{∥u_{n}∥} − \frac{b ∫_{Ω} e^{u_{n}} dx}{∥u_{n}∥}\}  (3.3)
= lim_{n→∞} b\frac{∫_{Ω}[e^{u_{n}}u_{n} − e^{u_{n}} − u_{n}] dx}{∥u_{n}∥}.

Combining (3.2) and (3.3) we get

lim_{n→∞} \frac{∫_{Ω}[e^{u_{n}}u_{n} − e^{u_{n}}] dx}{∥u_{n}∥} = 0.  (3.4)

In this way, (3.1) and (3.4) imply

lim_{n→∞} \frac{∫_{Ω} e^{u_{n}} dx}{∥u_{n}∥} = 0  (3.5)

and

lim_{n→∞} \frac{∫_{Ω} e^{u_{n}}u_{n} dx}{∥u_{n}∥} = 0.

Now compute f′(u_{n})(E_{i}), which goes to 0 for every i; in this way, we get

Λ_{i} \frac{∫_{Ω} u_{n}E_{i} dx}{∥u_{n}∥} − \frac{b ∫_{Ω}[e^{u_{n}}E_{i} − E_{i}] dx}{∥u_{n}∥} → 0.

But the second integral goes to 0 as n goes to ∞ by (3.5), since E_{i} is bounded for every i, so that

Λ_{i} ∫_{Ω} uE_{i} = 0 ∀ i;

that is, u ≡ 0, since Λ_{i} ≠ 0 (for c ≠ λ_{i} ∀ i).

On the other hand, computing \frac{f(u_{n})}{∥u_{n}∥^{2}} and passing to the limit, we get

1 − c ∫_{Ω} |Du|^{2} dx = 0;

that is, u ≢ 0, and a contradiction arises.

In this way, (u_{n})_{n} is bounded and the (P S)_{a} follows. □

We now want to prove that, if 0 < b < λ_{1}(λ_{1} − c), f satisfies the geometrical assumptions to exhibit a mountain-pass structure (see [1]). To this purpose it is enough to show that if |u| ≤ ρ, there exists a constant M_{ρ}, going to 1 as ρ → 0, such that

f(u) ≥ \frac{1}{2} ∫_{Ω} \{Δu|^{2} − c|Du|^{2} − bM_{ρ}u^{2}\} dx.

In fact, write u = ∑ α_{i}e_{i}, so that

f(u) ≥ \frac{1}{2} ∑ α_{i}^{2}(λ_{i}^{2} − cλ_{i} − bM_{ρ}) = \frac{1}{2} ∑ λ_{i}^{2}α_{i}^{2} \frac{λ_{i}^{2} − cλ_{i} − bM_{ρ}}{λ_{i}^{2}}

≥ \frac{1}{2}∥u∥^{2} inf_{i} \frac{λ_{i}^{2} − cλ_{i} − bM_{ρ}}{λ_{i}^{2}} = \frac{1}{2}∥u∥^{2}m_{ρ},

where

m_{ρ} = min \{1, \frac{λ_{1}^{2} − cλ_{1} − bM_{ρ}}{λ_{1}^{2}} \} > 0.

In particular, m_{ρ} = \frac{λ_{1}^{2}−cλ_{1}−bM_{ρ}}{λ_{1}^{2}} if c ≥ 0 or c < 0 and b ≥ −cλ_{1}, and m_{ρ} = 1 if c < 0 and b < −cλ_{1}.

In this way, if ρ is small enough, λ_{1}^{2} − cλ_{1} − bM_{ρ} > 0 and u = 0 is a strict local minimum for f.

Now take t > 0 and consider

f(te_{1}) = \frac{t^{2}}{2} (λ_{1}^{2} − cλ_{1}) − a ∫_{Ω} [e^{te_{1}} − te_{1} − 1] dx.

If t → +∞, it is clear that f(te_{1}) → −∞.

We have thus proved that f satisfies all the assumptions of the mountainpass theorem. Therefore f has a nontrivial critical point; that is, (P) has a nontrivial solution.

So, we only have to prove that, if |u| ≤ ρ, there exists M_{ρ} such that e^{u} − u − 1 ≤ M_{ρ}u^{2}/2. Let us define

M_{ρ} = sup_{|u|≤ρ} 2\frac{e^{u} − u − 1}{u^{2}}.

We want to show that M_{ρ} = 2(e^{ρ} − ρ − 1)/ρ^{2}.

Assume first 0 ≤ u ≤ ρ, and consider the function g(u) = \frac{e^{u}−u−1}{u^{2}} (where g(0) := lim_{u→0} g(u) = 1/2). Then g′(u) = \frac{(u−2)e^{u}+u+2}{u^{3}}. Setting m(u) = (u − 2)e^{u} + u + 2, we get m′′(u) = ue^{u}, so that m is convex on [0, ρ]. So m′ is increasing. But m′(0) = 0, so m is increasing for u positive; that is, m(u) > 0, since m(0) = 0. This means g′(u) > 0 for u > 0, and so g(u) ≤ \frac{e^{ρ}−ρ−1}{ρ^{2}}. Note that M_{ρ} ↓ 1 as ρ → 0.

Now assume u < 0. Set v = −u, and consider the function h(v) = \frac{e^{−v}+v−1}{v^{2}}. Reasoning as in the previous case, it is easy to show that h(v) ≤ h(0^{−}) = 1/2 ∀ v > 0. We have thus proved that

sup_{|u|≤ρ} \frac{e^{u} − u − 1}{u^{2}} = \frac{e^{ρ} − ρ − 1}{ρ^{2}}.

Theorem 2 is thus completely proved.

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