On the multiplicity of the solutions of the equation −Δu = λ.f(u)

Jean Saint Raymond

Equipe d’Analyse – Boîte 186 Université Paris VI 4, place Jussieu F - 75252 Paris cedex 05, France E-mail : jsr@ccr.jussieu.fr

ABSTRACT : Let Ω be a bounded open subset of ℝ^{n}. We discuss conditions on the continuous function f : ℝ → ℝ ensuring that the problem : −Δu = λ.f(u) admits unboundedly many weak solutions in H_{0}^{1}(Ω) for any positive real λ.

1991 AMS Classification : 35J20

Multiplicity of the solutions

1. Introduction and notations

We are mainly concerned in this paper by the following nonlinear eigenvalue problem : we look at weak solutions in H_{0}^{1}(Ω) for the problem −Δu = λ.f(u), where Ω is a bounded open subset of the euclidean space ℝ^{n}, λ > 0 and f : ℝ → ℝ continuous. It is a well-known fact that this problem is closely related to the following variational problem : Find critical points in H_{0}^{1}(Ω) for the functional

Φ_{λ′} : u ↦ ∫_{Ω} F ∘u(x) dx − λ′ ∫_{Ω} |∇u|^{2} dx

where F is the primitive of f vanishing at 0 and λ′ = \frac{1}{2λ}.

More generally, if f is a measurable function from Ω × ℝ to ℝ which is continuous with respect to the second variable, we can look at weak solutions for the equation −Δu = λf(., u(.)) as critical points for the functional

Φ_{λ′} : u ↦ ∫_{Ω} (F (x, u(x)) − λ′ |∇u(x)|^{2}) dx

where λ′ = \frac{1}{2λ} and F (x, t) = ∫_{0}^{t} f(x, s) ds.

In order to give sense to the previous formulas, we assume that for every x ∈ Ω and every t ∈ ℝ

|f(x, t)| ≤ a |t|^{p} + b(x)

where 1 < p < \frac{n + 2}{n − 2}, a > 0 and b is a positive element of L^{q′}(Ω), for q′ = \frac{p + 1}{p}. So we

have

|F (x, t)| ≤ a\frac{|t|^{q}}{q} + |t| b(x)

with 2 < q = p + 1 < \frac{2n}{n − 2}. Then H_{0}^{1}(Ω) embeds compactly in L^{q}(Ω) and the functional

J : u ↦ ∫_{Ω} F (x, u(x)) dx is sequentially weakly continuous. Moreover, J is Fréchetdifferentiable at u and for h ∈ H_{0}^{1}(Ω) we get :

J′(u).h = ∫_{Ω} f(x, ∘u(x)).h(x) dx

so that ∇J(u) = f(., u(.)) ∈ L^{q′}(Ω). It follows that ∇J is continuous from H_{0}^{1}(Ω) into L^{q′}(Ω) and that Φ_{λ} is weakly upper semi-continuous on each closed ball B(r), hence attains there its maximum since closed balls in H_{0}^{1}(Ω) are weakly compact.

Clearly, if the maximum is attained at an interior point u of the ball B(r), it is a local maximum, hence a critical point of Φ_{λ}, and we get

∇J(u) − λ∇ ∥.∥^{2} (u) = f(., u(.)) + 2λ.Δu = 0

it is −Δu = \frac{1}{2λ}f(., u(.)).

More precisely we look for conditions under which we can guarantee the set of weak solutions of this problem is unbounded. In [4], B. Ricceri introduced tools for the study of local minima of a functional of the type Φ + λΨ, where Φ and Ψ are weakly l.s.c. functionals on a reflexive Banach space. In the context of our problem Φ(u) = − ∫_{Ω} F (x, u(x)) dx and Ψ(u) = ∫_{Ω} |∇u(x)|^{2} dx, and the boundedness of the set of the solutions is controlled by the behaviour of the function

Q(r) = inf_{∥u∥<r} \frac{sup_{∥v∥<r} J(v) − J(u)}{r^{2} − ∥u∥^{2}}

On the other hand, P. Omari and F. Zanolin proved in [3] that the set of solutions of a similar problem is unbounded using convenient hypotheses on the potential F. And we refer to their paper for more details and for references on earlier works. Other important results about multiple solutions can be found in the paper [1] by Bartsch and Willem.

This paper is divided in three sections. In the first one we answer a conjecture of B. Ricceri. In the case where F (t) = |t|^{q}, the homogeneity of the function J allows an easy computation for the function Q above. And it follows that the conditions for the existence of infinitely many local maximum type solutions of the problem −Δu = λ′f(u) cannot be fulfilled. But if we add a small perturbation to the previous function F, the computation of Q becomes very difficult. Nevertheless B. Ricceri conjectured that the result concerning the multiplicity of the solutions was unchanged, and we prove that this is the case.

In the next two sections we restrict ourselves to the autonomous case : we consider only the case where the function (x, t) ↦ F (x, t) depends only on t.

In the first of these we study the case where the function F is not monotone on any neighborhood of +∞, and prove that under mild extra conditions the set of solutions is unbounded.

Finally, in the last section, we show that the oscillatory behaviour described in the previous section (with even a stronger property) is generic in some sense : precisely the set of those functions having this property is a comeager set in a Banach space of functions.

This work was performed in July 1999 during a stay of the author in Italy at the University of Catania, supported by the C.N.R. I am very grateful to Prof. B. Ricceri for his stimulating questions.

2. A conjecture of B. Ricceri

The aim of this section is to prove that for a function like F : t ↦ |t|^{q} for q > 2, the functional Φ_{λ} cannot have unboundedly local maxima for any λ > 0.

Lemma 2.1.

i) If Φ_{λ} attains its maximum over the ball B(r) at an interior point then Q(r) ≤ λ. ii) If Φ_{λ} attains its maximum over the ball B(r) on the sphere then Q(r) ≥ λ.

Proof : Notice that since J is weakly continuous on each ball and since the sphere is weakly dense in the ball we have :

μ(r) := sup{J(u) : ∥u∥ = r} = sup{J(u) : ∥u∥ < r}

We also have

Q(r) = inf_{∥u∥<r} \frac{sup_{∥v∥<r} J(v) − J(u)}{r^{2} − ∥u∥^{2}} = inf_{s<r} \frac{μ(r) − μ(s)}{r^{2} − s^{2}}

If there is a u_{0} such that s = ∥u_{0}∥ < r and Φ_{λ}(u) ≤ Φ_{λ}(u_{0}) for ∥u∥ < r, we have, for ∥u∥ = r,

Φ_{λ}(u) = J(u) − λr^{2} ≤ Φ_{λ}(u_{0}) ≤ μ(s) − λs^{2}

hence

μ(r) − λr^{2} ≤ μ(s) − λs^{2}

and Q(r) ≤ λ.

If Φ_{λ} attains its maximum at some point u_{0} of the sphere, we have, for s < r,

μ(s) − λs^{2} = sup_{∥v∥=s} Φ_{λ}(v) ≤ sup_{∥v∥≤r} Φ_{λ}(v) = Φ_{λ}(u_{0}) ≤ μ(r) − λr^{2}

hence Q(r) ≥ λ. ■

Corollary 2.2. If

lim inf_{r→∞} Q(r) < λ < lim sup_{r→∞} Q(r)

there is a sequence (u_{n}) such that ∥u_{n}∥ → ∞ and that each u_{n} is a local maximum of Φ_{λ},

hence satisfies −Δu_{n} + \frac{1}{2λ}f ∘u_{n} = 0.

Proof : One can choose an increasing sequence (r_{n}) such that r_{n+1} > r_{n} +1 for each n and that Q(r_{2n}) < λ < Q(r_{2n+1}). Since Q(r_{2n−1}) > λ the maximum of Φ_{λ} over the ball B(r_{2n}) cannot be attained in the ball B(r_{2n−1)}, but since Q(r_{2n}) < λ, it cannot be attained on the sphere S(r_{2n}). Then it is attained at some point u_{n} such that r_{2n−1} ≤ ∥u_{n}∥ < r_{2n}, and it is a local maximum. ■

Definition. We will say that f is of type LM_{∞} if

lim inf_{r→∞} Q(r) = 0  and  lim sup_{r→∞} Q(r) = +∞

If f is of type LM_{∞}, there are unboundedly many critical points for Φ_{λ} for each positive real λ. It is quite unclear whether “simple functions” can be of type LM_{∞}. First of all, we

show that it is not the case for the function f : t ↦ t. |t|^{p−1} (with 1 < p < \frac{n + 2}{n − 2}), even

with a small perturbation. The following theorem answers a conjecture of B. Ricceri.

Theorem 2.3. Let Ω be a bounded open subset of ℝ^{n}, 1 < p < \frac{n + 2}{n − 2}, F : Ω × ℝ → ℝ be

a measurable function, of class C^{1} with respect to the second variable, satisfying :

i) ξ \frac{∂F}{∂ξ} (x, ξ) ≥ ε |ξ|^{p+1} − b(x) |ξ|

ii) |\frac{∂F}{∂ξ} (x, ξ)| ≤ a |ξ|^{p} + b(x)

for some positive a and ε, and some positive function b ∈ L^{q′}(Ω) with q′ = \frac{p + 1}{p}. Then we have : lim_{r→∞} Q(r) = +∞.

Notice first that, up to replacing F by F_{0} : (x, t) ↦ F (x, t) − F (x, 0) which adds a constant to J, we can assume F (x, 0) = 0. Then, with f(x, ξ) = \frac{∂F}{∂ξ} (x, ξ), we have

F (x, t) = ∫_{0}^{1} tξ.f(x, tξ) \frac{dξ}{ξ} ≥ ε\frac{|t|^{p+1}}{p + 1} − |t| b(x)

and

|F (x, t)| ≤ ∫_{0}^{1} |tξ.f(x, tξ)| \frac{dξ}{ξ} ≤ a\frac{|t|^{p+1}}{p + 1} + |t| b(x)

In order to prove the theorem, we begin by proving several lemmas.

Let us denote by u_{0} a fixed non-negative function of class C^{1} with compact support in Ω and norm 1 in H_{0}^{1}(Ω). We also denote ∥v∥_{p} = (∫_{Ω} |v(x)|^{p} dx)^{1/p} for p ≥ 1 and v ∈ L^{p}(Ω). Lemma 2.4. For all λ > 0, there exist δ > 0 and R_{0} > 0 such that for every u in the unit sphere S(1) of H_{0}^{1}(Ω) the following holds :

t ≥ R_{0} and ∥u∥_{q} < δ ⟹ J(tu) − λt^{2} < \frac{1}{2}(J(tu_{0}) − λt^{2})

Proof : We have F (x, ξ) ≥ ε\frac{|ξ|^{q}}{q} − b(x) |ξ|, hence, for t ≥ 0

J(tu_{0}) − λt^{2} ≥ ∫_{Ω} (εt^{q} \frac{|u_{0}(x)|^{q}}{q} − tb(x)u_{0}(x)) dx − λt^{2} ≥ \frac{εt^{q}}{q} ∥u_{0}∥_{q}^{q} − t ∥b∥_{q′} ∥u_{0}∥_{q} − λt^{2}

and similarly, since |F (x, ξ)| ≤ a\frac{|ξ|^{q}}{q} + b(x) |ξ|,

|J(tu) − λt^{2}| ≤ ∫_{Ω} (at^{q} \frac{|u(x)|^{q}}{q} + tb(x)u(x)) dx + λt^{2} ≤ \frac{at^{q}}{q} ∥u∥_{q}^{q} + t ∥b∥_{q′} ∥u∥_{q} + λt^{2}

So, if ∥u∥_{q} < δ,

(J(tu_{0}) − λt^{2}) − 2 |J(tu) − λt^{2}| ≥ \frac{t^{q}}{q} (ε ∥u_{0}∥_{q}^{q} − 2aδ^{q}) − (t ∥b∥_{q′} (∥u_{0}∥_{q} + 2δ) + 3λt^{2})

and if we choose δ in such a way that 4aδ^{q} < ε ∥u_{0}∥_{q}^{q}, we get for a convenient a′ :

(J(tu_{0}) − λt^{2}) − 2 |J(tu) − λt^{2}| ≥ t^{q} \frac{ε}{2q} ∥u_{0}∥_{q}^{q} − a′(t + t^{2})

And since q > 2, there exists a R_{0} > 0 such that t^{q} \frac{ε}{2q} ∥u_{0}∥_{q}^{q} > a′(t + t^{2}) for t ≥ R_{0}. ■

Multiplicity of the solutions

Lemma 2.5. For λ > 0 and u ∈ S(1) the function g_{λ,u} : t ↦ J(tu) − λt^{2} is derivable, and its derivative satisfies :

g′_{λ,u}(t) ≥ εt^{q−1} ∥u∥_{q}^{q} − 2λt − ∥b∥_{q′} ∥u∥_{q}

and

|g′_{λ,u}(t)| ≤ at^{q−1} ∥u∥_{q}^{q} + 2λt + ∥b∥_{q′} ∥u∥_{q}

Proof : Since J(tu) = ∫_{Ω} F (x, tu(x)) dx we get

\frac{d}{dt}J(tu) = ∫_{Ω} \frac{∂F}{∂ξ} (x, tu(x)).u(x) dx = ∫_{Ω} f(x, tu(x)).u(x) dx

hence εt^{q−1} |u(x)|^{q} − b(x) |u(x)| ≤ f(x, u(x)).u(x) ≤ |f(x, u(x)).u(x)| ≤ at^{q−1} |u(x)|^{q} + b(x) |u(x)| and

εt^{q−1} ∥u∥_{q}^{q} − ∥b∥_{q′} ∥u∥_{q} ≤ \frac{d}{dt}J(tu) ≤ |\frac{d}{dt}J(tu)| ≤ at^{q−1} ∥u∥_{q}^{q} + ∥b∥_{q′} ∥u∥_{q}

and the result follows from these inequalities. ■ Lemma 2.6. For each λ > 0 there is some R ≥ R_{0} such that for u ∈ S(1) and t ≥ R :

J(tu) − λt^{2} ≥ \frac{1}{2}(J(tu_{0}) − λt^{2}) ⟹ g′_{λ,u}(t) > 0

Proof : It follows from Lemma 2.4 that conditions ∥u∥ = 1, t ≥ R_{0} and J(tu) − λt^{2} ≥ \frac{1}{2}(J(tu_{0}) − λt^{2}) imply ∥u∥_{q} ≥ δ. then by Lemma 2.5

g′_{λ,u}(t) ≥ εt^{q−1} ∥u∥_{q}^{q} − 2λt − ∥b∥_{q′} ∥u∥_{q}

≥ t^{q−1} ∥u∥_{q}^{q} (ε − 2\frac{λ}{∥u∥_{q}^{q}} t^{2−q} − \frac{∥b_{q′}∥}{∥u_{q}^{q−1}∥}t^{1−q})

≥ t^{q−1} ∥u∥_{q}^{q} (ε − 2\frac{λ}{δ^{q}} t^{2−q} − \frac{∥b_{q′}∥}{δ^{q−1}}t^{1−q})

and if R ≥ R_{0} is chosen large enough so that \frac{2λ}{δ^{q}R^{q−2}} + \frac{∥b_{q′}∥}{δ^{q−1}R^{q−1}} < \frac{ε}{2}, we get

g′_{λ,u}(t) ≥ \frac{δ^{q}ε}{2}t^{q−1} > 0

for t ≥ R. ■

Lemma 2.7. For λ > 0 and ρ > 0, there exists a constant β > 0 such that, for u ∈ S(1), t and t′ in [0, ρ], one has :

|g_{λ,u}(t′) − g_{λ,u}(t)| ≤ β |t′ − t|

Proof : Since q < \frac{2n}{n − 2}, H_{0}^{1}(Ω) embeds continuously into L^{q}(Ω). Thus there is a constant

γ such that, for u ∈ H_{0}^{1}(Ω), ∥u∥_{q} ≤ γ ∥u∥. Moreover by Lemma 2.5, one has

|g′_{λ,u}(t)| ≤ at^{q−1} ∥u∥_{q}^{q} + ∥b∥_{q′} ∥u∥_{q} + 2λt

then for 0 < t < ρ and ∥u∥ = 1 :

|g′_{λ,u}(t)| ≤ β := aρ^{q−1}γ^{q} + γ ∥b∥_{q′} + 2λρ

and the result follows from the mean value theorem. ■

Lemma 2.8. For λ > 0 and ρ > 0 the function : r ↦ μ(r) − λr^{2} is Lipschitz on [0, ρ].

Proof : From Lemma 2.7 we know that there is a β > 0 such that for t and s in [0, ρ] and u ∈ S(1) :

g_{λ,u}(s) ≤ g_{λ,u}(t) + β |s − t| ≤ (μ(t) − λt^{2}) + β |s − t|

hence μ(s)−λs^{2} ≤ (μ(t)−λt^{2})+β |s − t|. And this shows that the function : r ↦ μ(r)−λr^{2} is β-Lipschitz on [0, ρ]. ■

For fixed λ, we consider the sets

T = {r ∈ ℝ^{+} : μ(r) − λr^{2} = sup_{s≤r}(μ(s) − λs^{2}) }

T′ = {r ∈ ℝ^{+} : ∀s < r μ(s) − λs^{2} < μ(r) − λr^{2} }

Lemma 2.9. The set T is closed in ℝ^{+} and T′ is unbounded.

Proof : The function : r ↦ μ(r) − λr^{2} is locally Lipschitz, hence continuous. Thus the set

T = ⋂_{s∈ℝ^{+}}({r : μ(r) − λr^{2} ≥ μ(s) − λs^{2}} ∪ [0, s])

is closed.

Let θ > 0. We show that T′ meets ]θ, +∞[. Since μ is continuous, we have :

A_{θ} := sup_{s≤θ}(μ(s) − λs^{2}) < +∞

From the proof of Lemma 2.4 , we know that

(μ(r) − λr^{2}) − r^{2} ≥ g_{λ,u_{0}}(r) − r^{2} = J(ru_{0}) − (λ + 1)r^{2} ≥ r^{q} \frac{ε}{q} ∥u_{0}∥_{q}^{q} − r ∥u_{0}∥_{q} − (λ + 1)r^{2}

thus that there exists a R′ such that g_{λ,u_{0}}(r) − r^{2} > 0 for r ≥ R′, that is μ(r) − λr^{2} > r^{2} for r ≥ R′.

Then with r > max(R′, \sqrt{A_{θ}}) we get M := μ(r) − λr^{2} > A_{θ}. And the closed set F = {s ∈ ℝ^{+} : μ(s) − λs^{2} ≥ M} which is non-empty since r ∈ F, has a least element r_{0}. Since A_{θ} < M ≤ μ(r_{0}) − λr_{0}^{2} we get r_{0} > θ. And for every s < r_{0} we have s_{0} ∉ F, hence μ(s) − λs^{2} < M ≤ μ(r_{0}) − λr_{0}^{2} and r_{0} ∈ T′. Thus r_{0} ∈ T′∩]θ, +∞[.

Since θ is arbitrary, this shows that T′ is unbounded. ■

Multiplicity of the solutions

Lemma 2.10. The set T′ contains a half-line [R_{1}, +∞[. Proof : Let λ be fixed. By Lemmas 2.4 and 2.6, there is a R > 0 such that for r ≥ R and ∥u∥ = 1 we have g_{λ,u_{0}}(r) > 0 and

g_{λ,u}(r) > \frac{1}{2}g_{λ,u}(r_{0}), ⟹ g′_{λ,u}(r) > 0

Moreover, by Lemma 2.9 there is a R_{1} > R in T′. We show that [R_{1}, +∞[⊂ T′. Assume by contradiction that r ∉ T′ for some r > R_{1}. Then the function u ↦ J(u) − λ ∥u∥^{2} attains its maximum α on the closed ball B(r) at some point u^{∗} with s = ∥u^{∗}∥ < r.

Since R_{1} < r, we have

μ(R_{1}) − λR_{1}^{2} = sup_{∥u∥=R_{1}} J(u) − λ ∥u∥^{2} ≤ α = J(u^{∗}) − λ ∥u^{∗}∥^{2} ≤ μ(s) − λs^{2}

hence s ≥ R_{1} ≥ R. And for u^{∗∗} = \frac{1}{s}u^{∗}, we get

g_{λ,u^{∗∗}}(s) = α ≥ g_{λ,u_{0}}(s) > 0

which implies g_{λ,u^{∗∗}}(s) > \frac{1}{2}g_{λ,u_{0}}(s), hence g′_{λ,u^{∗∗}}(s) > 0. We conclude that there is s′ with s < s′ < r such that

g_{λ,u^{∗∗}}(s′) − g_{λ,u^{∗∗}}(s) > \frac{1}{2}(s′ − s)g_{λ,u^{∗∗}}(s) > 0

and finally

α = g_{λ,u^{∗∗}}(s) < g_{λ,u^{∗∗}}(s′) ≤ sup_{∥u∥≤r} J(u) − λ ∥u∥^{2} = α

and this contradiction completes the proof. ■ Proof of Theorem 2.3

It follows from the previous Lemma that for every λ > 0 there is a R_{1} > 0 such that r ≥ R_{1} and s < r ⟹ μ(s) − λs^{2} < μ(r) − λr^{2}

thus by Lemma 2.1 Q(r) ≥ λ for r ≥ R_{1}. And since λ is arbitrary we get

lim_{r→∞} Q(r) = +∞

■

3. Oscillating functions A consequence of Theorem 2.3 is that in order to find functions of type LM_{∞} we have

to look at more oscillating functions, like in the paper [3] by Omari and Zanolin.

Theorem 3.1. Assume 1 < p < \frac{n + 2}{n − 2} and let F : ℝ → ℝ be of class C^{1} and satisfy :

i) |F′(t)| ≤ a (1 + |t|^{p}) ii) There exists a constant M > 0 such that for every ρ > 0 there exists a t > 0 satsisfying

F (t) ≥ ρ(1 + t^{2}) and F (s) ≥ −M.F (t) for 0 ≤ s ≤ t. iii) sup{t : F′(t) < 0} = +∞ iv) inf{t : F′(t) > 0} < 0 Then there are unboundedly many local maxima of Φ_{λ} for every positive λ.

Put f = F′. By hypothesis there is an increasing sequence (θ_{n})_{n≥1} converging to +∞ such that f(θ_{n}) < 0 . Thus we can find a sequence θ′_{n} such that θ_{n} < θ′_{n} < θ_{n+1} and

∀t ∈]θ_{n}, θ′_{n}[ F (t) < F (θ_{n})

Similarly there are θ_{0} and θ′_{0} such that θ′_{0} < θ_{0} < 0 and

∀t ∈]θ′_{0}, θ_{0}[ F (t) < F (θ_{0})

Consider for n ≥ 1 the set

E_{n} = {u ∈ H_{0}^{1}(Ω) : ∀x ∈ Ω θ′_{0} ≤ u(x) ≤ θ′_{n} }

Since H_{0}^{1}(Ω) embeds in L^{q}(Ω), the mapping u ↦ F ∘u is continuous from H_{0}^{1}(Ω) into L^{1}(Ω). Thus the set E_{n} is closed in H_{0}^{1}(Ω), and clearly convex. Hence E_{n} is weakly closed in H_{0}^{1}(Ω). Let us denote by M_{n} the supremum of F on the compact interval [θ′_{0}, θ′_{n}].

Lemma 3.2. The functional Φ_{λ} is bounded from above on E_{n} and attains its maximum.

Proof : For u ∈ E_{n} and x ∈ Ω we have F ∘u(x) ≤ M_{n}, hence

Φ_{λ}(u) ≤ J(u) = ∫_{Ω} F ∘u(x) dx ≤ M_{n} |Ω|

Then, with α_{n} = sup_{u∈E_{n}} Φ_{λ}(u), the set

A_{n,k} = {u ∈ E_{n} : Φ_{λ}(u) ≥ α_{n} − 2^{−k} }

is bounded : indeed, for u ∈ A_{n,k} we have

∥u∥^{2} = \frac{1}{λ}(J(u) − Φ_{λ}(u)) ≤ \frac{1}{λ}(M_{n} |Ω| − α_{n} + 2^{−k})

and since Φ_{λ} is sequentially weakly u.s.c. the sets A_{n,k} are bounded and weakly closed, hence weakly compact. They are non-empty by definition of α_{n}. Thus their intersection K_{n} is non-empty, and Φ_{λ} attains its maximum α_{n} at each point u^{∗} of K_{n}. ■

It is worth noticing that the set K_{n} is norm-compact in H_{0}^{1}(Ω). Indeed, if (u_{j}) is a sequence in K_{n} there is a subsequence (u_{j})_{j∈I} which converges weakly to some u ∈ K_{n}. Then

∥u_{j}∥^{2} = \frac{1}{λ}(J(u_{j}) − Φ_{λ}(u_{j})) = \frac{1}{λ}(J(u_{j}) − α_{n})

and since J is weakly continuous this converges (for j ∈ I) to \frac{1}{λ}(J(u) − α_{n}) = ∥u∥^{2}. Then

∥u − u_{j}∥^{2} = ∥u_{j}∥^{2} + ∥u∥^{2} − 2⟨u, u_{j}⟩ → ∥u∥^{2} + ∥u∥^{2} − 2⟨u, u⟩ = 0

So the sequence (u_{j})_{j∈I} converges strongly to u.

Lemma 3.3. The sequence α_{n} grows to the infinity.

Proof : Let T be fixed and choose ρ_{0} such that ρ_{0} |Ω| ≥ 2T. For ε > 0, denote by Ω_{ε} the open subset {x ∈ ℝ^{n} : d(x, ℝ^{n} \ Ω) < ε } of Ω. We have ⋂_{ε>0} Ω_{ε} = ∅, thus we can find some ε > 0 such that

|Ω_{ε}| < \frac{1}{2(M + 2)} |Ω|

Then, there is some ρ ≥ ρ_{0} such that ρε^{2} > 1, some β such that F (β) > ρ(1 + β^{2}) and F (s) ≥ −M.F (β) for 0 ≤ s ≤ β, and some integer n such that β ≤ θ_{n}. We put

u_{0}(x) = β inf(1, ε^{−1}.d(x, ℝ^{n} \ Ω))

It is easy to check that u_{0} is \frac{β}{ε} -Lipschitz hence that |∇u_{0}| ≤ \frac{β}{ε} a.e. Then u_{0} ∈ E_{n} and u_{0}(x) < β for x ∈ Ω_{ε}. Hence,

J(u_{0}) = ∫_{Ω_{ε}} F ∘u_{0}(x) dx + ∫_{Ω\Ω_{ε}} F ∘u_{0}(x) dx ≥ −M.F (β) |Ω_{ε}| + F (β)(|Ω| − |Ω_{ε}|)
≥ F (β) (|Ω| − (M + 1) |Ω_{ε}|) ≥ ρ(1 + β^{2}) |Ω| − (M + 1)ρ(1 + β^{2}) |Ω_{ε}|

Moreover we have |∇u_{0}(x)| = 0 a.e. on Ω \ Ω_{ε} and |∇u_{0}(x)| ≤ \frac{β}{ε} a. e. on Ω_{ε}, hence

∥u_{0}∥^{2} = ∫_{Ω} |∇u_{0}(x)|^{2} dx ≤ (\frac{β}{ε} )^{2} |Ω_{ε}|

We conclude that

Φ_{ρ}(u_{0}) ≥ ρ(1 + β^{2}) (|Ω| − (M + 1 + \frac{1}{ρε^{2}} ) |Ω_{ε}|)

≥ ρ(1 + β^{2})(|Ω| − (M + 2) |Ω_{ε}|) ≥ \frac{ρ(1 + β^{2})}{2} |Ω|

≥ \frac{ρ_{0}}{2} |Ω| ≥ T

Then we have

α_{n} = sup_{u∈E_{n}} Φ_{λ}(u) ≥ Φ_{λ}(u_{0}) ≥ T

and this shows that sup_{n} α_{n} = +∞. ■

The following well-known superposition lemma can be found in [2] :

Lemma 3.4. (Marcus and Mizel) Let Ω be an open subset of ℝ^{n}, 1 ≤ p < ∞ and φ : ℝ → ℝ a Lipschitz mapping satisfying φ(0) = 0. Then φ∘u belongs to W^{1,p} for any u ∈ W^{1,p}(Ω). Furthermore the mapping T : u ↦ φ∘u is continuous from W^{1,p} into itself.

In particular, if φ is piecewise affine and vanishes at 0 the mapping u ↦ φ∘u is continuous from H_{0}^{1}(Ω) to H_{0}^{1}(Ω).

Lemma 3.5. If u^{∗} belongs to K_{n} then θ_{0} ≤ u^{∗}(x) ≤ θ_{n} a.e. on Ω.

Let h : ℝ → ℝ be the function defined by :

h(t) = \{ \begin{matrix}θ_{0} & if t ≤ θ_{0} \\ t & if θ_{0} ≤ t ≤ θ_{n} \\ θ_{n} & if t ≥ θ_{n} \end{matrix}

Then by Lemma 3.4 v^{∗} = h∘u^{∗} belongs to E_{n} too and if X denotes the measurable set {x ∈ Ω : u^{∗}(x) ∉ [θ_{0}, θ_{n}] } we have for x ∈ X :

- either θ′_{0} ≤ u^{∗}(x) < θ_{0} and F ∘v^{∗}(x) = F (θ_{0}) > F ∘u^{∗}(x)

- or θ_{n} ≤ u^{∗}(x) < θ′_{n} and F ∘v^{∗}(x) = F (θ_{n}) > F ∘u^{∗}(x)

Then F ∘v^{∗} > F ∘u^{∗} a.e. on X and F ∘v^{∗} = F ∘u^{∗} a.e. on Ω \ X. Moreover we have ∇v^{∗} = 0 a.e. on X and ∇v^{∗} = ∇u^{∗} a.e. on Ω \ X. It follows that

Φ_{λ}(v^{∗}) − Φ_{λ}(u^{∗}) = ∫_{X}(F ∘v^{∗}(x) − F ∘u^{∗}(x)) dx + λ ∫_{X} |∇u^{∗}(x)|^{2} dx

and since u^{∗} ∈ K_{n} and v^{∗} ∈ E_{n}, we have Φ_{λ}(v^{∗}) − Φ_{λ}(u^{∗}) ≤ 0 . This implies :

∫_{X}(F ∘v^{∗}(x) − F ∘u^{∗}(x)) dx = ∫_{X} |∇u^{∗}(x)|^{2} dx = 0

Since F ∘v^{∗}(x) > F ∘u^{∗}(x) a.e. on X, this implies in turn that X is a null set, hence

∥v^{∗} − u^{∗}∥^{2} = ∫_{X} |∇u^{∗}(x)|^{2} dx = 0

and u^{∗} = v^{∗}, that means u^{∗}(x) = v^{∗}(x) ∈ [θ_{0}, θ_{n}] for almost all x in Ω. ■

Lemma 3.6. Every element of K_{n} is a local maximum of Φ_{λ}.

Proof : Let u^{∗} ∈ K_{n} and h be as above. Then the mapping T : u ↦ h∘u is continuous from H_{0}^{1}(Ω) to H_{0}^{1}(Ω). Let

G(ξ) = F (ξ) − F ∘h(ξ)

For u ∈ H_{0}^{1}(Ω), we have T u ∈ E_{n}, hence Φ_{λ}(T u) ≤ Φ_{λ}(u^{∗}) and :

Φ_{λ}(u) − Φ_{λ}(T u) = ∫_{Ω} G∘u(x) dx − λ ∫_{Ω} (|∇u(x)|^{2} − |∇(T u)(x)|^{2}) dx

As in the previous lemma, if X denotes the set {x ∈ Ω : u(x) ∉ [θ_{0}, θ_{n}] }, we have ∇u(x) = ∇(T u)(x) a.e. on Ω \ X and ∇(T u)(x) = 0 a.e. on X, hence

∫_{Ω} (|∇u(x)|^{2} − |∇(T u)(x)|^{2}) dx = ∫_{X} |∇u(x)|^{2} dx = ∫_{Ω} |∇(u − T u)(x)|^{2} dx = ∥u − T u∥^{2}

Since by condition i) we have

|F′(t)| ≤ a(1 + |t|^{p})

for all t ∈ ℝ, we have

|F (ξ)| = |F (0) + ∫_{0}^{ξ} F′(t) dt| ≤ |F (0)| + ∫_{0}^{|ξ|} a(1 + t^{p}) dt ≤ a_{1}(1 + |ξ|^{p+1})

for some constant a_{1}.

Then the function G satisfies :

G(ξ) \{ \begin{matrix}= 0 & if θ_{0} ≤ ξ ≤ θ_{n} \\ < 0 & if θ′_{0} < ξ < θ_{0} or θ_{n} < ξ < θ′_{n} \\ ≤ 2a_{1}(1 + |ξ|^{p+1}) & for all ξ ∈ ℝ \end{matrix}

Thus

C = sup \{\frac{|G(ξ)|}{|ξ − h(ξ)|^{p+1}} : ξ ∉]θ′_{0}, θ′_{n}[ \} < +∞

and it follows that for every ξ ∈ ℝ the following holds :

G(ξ) ≤ C |ξ − h(ξ)|^{p+1}

Since q = p + 1 < \frac{2n}{n − 2}, H_{0}^{1}(Ω) embeds continuously in L^{q}(Ω) and there is some γ such

that ∥u∥_{q} ≤ γ ∥u∥ for every u ∈ H_{0}^{1}(Ω). Hence

∫_{Ω} G∘u(x) dx ≤ C ∫_{Ω} |(u − T u)(x)|^{q} dx = C ∥u − T u∥_{q}^{q} ≤ Cγ^{q} ∥u − T u∥^{q}

and

Φ_{λ}(u) − Φ_{λ}(T u) ≤ Cγ^{q} ∥u − T u∥^{q} − λ ∥u − T u∥^{2}

Φ_{λ}(u) ≤ Φ_{λ}(u^{∗}) − λ ∥u − T u∥^{2} (1 − \frac{Cγ^{q}}{λ} ∥u − T u∥^{p−1})

Since T is continuous and T u^{∗} = u^{∗}, there is some α > 0 such that

∥u − u^{∗}∥ < α ⟹ ∥u − T u∥^{p−1} < \frac{λ}{2Cγ^{q}}

Then for u ∈ B(u^{∗}, α) we get

Φ_{λ}(u) ≤ Φ_{λ}(u^{∗}) − \frac{λ}{2} ∥u − T u∥^{2} ≤ Φ_{λ}(u^{∗})

and this shows that Φ_{λ} has a local maximum at u^{∗}. ■

Proof of Theorem 3.1

Let R > 0. We have to prove the existence of some local maximum of Φ_{λ} whose norm is larger than R. As in the previous lemma we have |F (ξ| ≤ a_{1}(1 + |ξ|^{p+1}). If u ∈ H_{0}^{1}(Ω) has norm less than R, we have, for q = p + 1 :

J(u) = ∫_{Ω} F ∘u(x) dx ≤ ∫_{Ω} a_{1}(1 + |u(x)|^{q}) dx ≤ a_{1}(|Ω| + ∥u∥_{q}^{q}) ≤ a(|Ω| + γ^{q}R^{q})

hence

Φ_{λ}(u) = J(u) − λ ∥u∥^{2} ≤ J(u) ≤ C_{R} := a(|Ω| + γ^{q}R^{q})

Since the sequence (α_{n}) goes to the infinity, there is some n such that α_{n} > C_{R}. Then no element of K_{n} can belong to the ball B(R). Thus each point of K_{n} is a local maximum of Φ_{λ} and has a norm larger than R. ■

Corollary 3.7. Assume p < \frac{n + 2}{n − 2} and 2 < q ≤ p + 1. Let F : ℝ → ℝ be of class C^{1} and satisfy :

i) |F′(t)| ≤ a (1 + |t|^{p}) ii’) |F (t)| ≤ a (1 + t^{q}) for t ≥ 0 and lim sup_{t→+∞} t^{−q}F (t) > 0 iii) sup{t : F′(t) < 0} = +∞ iv) inf{t : F′(t) > 0} < 0 Then there are unboundedly many local maxima of Φ_{λ} for every positive λ.

Proof : We have only to prove that condition ii) of Theorem 3.1 is fulfilled. Choose η > 0

such that η < lim sup_{t→+∞} t^{−q}F (t) then put M = \frac{2a}{η}. If ρ is chosen, there is some t_{0} > 1

such that ηt_{0}^{q} > ρ(1 + t_{0}^{2}), hence a β > t_{0} such that

F (β) > ηβ^{q} = (\frac{β}{t_{0}} )^{q} ηt_{0}^{q} > (\frac{β}{t_{0}} )^{2} ρ(1 + t_{0}^{2}) > ρ(1 + β^{2})

while for 0 ≤ s ≤ β we have

F (s) ≥ − |F (s)| ≥ −a(1 + s^{q}) ≥ −a(1 + β^{q}) ≥ −\frac{M}{2} η(1 + β^{q}) ≥ −Mηβ^{q} ≥ −M.F (β)

and this completes the proof. ■

Corollary 3.8. Assume 1 < p < \frac{n + 2}{n − 2}. Let F : ℝ → ℝ be of class C^{1} and satisfy :

i ) |F′(t)| ≤ a (1 + |t|^{p}) ii”) lim inf_{t→+∞} t^{−2}F (t) > −∞ and lim sup_{t→+∞} t^{−2}F (t) = +∞ iii ) sup{t : F′(t) < 0} = +∞ iv ) inf{t : F′(t) > 0} < 0 Then there are unboundedly many local maxima of Φ_{λ} for every positive λ.

As above we have only to prove that condition ii) of Theorem 3.1 holds. Let M > 0

such that inf_{t≥0} \frac{F (t)}{1 + t^{2}} > −M. Let ρ > 1. There is some β > 1 such that β^{−2}F (β) > 2ρ. Then we have

F (β) ≥ 2ρβ^{2} ≥ ρ(1 + β^{2})

and, for 0 ≤ s ≤ β,

F (s) ≥ −M(1 + s^{2}) ≥ −Mρ(1 + s^{2}) ≥ −Mρ(1 + β^{2}) ≥ −M.F (β)

and this completes the proof. ■

Example 3.9. Assume 2 < q < q + δ < \frac{2n}{n − 2} and φ is a C^{1} periodic non constant function on ℝ. Then if φ is not everywhere negative the function

F : ξ ↦ |ξ|^{q} φ(|ξ|^{δ})

satisfies the conditions of Corollary 3.7.

We have, for ξ ≠ 0,

f(ξ) = F′(ξ) = ξ |ξ|^{q+δ−2} (δφ′(|ξ|^{δ}) + \frac{q}{|ξ|^{δ}} φ(|ξ|^{δ})) = O(|ξ|^{q+δ−1})

Hence condition i) is satisfied. Let T be the period of φ. Since φ is not constant and

∫_{0}^{T} φ′(t) dt = φ(T) − φ(0) = 0

there are ξ_{0} and ξ_{1} such that 0 < ξ_{0} < ξ_{1} < T and φ′(ξ_{0}).φ′(ξ_{1}) < 0. Then for every large enough integer k, F′((ξ_{0} + kT)^{1/δ}) has the same sign as φ′(ξ_{0}) and F′((ξ_{1} + kT)^{1/δ}) the opposite sign. Moreover F′(−ξ) = −F′(ξ). And this ensures conditions iii) and iv).

Finally, if 0 ≤ ξ_{2} ≤ T and η = φ(ξ_{2}) > 0, we have, for t_{k} = (ξ_{2} + kT)^{1/δ},

F (t_{k}) = t_{k}^{q}φ(ξ_{2}) = ηt_{k}^{q}

and condition ii’) follows since |F (ξ)| ≤ ∥φ∥_{∞} . |ξ|^{q}. ■

In particular, for n ≤ 5 and δ = 1, we get the following

Example 3.10. If φ is a non constant C^{1} periodic function taking positive values and

2 < q < \frac{n + 2}{n − 2} then the function

F : ξ ↦ |ξ|^{q} φ(ξ)

satisfies the conditions of Corollary 3.7.

If one wants criteria related on the continuous function f = F′, one gets the following

Example 3.11. Let 0 < δ ≤ 1 and φ ≠ 0 be a continuous periodic function with null mean

on a period. If 1 + δ < p < \frac{n + 2}{n − 2}, the function defined by

F (ξ) = ∫_{0}^{ξ} |t|^{p} φ(|t|^{δ}) dt

satisfies the conditions of Corollary 3.7.

Proof : Let T be the period of φ. Since φ ≠ 0 and ∫_{0}^{T} φ(t) dt = 0, the function φ takes on each period positive and negative values. And this ensures conditions iii) and iv). Since φ is bounded, condition i) is satisfied.

Since ∫_{0}^{T} φ(t) dt = 0, all primitives of φ are periodic and one of them has a null mean on [0, T]. This latter is the derivative of a T -periodic function : hence there is a C^{2} periodic function ψ on ℝ such that ψ′′ = φ. Then, for ξ > 0,

F (ξ) = ∫_{0}^{ξ} t^{p}ψ′′(t^{δ}) dt = ∫_{0}^{ξ} \frac{t^{p+1−δ}}{δ} (δt^{δ−1}ψ′′(t^{δ})) dt

= [\frac{t^{p+1−δ}}{δ} ψ′(t^{δ})]_{0}^{ξ} − \frac{p + 1 − δ}{δ^{2}} ∫_{0}^{ξ} t^{p+1−2δ}(δt^{δ−1}ψ′(t^{δ})) dt

= [\frac{t^{p+1−δ}}{δ} ψ′(t^{δ}) − \frac{p + 1 − δ}{δ^{2}} t^{p+1−2δ}ψ(t^{δ})]_{0}^{ξ} + \frac{p + 1 − δ}{δ^{2}} ∫_{0}^{ξ} \frac{d}{dt}(t^{p+1−2δ})ψ(t^{δ}) dt

and since

∫_{0}^{ξ} |\frac{d}{dt}(t^{p+1−2δ})ψ(t^{δ}) dt| ≤ ∥ψ∥_{∞} ∫_{0}^{ξ} |\frac{d}{dt}(t^{p+1−2δ})| dt = ∥ψ∥_{∞} ξ^{p+1−2δ}

we get

F (ξ) = \frac{ξ^{p+1−δ}}{δ} (ψ′(ξ^{δ}) + O(ξ^{−δ}))

Since ψ′ is bounded and F is odd this ensures the first part of condition ii’) with q = p + 1 − δ. And since ψ′ is periodic, not zero and satisfies ∫_{0}^{T} ψ′(t) dt = 0, there is some t^{∗} ∈ [0, T] such that ψ′(t^{∗}) > 0. Then, for t_{k} = (t^{∗} + kT)^{1/δ},

lim sup_{ξ→+∞} ξ^{−q}F (ξ) ≥ lim_{k→∞} t_{k}^{−p−1+δ}F (t_{k}) = \frac{ψ′(t^{∗})}{δ} > 0

which ensures the last part of condition ii’). ■

In particular, for n ≤ 5 and δ = 1, we get the following

Example 3.12. Let φ be a continuous T -periodic function on ℝ with null mean on [0, T].

If 2 < p < \frac{n + 2}{n − 2}, the function

F (ξ) = ∫_{0}^{ξ} |t|^{p} φ(t) dt

satisfies the hypotheses of Corollary 3.7.

Notice that condition iii) of Theorem 3.1 and condition ii) of Theorem 2.3 are incompatible. Thus it would be an interesting question to know wether there exists functions f satisfying condition iii) of Theorem 3.1 and for which we have lim_{r→+∞} Q(r) = +∞. But this seems to be quite difficult since the behaviour of the function Q at the infinity is not easy to derive from this of the function F.

4. Genericity of functions of type LM_{∞}

For the oscillating functions we looked in the previous section, the functional Φ_{λ} has for each λ > 0 unboundedly many local maxima. Nevertheless these functions are not necessarily of type LM_{∞} since we dont have any control on the function Q. Although it is probably hard to produce simple explicit examples for n ≥ 2, we shall show that in some sense “almost all” function f is of type LM_{∞}.

Assume p < \frac{n + 2}{n − 2} and define the Banach space of continuous functions

E_{p} = \{f ∈ C(ℝ) : lim_{t→∞} \frac{|f(t)|}{1 + |t|^{p}} = 0 \}

equipped with the norm | ; | ; |f| ; | ; | = sup_{t∈ℝ} \frac{|f(t)|}{1 + |t|^{p}}, which is isomorphic to the space C_{0}(ℝ) of

continuous functions on ℝ vanishing at the infinity.

Multiplicity of the solutions

Theorem 4.1. The subset M_{p} of E_{p} consisting of the elements of type LM_{∞} is a residual set in E_{p}.

Define for m ∈ ℕ the subsets U_{m} and V_{m} of E_{p} by

U_{m} = \{f ∈ E_{p} : ∃r > m μ_{f}(r) < \frac{1}{m}r^{2} \}

V_{m} = {f ∈ E_{p} : ∃r > m μ_{f}(r) > mr^{2} }

where μ_{f}(r) = sup_{∥u∥=r} J_{f}(u) and J_{f}(u) denotes, as at the beginning of this paper,

J_{f}(u) = ∫_{Ω} (∫_{0}^{u(x)} f(s) ds) dx

Lemma 4.2. The sets U_{m} and V_{m} are open in E_{p}. Proof : The function f ↦ J_{f}(u) is clearly linear for each u ∈ H_{0}^{1}(Ω) from E_{p} to ℝ, and since p + 1 < \frac{2n}{n − 2} there is a constant γ such that ∥u∥_{p+1} ≤ γ ∥u∥ for each u ∈ H_{0}^{1}(Ω). We have by an easy computation :

|∫_{0}^{ξ} f(s) ds| ≤ | ; | ; |f| ; | ; | ∫_{0}^{|ξ|}(1 + |s|^{p}) ds = | ; | ; |f| ; | ; | (|ξ| + \frac{1}{p + 1} |ξ|^{p+1}) ≤ | ; | ; |f| ; | ; |(1 + |ξ|^{p+1})

hence

|J_{f}(u)| ≤ | ; | ; |f| ; | ; | ∫_{Ω}(1 + |u(x)|^{p+1}) dx ≤ | ; | ; |f| ; | ; | (|Ω| + γ^{p+1} ∥u∥^{p+1})

from which we deduce :

|μ_{f}((r) − μ_{g}(r)| ≤ sup_{∥u∥=r} |J_{f}(u) − J_{g}(u)| ≤ | ; | ; |f − g| ; | ; | (|Ω| + γ^{p+1}r^{p+1})

Thus for each r the function f ↦ μ_{f}(r) is continuous, and it follows that the sets

U_{m} = ⋃_{r>m}\{f ∈ E_{p} : μ_{f}(r) < \frac{1}{m}r^{2} \}

and

V_{m} = ⋃_{r>m}{f ∈ E_{p} : μ_{f}(r) > mr^{2} }

are open in E_{p}. ■ Lemma 4.3. The sets U_{m} are dense in E_{p}. Proof : Let f_{0} ∈ E_{p} and ε > 0. Since the set K(ℝ) of continuous functions on ℝ with compact support is dense in C_{0}(ℝ), there is g_{1} ∈ K(ℝ) such that

sup_{t∈ℝ} |g_{1}(t) − \frac{f_{0}(t)}{1 + |t|^{p}} | < ε

Multiplicity of the solutions

hence, with f_{1}(t) = (1 + |t|^{p})g_{1}(t), f_{1} ∈ K(ℝ) and | ; | ; |f_{1} − f_{0}| ; | ; | < ε. Moreover, for every ξ ∈ ℝ, we have

|∫_{0}^{ξ} f_{1}(s) ds| ≤ ∥f_{1}∥_{1} = ∫_{−∞}^{+∞} |f_{1}(s)| ds < +∞

It follows that for each u ∈ H_{0}^{1}(Ω) :

J_{f}(u) = ∫_{Ω} (∫_{0}^{u(x)} f_{1}(s) ds) dx ≤ ∫_{Ω} ∥f_{1}∥_{1} dx = ∥f_{1}∥_{1} |Ω|

hence

μ_{f}(r) ≤ ∥f_{1}∥_{1} |Ω| − λr^{2}

Thus f_{1} ∈ U_{m} for every m, and U_{m} ∩ B(f_{0}, ε) ≠ ∅ for each m ∈ ℕ and each f_{0} ∈ E_{p}. ■ Lemma 4.4. The sets V_{m} are dense in E_{p} Proof : Let f_{0} ∈ E_{p} and ε > 0. As above, there exists a f_{1} ∈ K(ℝ) such that | ; | ; |f_{0} − f_{1}| ; | ; | < \frac{ε}{2}. Let q be such that 1 < q < p. Then the function f^{∗} defined by

f^{∗}(t) = t. |t|^{q−1}

belongs to E_{p} and we have |||f^{∗}||| = sup_{t>0} \frac{t^{q}}{1 + t^{p}}.

For t ≤ 1, we have t^{q} ≤ 1 ≤ 1 + t^{p}, hence \frac{t^{q}}{1 + t^{p}} ≤ 1. And for t ≥ 1, we have

t^{q} ≤ t^{p} ≤ 1+t^{p}, hence \frac{t^{q}}{1 + t^{p}} ≤ 1. We conclude that |||f^{∗}||| ≤ 1. So f_{2} = f_{1} + \frac{ε}{2}f^{∗} ∈ B(f_{0}, ε).

As in the lemma above, we have :

|∫_{0}^{ξ} f_{1}(t) dt| ≤ ∫_{−∞}^{+∞} |f_{1}(t)| dt = ∥f_{1}∥_{1}

thus

|∫_{0}^{ξ} f_{2}(t) dt| ≥ \frac{ε}{2} ∫_{0}^{ξ} f^{∗}(t) dt − |∫_{0}^{ξ} f_{1}(t) dt| ≥ \frac{ε}{2(q + 1)} |ξ|^{q+1} − ∥f_{1}∥_{1}

And it follows that

\begin{aligned}J_{f_{2}}(u) & = ∫_{Ω} (∫_{0}^{u(x)} f_{2}(t) dt) dx ≥ \frac{ε}{2(q + 1)} ∫_{Ω} |u(x)|^{q+1} dx − ∥f_{1}∥_{1} |Ω| \\ & ≥ \frac{ε}{2(q + 1)} ∥u∥_{q+1}^{q+1} − ∥f_{1}∥_{1} |Ω| \end{aligned}

If we choose some u_{0} in H_{0}^{1}(Ω) such that ∥u_{0}∥ = 1, we get

μ_{f_{2}}(r) − mr^{2} ≥ J_{f_{2}}(r.u_{0}) − mr^{2} ≥ \frac{εr^{q+1}}{2(q + 1)} ∥u_{0}∥_{q+1}^{q+1} − ∥f_{1}∥_{1} |Ω| − mr^{2}

and since q +1 > 2 we can easily find a r > m such that ∥f_{1}∥_{1} |Ω|+mr^{2} < \frac{εr^{q+1}}{2(q + 1)} ∥u_{0}∥_{q+1}^{q+1}.

This shows that f_{2} ∈ V_{m}, thus that V_{m} ∩ B(f_{0}, ε) ≠ ∅. And the proof of the lemma is complete. ■

Proof of Theorem 4.1

The set

G = ⋂_{m} U_{m} ∩ ⋂_{m} V_{m}

is the intersection of a countable family of dense open sets ; thus it is comeager in the Banach space E_{p}, hence dense. We shall prove that G ⊂ M_{p}.

Let f ∈ G. We can choose an increasing sequence (r_{m})_{m≥1} such that r_{m} ≥ m and

μ_{f}(r_{2m−1}) < \frac{1}{2m − 1}r_{2m−1}^{2}

μ_{f}(r_{2m}) > 2m.r_{2m}^{2}

Then

Q(r_{2m+1}) = inf_{s<r_{2m+1}} \frac{μ(r_{2m+1}) − μ(s)}{r_{2m+1}^{2} − s^{2}} ≤ \frac{μ(r_{2m+1}) − μ(r_{2m})}{r_{2m+1}^{2} − r_{2m}^{2}}

≤ \frac{1}{r_{2m+1}^{2} − r_{2m}^{2}} (\frac{r_{2m+1}^{2}}{2m + 1} − 2m.r_{2m}^{2}) ≤ \frac{1}{r_{2m+1}^{2} − r_{2m}^{2}} (\frac{r_{2m+1}^{2}}{2m + 1} − \frac{r_{2m}^{2}}{2m + 1})

≤ \frac{1}{2m + 1}

And this shows that lim inf_{r→∞} Q(r) ≤ 0. Moreover Φ_{2m−1} attains its maximum on B(r_{2m+1}) at a point u_{m}, and we have

Φ_{2m−1}(u_{m}) = sup_{u∈B(r_{2m+1})} Φ_{2m−1}(u) ≥ sup_{∥u∥=r_{2m}} Φ_{2m−1}(u) = μ_{f}(r_{2m})−(2m−1)r_{2m}^{2} ≥ r_{2m}^{2} > 0

while for each u ∈ S(r_{2m+1}) we have

Φ_{2m−1}(u) ≤ μ_{f}(r_{2m+1}) − (2m − 1)r_{2m+1}^{2} < (\frac{1}{2m + 1} − 2m)r_{2m}^{2} < 0

Thus u_{m} lies in the interior of the ball B(r_{2m+1}) and is a critical point of Φ_{λ}. Moreover, since S(r_{2m−1}) is weakly dense in B(r_{2m−1}) we have

sup_{∥u∥≤r_{2m−1}} J_{f}(u) = sup_{∥u∥=r_{2m−1}} J_{f}(u) = μ_{f}(r_{2m−1}) < \frac{1}{2m − 1}r_{2m−1}^{2} < \frac{1}{2m − 1}r_{2m}^{2}

hence, for ∥u∥ ≤ r_{2m−1}

Φ_{2m−1}(u) ≤ J_{f}(u) < \frac{1}{2m − 1}r_{2m}^{2} ≤ r_{2m}^{2} ≤ Φ_{2m−1}(u_{m})

and this implies that ∥u_{m}∥ > r_{2m−1} ≥ 2m − 1. Finally, since Φ_{2m−1} attains its maximum over B(∥u_{m}∥) on S(∥u_{m}∥), we have Q(∥u_{m}∥) ≥ 2m − 1 by Lemma 2.1.

So lim sup_{r→∞} Q(r) = +∞, and we are done. ■

Remark 4.5.

We could also replace the Banach space E_{p} by

Ẽ_{p} := \{f ∈ C(ℝ) : sup_{t∈ℝ} \frac{|f(t)|}{1 + |t|^{p}} < +∞ \}

equipped with the same norm | ; | ; |f| ; | ; | := sup_{t∈ℝ} \frac{|f(t)|}{1 + |t|^{p}}. Then Ẽ_{p} is a Banach space isomorphic

to the space C_{b}(ℝ) of bounded continuous functions on ℝ. But in this case the previous result is not longer true : the set M̃_{p} of functions of type LM_{∞} is not dense in Ẽ_{p}.

Indeed, if f^{∗} denotes the function : t ↦ t. |t|^{p−1} which belongs to Ẽ_{p}, and ρ < |||f^{∗}||| = 1, for every function f of Ẽ_{p} satisfying |||f − f^{∗}||| < ρ we will have for every t ∈ ℝ :

\{ \begin{matrix}t.f(t) & ≥ (1 − ρ) |t|^{p+1} − ρ |t| \\ |f(t)| & ≤ (1 + ρ) |t|^{p} + ρ \end{matrix}

Then it follows from Theorem 2.3 that

lim inf_{r→∞} Q_{f}(r) = +∞

hence that f is not of type LM_{∞}. So M̃_{p} ∩ B(f^{∗}, ρ) = ∅, that is f^{∗} does not belong to the closure in Ẽ_{p} of the set M̃_{p}. ■

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