In this article we present some existence and multiplicity results for critical points of functionals of the form F(u

ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu

MULTIPLICITY OF CRITICAL POINTS FOR THE FRACTIONAL ALLEN-CAHN ENERGY

DAYANA PAGLIARDINI

Abstract. In this article we study the fractional analogue of the Allen-Cahn energy in bounded domains. We show that it admits a number of critical points which approaches infinity as the perturbation parameter tends to zero.

1. Introduction

The problems involving fractional operators attracted great attention during the previous years. Indeed these problems appear in areas such as optimization, finance, crystal dislocation, minimal surfaces, water waves, fractional diffusion; see for example [8, 6, 3, 4, 7, 19, 18]). In particular, from a probabilistic point of view, the fractional Laplacian is the infinitesimal generator of a L´evy process, see e.g. [2].

In this article we present some existence and multiplicity results for critical points of functionals of the form

F(u) = Z

Ω

Z

Ω

|u(x)−u(y)|²

|x−y|^n+2s dx dy+ 1 ^2s

Z

Ω

W(u)dx, if s∈(0,1/2), (1.1) F(u) = 1

|log| Z

Ω

Z

Ω

|u(x)−u(y)|²

|x−y|ⁿ⁺¹ dx dy+ 1

|log| Z

Ω

W(u)dx, ifs= 1/2, (1.2) F(u) = ^2s−1

2 Z

Ω

Z

Ω

|u(x)−u(y)|²

|x−y|^n+2s dx dy+1 Z

Ω

W(u)dx, ifs∈(1/2,1), (1.3) where Ω is a smooth bounded domain ofRⁿ, u∈H^s(Ω;R),W ∈C²(R;R⁺) is the well known double well potential (see Section 2), and∈R⁺.

Fis the fractional energy of the Allen-Cahn equation. It is the fractional coun- terpart of the functionals studied by Modica-Mortola in [14, 15] where they proved the Γ-convergence of the energy to De Giorgi’s perimeter. In the same way, func- tionals (1.1),(1.2),(1.3) have been also considered by Valdinoci-Savin in [17], where it is discussed their Γ-convergence.

Moreover, as proved in [13] for the functional Z

Ω

[|Du|²+⁻¹(u²−1)²]dx,

2010Mathematics Subject Classification. 35R11, 58J37.

Key words and phrases. Allen-Cahn energy; fractional PDE; critical point; genus.

c

2016 Texas State University.

Submitted March 4, 2016. Published May 13, 2016.

1

we expect that the solutions have interesting geometric properties related to the interface minimality.

Some authors investigated multiplicity results of nontrivial solution for ^2s(−∆)^su+u=h(u) in Ω

u >0 u= 0 on∂Ω

(1.4) where Ω is a bounded domain inRⁿ,n >2s, andh(u) has a subcritical growth (see [12]), or for

^2s(−∆)^su+V(z)u=f(u) in Rⁿ, n >2s u∈H^s(Rⁿ)

u(z)>0 z∈Rⁿ

(1.5) where the potentialV :Rⁿ →R and the nonlinearity f : R→R satisfy suitable assumptions (see [11]).

Then Cabr´e and Sire in [5] studied the equation (−∆)^su+G⁰(u) = 0 inRⁿ

where G denotes the potential associated to a nonlinearity f, and they proved existence, uniqueness and qualitative properties of solutions.

Indeed, Passaseo in [16] studied the analogue of our functional, with the classical Laplacian instead of the fractional one, i.e.,

f(u) = Z

Ω

|Du|²dx+1

Z

Ω

G(u)dx (1.6)

where Ω is a bounded domain ofRⁿ,u∈H^1,2(Ω;R),G∈C²(R;R⁺) is a nonnega- tive function having exactly two zeros, αand β, and is a positive parameter: he proved that the number of critical points forf goes to∞as→0.

Passaseo was motivated by De Giorgi’s idea, contained in [9], i.e. ifu →u0 in L¹(Ω) as→0 and lim→0f(u)<∞, then the functionU(t), defined as steepest descent curves forfstarting fromu, converge to a curveU0(t) inL¹(Ω) such that U0(t) is a function with values in{α, β} for everyt≥0 and the interface between the sets E_t={x∈Ω :U₀(t)(x) =α} and Ω\E_t moves by mean curvature. As a consequence the critical pointsu off which satisfy

lim inf

→∞ f(u)<+∞ (1.7)

converge in L¹(Ω) to a functionu0 taking values in {α, β}. De Giorgi considered also the problem of existence and multiplicity for nontrivial critical points of f

with the property (1.7), and Passaseo’s critical points verify this property and lim inf

→∞ f(u)>0, so he can say thatu0 is nontrivial.

In this article we want to extend Passaseo’s results by replacing the functionG in (1.6) with the double well potential W, and Passaseo’s functional f with its fractional counterpart.

The article is organized as follows: in Section 2 we give some preliminaries definitions and results. In Section 3, we define suitable functions and sets, then most of the work is dedicated to prove nonlocal estimates needful to obtain the bound from above ofF, (see Lemma 3.5), and the (PS)-condition, Lemma 3.7. In

fact in particular for the first of these results, we had to split the domain in two types of regions and estimateF in the three possible interactions.

Finally, after recalling a technical result, Lemma 3.6, we can apply a classical Krasnoselsii’s genus tool to show the existence and multiplicity results for solutions.

Hence, knowing that minimizers of F Γ-converge to minimizers of the area functional, we hope that also min-max solutions can pass to the limit as→0 in a suitable sense, producing critical points of positive index for local, ifs∈[1/2,1), or nonlocal, ifs∈(0,1/2), area functional.

2. Notation and preliminary results

In this section we introduce the framework that we will be used throughout this article.

Let Ω be a bounded domain of Rⁿ, denote by |Ω| its Lebesgue measure and considerW the double well potential, that is an even function such that

W :R→[0,+∞) W ∈C²(R;R⁺) W(±1) = 0

W >0 in (−1,1) W⁰(±1) = 0 W⁰⁰(±1)>0. (2.1) Now we fix the fractional exponents∈(0,1). For anyp∈[1,+∞), we define

W^s,p(Ω) :=n

u∈L^p(Ω) : |u(x)−u(y)|

|x−y|^s+n/p ∈L^p(Ω×Ω)o

;

i.e. an intermediary Banach space betweenL^p(Ω) and W^1,p(Ω), endowed with the natural norm

kukW^s,p(Ω):=Z

Ω

|u|^pdx+ Z

Ω

Z

Ω

|u(x)−u(y)|^p

|x−y|^n+sp dx dy^1/p .

Ifp= 2 we defineW^s,2(Ω) =H^s(Ω) and it is a Hilbert space. Now letS⁰(Rⁿ) be the set of all temperated distributions, that is the topological dual ofS(Rⁿ). As usual, for anyϕ∈S(Rⁿ), we denote by

Fϕ(ξ) = 1 (2π)^n/2

Z

Rⁿ

e^−iξ·xϕ(x)dx

the Fourier transform of ϕand we recall that one can extend F from S(Rⁿ) to S⁰(Rⁿ). At this point we can define, for any u ∈ S(Rⁿ) and s ∈ (0,1), the fractional Laplacian operator as

(−∆)^su(x) =C(n, s)P.V.

Z

Rⁿ

u(x)−u(y)

|x−y|^n+2s dy.

Here P.V. stands for the Cauchy principal value andC(n, s) is a normalizing con- stant (see [10] for more details). It is easy to prove that this definition is equivalent to the following two:

(−∆)^su(x) =−1 2C(n, s)

Z

Rⁿ

u(x+y) +u(x−y)−2u(x)

|y|^n+2s dy ∀x∈Rⁿ, and

(−∆)^su(x) =F⁻¹(|ξ|^2s(Fu)) ∀ξ∈Rⁿ. Now we recall some embedding’s results for the fractional spaces:

Proposition 2.1 ([10]). Let p∈[1,+∞) and 0< s ≤s⁰ ≤1. LetΩ be an open set ofRⁿ andu: Ω→Rbe a measurable function. Then W^s0,p(Ω) is continuously embedded inW^s,p(Ω), denoted byW^s0,p(Ω),→W^s,p(Ω), and the following inequality holds

kuk_Ws,p(Ω)≤Ckuk_Ws0,p(Ω)

for some suitable positive constant C=C(n, s, p)≥1.

Moreover, if also Ω is an open set of Rⁿ of class C^0,1 with bounded boundary, thenW^1,p(Ω),→W^s,p(Ω) and we have

kuk_Ws,p(Ω)≤Ckuk_W1,p(Ω)

for some suitable positive constant C=C(n, s, p)≥1.

Definition 2.2([10]). For anys∈(0,1) and anyp∈[1,+∞), we say that an open set Ω⊆Rⁿis an extension domain forW^s,p if there exists a positive constantC= C(n, p, s,Ω) such that: for every functionu∈W^s,p(Ω) there exists ˜u∈W^s,p(Rⁿ) with ˜u(x) =u(x) for allx∈Ω andk˜uk_Ws,p(Rⁿ)≤Ckuk_Ws,p(Ω).

Theorem 2.3 ([10]). Let s ∈ (0,1) and p ∈ [1,+∞) be such that sp < n. Let q ∈[1, p^∗), where p^∗ =p^∗(n, s) =np/(n−sp) is the so-called “fractional critical exponent”. Let Ω ⊆ Rⁿ be a bounded extension domain for W^s,p and I be a bounded subset of L^p(Ω). Suppose that

sup

f∈I

Z

Ω

Z

Ω

|f(x)−f(y)|^p

|x−y|^n+sp dx dy <∞.

ThenI is pre-compact in L^q(Ω).

We recall also the notion of Krasnoselskii’s genus, useful in the sequel.

Definition 2.4([1]). LetH be a Hilbert space andEbe a closed subset ofH\ {0}, symmetric with respect to 0 (i.e. E=−E).

We call genus ofE inH, indicated with gen_H(E), the least integermsuch that there existsφ∈C(H;R) such thatφis odd andφ(x)6= 0 for allx∈E.

We set gen_H(E) = +∞ if there are no integer with the above property and gen_H(∅) = 0.

It is well known that gen_H(S^k) =k+ 1 ifS^k is ak-dimensional sphere ofH with centre in zero.

Finally we recall a well known result:

Theorem 2.5 ([1]). Let H be a Hilbert space and f : H → R be an even C²- functional satisfying the following Palais-Smale condition: given a sequence (ui)i

in H such that the sequence (f(ui))i is bounded and f⁰(ui)→0,(ui)i is relatively compact inH.

Set f^c ={u∈H :f(u)≤c} for all c ∈R. Then, for allc1,c2∈R, such that c1≤c2< f(0), we have

gen_H(f^c2)≤gen_H(f^c1) + #{(−ui, ui) :c1≤f(ui)≤c2, f⁰(ui) = 0}, (2.2) where, ifA is a set, we indicate with #A the cardinality ofA.

For the rest of this article, we considerH^s(Ω) as Hilbert space and we shall write simply gen(E) instead of gen_Hs(Ω)(E); then we refer to the Palais-Smale condition with the symbol (P S)-condition.

3. Multiplicity of critical points Let us state the fundamental result of the paper.

Theorem 3.1. Let Ω be a smooth bounded domain of Rⁿ and W be a function satisfying (2.1). Then there exist two sequences of positive numbers (k)k, (ck)k

such that for every∈(0, k), the functionalF has at least kpairs (−u_1,, u_1,), . . . ,(−u_k,, u_k,)

of critical points, all of them different from the constant pair(−1,1)satisfying

−1≤ui,(x)≤1 ∀x∈Ω, ∀∈(0, k), i= 1, . . . k;

F(ui,)≤ck ∀∈(0, k), i= 1, . . . , k.

Moreover, for all ∈(0, _k)and all i= 1, . . . , k we have F(u)≥min

F(u) :u∈H^s(Ω),−1≤u(x)≤1 forx∈Ω, Z

Ω

u dx= 0 . (3.1) Remark 3.2. The constant function u ≡ 0 is obviously a critical point for the functional F for every > 0 but it is not included among the ones given by Theorem 3.1. Instead ifs∈(1/2,1), but for the other cases it is similar,

F(0) = 1

W(0)|Ω| →+∞ as→0.

Moreover, since inf{W(t) : W⁰(t) = 0,−1 < t <1} >0, one can say that the critical points given by Theorem 3.1 are not constant functions. In fact, ifu=c

is a constant critical point for F (distinct from −1 and 1), it must be W⁰(c) = 0 and−1< c<1; therefore

W(c)≥inf{W(t) :W⁰(t) = 0,−1< t <1}>0 (3.2) and so, for example by considering the functional related tos ∈(1/2,1), but the other cases are similar,

F(c) = 1

W(c)|Ω| →+∞ as→0, (3.3)

in contradiction withF(c)≤ck for all∈(0, k).

Notice that for all >0, min

F(u) :u∈H^s(Ω),−1≤u(x)≤1 ∀x∈Ω, Z

Ω

u dx= 0 >0 (3.4) if we assume, without loss of generality, that Ω is a connected domain.

Let ¯ube a minimizing function; if we assumeF(¯u) = 0, then Z

Ω

Z

Ω

|¯u(x)−u(y)|¯ ²

|x−y|^n+2s dx dy≡0

andW(¯u)≡0. Therefore we must have ¯u≡0 in contradiction withW(0)>0.

Definition 3.3. Letk be a fixed positive integer; for every λ= (λ⁽⁰⁾, . . . , λ^(k))∈ R^k+1 define the function ϕλ:R→Rby

ϕ_λ(t) =

k

X

m=0

λ^(m)cos(mt).

For everyλ∈R^k+1with|λ|_Rk+1 = 1 and >0, letL(ϕλ) :R→Rbe the function defined by

L(ϕλ)(t) = 1 2

Z t+

t−

ϕλ(τ)

|ϕλ(τ)|dτ;

notice that L(ϕ_λ) is well defined because ϕ_λ has only isolated zeros ∀λ ∈ R^k+1 with|λ|_Rk+1 = 1.

Forx= (x₁,· · ·, x_n)∈Ω⊂Rⁿ we consider the projection onto the first compo- nent,P1(x) =x1, and the set

S^k ={L(ϕλ)◦P1:λ∈R^k+1,|λ|_Rk+1 = 1}.

Lemma 3.4. Let us fixa, b∈Rwith a < b and set χ(ϕλ) = #{t∈[a, b] :ϕλ(t) = 0}

forλ∈R^k+1 with |λ|_Rk+1 = 1. Then for every k∈Nwe have sup{χ(ϕ_λ) :λ∈R^k+1,|λ|_Rk+1 = 1}<+∞.

For a proof of the above lemma, see [16].

Lemma 3.5. Let Ω be a bounded domain of Rⁿ and W be a function satisfying (2.1). Then, for everyk∈Nthere exists a positive constantck such that

maxF(f)≤ck ∀f ∈S^k. (3.5) Proof. Letuλ,=L(ϕλ)◦P1∈S^k and set

a= infP1(Ω), b= supP1(Ω), Z_λ={t∈[a, b] :ϕ_λ(t) = 0}, Zλ,={t∈R: dist(t, Zλ)< }.

Note that

(i) IfP1(x)∈/ Zλ,, then|uλ,(x)|= 1 andDuλ,(x) = 0, while (ii) ifP₁(x)∈Z_λ,, then|u_λ,(x)| ≤1 and|Du_λ,(x)| ≤ ¹.

Since Ω is bounded, we can suppose it is included in a cubeQof side large enough.

We will denote withY_λ,=Z_λ,^C the complement toZ_λ,, then we have to distinguish three cases:

(a) if x∈Y_λ, andy∈Y_λ,; (b) ifx∈Z_λ,andy∈Y_λ,; (c) if x∈Z_λ,andy∈Z_λ,.

We setk= max{χ(ϕλ) :λ∈R^k+1,|λ|_Rk+1= 1}, then Zλ,=

k

X

i=1

Z_λ,ⁱ and Yλ,=

k

X

i=1

Y_λ,ⁱ .

Now we call ˇZλ,=P₁⁻¹(Zλ,)∩Ω, ˇYλ,=P₁⁻¹(Yλ,)∩Ω and we observe that Z

Yˇ_λ,

W(u_λ,)dx= 0, (3.6)

while if we setρ= sup{|x|:x∈Ω},M = max{W(t) :|t| ≤1}and denote byω_n−1 the (n−1)-dimensional measure of the unit sphere ofRⁿ⁻¹, it results

Z

Zˇ_λ,

W(uλ,)dx≤M|Zˇλ,| ≤2M ω_n−1ρⁿ⁻¹. (3.7)

At this point it remains to analyze R

Ω

R

Ω

|uλ,(x)−uλ,(y)|²

|x−y|^n+2s dxdy. We split it in three cases:

Case (a). We have Z

Yˇ_λ,

Z

Yˇ_λ,

|u_λ,(x)−u_λ,(y)|²

|x−y|^n+2s dxdy=

k

X

i,j=1 i6=j

Z

Yˇ_λ,ⁱ

Z

Yˇ_λ,^j

|u_λ,(x)−u_λ,(y)|²

|x−y|^n+2s dx dy

(3.8) We denoteQ₋ =Q∩P₁⁻¹({x1<0}),Q₊ =Q∩P₁⁻¹({y1>2}) and we splitQ₋ inN strips of width, with N of order 1/, so we obtain

k

X

i,j=1 i6=j

Z

Yˇ_λ,ⁱ

Z

Yˇ_λ,^j

|uλ,(x)−uλ,(y)|²

|x−y|^n+2s dx dy

≤k² Z

Q−

Z

Q₊

4

|x−y|^n+2sdx dy

≤4N k² Z −2

−

Z +∞

−2x₁

r^−2s−1dr dx₁

= 2 sN k²

Z −2

−

(−2x1)^−2sdx1.

(3.9)

Now we distinguish two cases:

(j) if s6= 1/2, we have 2

sN k² Z −2

−

(−2x1)^−2sdx1= 2^1−2sN k²

s(1−2s)^1−2s(2^1−2s−1); (3.10) (jj) while, ifs= 1/2,

2 sN k²

Z −2

−

(−2x₁)^−2sdx₁= 2^1−2s

s N k²log 2. (3.11) Case (b). We note that ˇY_λ,ⁱ ⊆Q\Zˇ_λ,ⁱ , so

Z

Zˇ_λ,

Z

Yˇ_λ,

|u_λ,(x)−u_λ,(y)|²

|x−y|^n+2s dxdy

≤

k

X

i=1

Z

Zˇⁱ_λ,

Z

Q\Zˇ_λ,ⁱ

|uλ,(x)−u_λ,(y)|²

|x−y|^n+2s dxdy

≤2ω_n−1ρⁿ⁻¹

k

X

i=1

sup

x∈Zˇ_λ,ⁱ

Z

Q\Zˇⁱ_λ,

min{1/²|x−y|²,4}

|x−y|^n+2s dy

≤2kωn−1ρⁿ⁻¹Z 2 0

1

²r^1−2sdr+ Z +∞

2

4r^−1−2sdr

=k2

r^2−2s 2−2s

2 0

+ 8r^−2s

−2s

+∞

2

ω_n−1ρⁿ⁻¹

=k^1−2s2^2−2s

1−s +2^2−2s s

ω_n−1ρⁿ⁻¹.

(3.12)

Case (c). It results Z

Zˇ_λ,

Z

Zˇ_λ,

|uλ,(x)−uλ,(y)|²

|x−y|^n+2s dxdy

=

k

X

i=1

Z

Zˇ_λ,ⁱ

Z

Zˇ_λ,ⁱ

|uλ,(x)−uλ,(y)|²

|x−y|^n+2s dxdy

+

k

X

i,j=1 i6=j

Z

Zˇ^j_λ,

Z

Zˇ_λ,ⁱ

|uλ,(x)−uλ,(y)|²

|x−y|^n+2s dxdy.

(3.13)

Concerning the first term of the right-hand side, we have

k

X

i=1

Z

Zˇ_λ,ⁱ

Z

Zˇ_λ,ⁱ

|uλ,(x)−uλ,(y)|²

|x−y|^n+2s dxdy

≤ 1 ²

k

X

i=1

|Zˇ_λ,ⁱ | Z 2

0

r^1−2sdr≤kω_n−1ρⁿ⁻¹2^2−2s 1−s^1−2s.

(3.14)

The other term is estimated as in Case (b).

So we can obtain the estimates for the functionalsF. In fact, by (3.9), (3.10), (3.11), (3.12) and (3.14), ifs∈(0,1/2) we have

F(u_λ,) = Z

Ω

Z

Ω

|uλ,(x)−u_λ,(y)|²

|x−y|^n+2s dx dy+ 1 ^2s

Z

Ω

W(u_λ,)dx

≤2kωn−1ρⁿ⁻¹2^2−2s

1−s^1−2s+2^2−2s s ^1−2s

+^1−2s2^2−2s

1−skω_n−1ρⁿ⁻¹+kM

^2s2ω_n−1ρⁿ⁻¹ +2^1−2sN k²

s(1−2s)^1−2s(2^1−2s−1)

≤k2^3−2s

1−s +2^3−2s

s +2^2−2s

1−s + 2M

ω_n−1ρⁿ⁻¹

+2^1−2sN k²

s(1−2s)(2^1−2s−1);

(3.15)

ifs= 1/2 we have F(u_λ, ) = 1

|log| Z

Ω

Z

Ω

|u_λ,(x)−u_λ,(y)|²

|x−y|ⁿ⁺¹ dx dy+ 1

|log| Z

Ω

W(u_λ,)dx

≤ 20k

|log|+ 2kM

|log|

ω_n−1ρⁿ⁻¹+ 1

s|log|N k²log 2

≤k(20 + 2M)ω_n−1ρⁿ⁻¹+1

sN k²log 2;

(3.16)

and, ifs∈(1/2,1) we get F(uλ, ) = ^2s−1

2 Z

Ω

Z

Ω

|uλ,(x)−uλ,(y)|²

|x−y|^n+2s dx dy+1

Z

Ω

W(uλ,)dx

≤k2^2−2s

1−s +2^2−2s

s +2^1−2s

1−s + 2M

ω_n−1ρⁿ⁻¹

+2^−2sN k²

s(1−2s)(2^1−2s−1).

(3.17)

Now we show a technical lemma, that we will used for proving our main result.

Lemma 3.6. For every >0andk∈Nthe setS^kverifies the following properties:

(a) S^k is a compact subset ofH^s(Ω);

(b) S^k =−S^k;

(c) for allk∈Nthere exists¯k >0 such that 0∈/S^k ∀∈(0,¯k);

(d) for allk∈Nand∀ >0 such that0∈/ S^k, it results gen(S^k)≥k+ 1.

Proof. The points (b), (c) and (d) are proved in [16]. For (a) we use [16, Lemma 2.8]

and the continuous embedding ofH¹(Ω) inH^s(Ω) for alls∈(0,1), see Proposition

2.1.

Before proving the main theorem of this work, we point out a useful property of F.

Lemma 3.7. The functionals (1.1),(1.2),(1.3)satisfy the (PS)-condition.

Proof. We will prove the lemma fors∈(1/2,1) being the other cases analogue. If W is quadratic, in particular there existα,β >0 such that

W(u)≥αu+β ∀u∈R. (3.18)

Since (F(un))nis bounded, (3.18) implies thatkunk_Hs(Ω)is bounded, henceun* uinH^s(Ω),un→uinL^q,∀q∈[1,2^∗= _n−2s²ⁿ ) from Theorem 2.3, thereforeun→u a.e. x∈Ω.

We claim thatuis a critical point of F. In fact for allv∈H^s(Ω), F⁰(u)(v) =^2s−1

Z

Ω

Z

Ω

u(x)−u(y)

|x−y|^n+2s(v(x)−v(y))dx dy +1

Z

Ω

W⁰(u)v dx

=^2s−1 lim

n→∞

Z

Ω

Z

Ω

un(x)−un(y)

|x−y|^n+2s (v(x)−v(y))dx dy +1

lim

n→∞

Z

Ω

W⁰(u_n)v dx

(3.19)

sinceu_n* uin H^s(Ω),u_n→uinL²(Ω) and, by hypothesis,F⁰(u_n)→0.

This implies thatF⁰(un)(un−u) +F⁰(u)(un−u)→0, but on the other hand F⁰(u_n)(u_n−u) +F⁰(u)(u_n−u)

=^2s−1 Z

Ω

Z

Ω

un(x)−un(y)

|x−y|^n+2s (un(x)−u(x)−un(y) +u(y))dx dy

−^2s−1 Z

Ω

Z

Ω

u(x)−u(y)

|x−y|^n+2s(un(x)−u(x)−un(y) +u(y))dx dy +1

Z

Ω

[W⁰(u_n)−W⁰(u)(u_n−u)]dx,

(3.20)

and the second term on the right hand side appraoches 0. In particular we obtain Z

Ω

Z

Ω

|un(x)−un(y)|²

|x−y|^n+2s dx dy→ Z

Ω

Z

Ω

|u(x)−u(y)|²

|x−y|^n+2s dx dy.

Hencekunk_Hs(Ω)→ kuk_Hs(Ω) and sinceun* uinH^s(Ω), we have the result.

We are now able to prove our main result.

Proof of Theorem 3.1. As usual we prove the theorem only fors∈(1/2,1). Con- siderW ∈C²(R;R⁺) another even function, which satisfies the following properties:

W =W ∀t∈[−1,1]; tW⁰(t)>0 for|t|>1, and the asymptotic behaviour guaranteeing that

F(u) = ^2s−1 2

Z

Ω

Z

Ω

|u(x)−u(y)|²

|x−y|^n+2s dx dy+1 Z

Ω

W(u)dx is aC²-functional satisfying the (PS)-condition.

We prove now that for every critical point u∈H^s(Ω) which is a critical point for the functionalF, it results|u(x)| ≤1 for allx∈Ω, and souis a critical point for the functionalF too: indeed we have that for allv∈H^s(Ω),

^2s−1 Z

Ω

Z

Ω

u(x)−u(y)

|x−y|^n+2s(v(x)−v(y))dxdy+1

Z

Ω

W⁰(u)v dx= 0.

In particular, if we set ˆu= max{min{u,1},−1}, by choosingv=u−u,ˆ ^2s−1

Z

Ω

Z

Ω

u(x)−u(y)

|x−y|^n+2s(u(x)−u(x)ˆ −u(y) + ˆu(y))dx dy +1

Z

Ω

W⁰(u)(u−u)ˆ dx= 0 with

Z

Ω

Z

Ω

u(x)−u(y)

|x−y|^n+2s(u(x)−u(x)ˆ −u(y) + ˆu(y))dx dy

= Z

Ω

Z

Ω

|u(x)−u(y)|²

|x−y|^n+2s dx dy≥0

(3.21)

and

Z

Ω

W⁰(u)(u−u)ˆ dx >0 ifu−uˆ6≡0 in Ω

sincetW⁰(t)>0 for|t|>1. It follows thatu= ˆu, i.e.,|u(x)| ≤1 for almost every x∈Ω.

Let k >0 be such thatk < _c¹

kW(0)|Ω|, where ck is the constant introduced in Lemma 3.5. Then, for every ∈(0, k) we can apply Theorem 2.5 to the func- tional F with c1 < 0 and c2 = ck, because F(0) = ¹W(0)|Ω| > ck for all ∈ (0, _k). In this way we can prove that for every ∈ (0, _k), F has at least (k+ 1) pairs (−u0,, u0,), . . . ,(−uk,, uk,) of critical points withF(ui,)≤ck for alli= 0,1, . . . , k. In fact gen(F^c1) = gen(∅) = 0, while gen(F^ck)≥gen(S^k)≥k+ 1 becauseS^k ⊆F^ck⊆H^s(Ω)\ {0}.

Note that these (k+ 1) pairs of critical points include also the one implied by the minimizers±1; so we can assume that (−u0,, u0,) = (−1,+1).

On the contrary, the other solutions are not minimizers for the functionalF if Ω is a connected domain. Indeed it results

F(ui,)>0 ∀∈(0, k) andi= 0,1, . . . , k because ifF(u_i,) =F(u_i,) = 0, then we should have

Z

Ω

Z

Ω

|ui,(x)−ui,(y)|²

|x−y|^n+2s dx dy= 0 and W(ui,)≡0 in Ω and soui,should be a constant function with value +1 or−1.

Moreover let us remark that for all∈(0, k) andi= 1, . . . kwe have F(ui,)≥min

F(u) :u∈H^s(Ω), Z

Ω

u dx= 0 . (3.22) In fact, we assume that

min

F(u) :u∈H^s(Ω), Z

Ω

u dx= 0 >0, otherwise (3.22) would be obvious. Then, for everyc₁>0 such that

c₁<min

F(u) :u∈H^s(Ω), Z

Ω

u dx= 0 ,

we would have clearly gen(F^c1) = 1 because belowc₁ the mean is non zero and we can use it as odd function intoR¹in the genus definition, see Definition 2.4; thus, if (3.22) were false, the solutions would belong to a set of genus one, in contradiction with their construction in Theorem 2.5.

Now, to prove (3.1), let us replace the function W appearing in the definition of functional F by a sequence of functions (Wj)j and denote by (F^j)j the cor- responding sequence of new functionals. Assume moreover that the functions Wj

satisfy the same properties asW for allj∈Nand that

j→∞lim Wj(t) = +∞ for|t|>1. (3.23) Then property (3.22) holds for the higher critical values of the functionalF^j for all j∈Nand so (3.1) follows forj large enough, taking into account that

j→∞lim min

F^j(u) :u∈H^s(Ω), Z

Ω

u dx= 0

= min

F(u) :u∈H^s(Ω),|u(x)| ≤1∀x∈Ω, Z

Ω

u dx= 0

because of (3.23).

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Dayana Pagliardini

Scuola Normale Superiore, Piazza dei Cavalieri 7, 56126 Pisa, Italy E-mail address:[email protected]