Citation: Nicola Soave. Saddle-shaped positive solutions for elliptic systems with bistable nonlinearity[J]. Mathematics in Engineering, 2020, 2(3): 423-437. doi: 10.3934/mine.2020019
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This paper concerns existence of multiple positive solutions for certain non-cooperative nonlinear elliptic systems with bistable nonlinearities, whose prototype is
{−Δu=u−u3−Λuv2−Δv=v−v3−Λu2vu,v>0in RN, with Λ>1. | (1.1) |
This system arises in the study of domain walls and interface layers for two-components Bose-Einstein condensates [4]. Domain walls solutions satisfying asymptotic conditions
{(u,v)→(1,0)as xN→+∞,(u,v)→(0,1)as xN→−∞,, | (1.2) |
in dimension N=1 have been carefully studied in [2,4], where in particular it is shown the existence of such a solution for every Λ>1 [4], and its uniqueness in the class of solutions with one monotone component [2]. In fact, uniqueness holds also without such assumption, and even in higher dimension [9]; precisely, in [9] it is shown that a solution to (1.1)–(1.2) (with the limits being uniform in x′∈RN−1) in RN with Λ>1 is necessarily montone in both the components with respect to xN, and 1-dimensional. The assumption Λ>1 is natural, since (1.1)–(1.2) has no solution at all when Λ∈(0,1]. Indeed, it is proved that (1.1) has only constant solutions for both Λ∈(0,1) [9], and Λ=1 [9,13].
We also refer to [1,3] for recent results regarding a system obtained from (1.1) adding in each equation an additional term representing the spin coupling.
To sum up, up to now it is known that (1.1) has only constant solutions for Λ∈(0,1], and at least one 1-dimensional non-constant solution for Λ>1. Moreover, solutions with uniform limits as in (1.2) are necessarily 1-dimensional, and unique modulo translations. In this paper we prove the existence of infinitely many geometrically distinct solutions to (1.1) in any dimension N≥2, for any Λ>1. This result enlightens once more the dichotomy Λ∈(0,1] vs. Λ>1. While for Λ∈(0,1] problem (1.1) is rigid in itself, and only possesses constant solutions, for Λ>1 we have multiplicity of non-constant solutions, and rigidity results can be recovered only with some extra assumption, such as (1.2).
Our result is based upon variational methods, and strongly exploits the symmetry of the problem. Roughly speaking, we shall construct solutions to (1.1) such that u−v "looks like" a sing changing solution of the Allen-Cahn equation −Δw=w−w3, with u≃w+, and v≃w−. The building blocks w in our construction will be both the saddle-type planar solutions (also called "pizza solutions") [5], and the saddle solutions in R2m [7,8].
We consider the following general version of (1.1):
{−Δu=f(u)−Λupvp+1inRN−Δv=f(v)−Λup+1vpinRNu,v>0inRN,with Λ>0, | (1.3) |
where N≥2, p≥1, and f is of bistable type; more precisely, let f:R→R be a locally Lipschitz continuous and odd nonlinearity. For a value M>0, we define the potential
F(t)=∫Mtf(s)ds, |
so that F∈C1,1(R), and F′=−f. We suppose that:
F≥0=F(±M)in R,andF>0in (−M,M). | (1.4) |
Note that in this case f(0)=f(±M)=0. F is often called a double well potential, and f is called bistable nonlinearity. A simple example is f(t)=t−t3.
With the above notation, we introduce
W(s,t)=F(s)+F(t)+Λp+1|s|p+1|t|p+1,(s,t)∈R2. |
The first of our main result concerns the existence of infinitely many geometrically distinct solutions for problem (1.3) in the plane. We consider polar coordinates (r,θ)∈[0,+∞)×[0,2π) in the plane. For any positive integer k, we define:
Rk, the rotation of angle π/k in counterclockwise sense;
Rik, the rotation of angle iπ/k in counterclockwise sense, with i=1,…,2k;
ℓ0, the line of equation x2=tan(π/2k)x1 in R2;
ℓi, the line Rik(ℓ1), i=1,…,k−1;
Ti, the reflection with respect to ℓi;
αk=tan(π/(2k));
Sk, the open circular sector {−π/(2k)<θ<π/(2k)}={αkx1>|x2|}⊂R2.
Theorem 1.1 (Saddle-type solutions in the plane). Let p≥1, f∈C0,1(R) be odd, and suppose that its primitive F satisfies (1.4). Suppose moreover that:
infs∈[0,M]W(s,s)>F(0). | (1.5) |
Then, for every positive integer k, there exists a positive solution (uk,vk) to system (1.3) in R2 having the following properties:
(i) 0<uk,vk<M in R2;
(ii) vk=uk∘Ti for every i=1,…,k, and uk(x1,−x2)=uk(x1,x2) in R2;
(iii) uk−vk>0 in Sk.
Notice in particular that {uk−vk=0}=⋃k−1i=0ℓi, which implies that (uk,vk)≠(uj,vj) if j≠k. Regarding assumption (1.5), we stress that for any bistable f it is satisfied provided that Λ>0 is sufficiently large, and can be explicitly checked in several concrete situations. In particular:
Corollary 1.2. For every Λ>1, problem (1.1) has infinitely many geometrically distinct non-constant solutions.
The corollary follows from the theorem, observing that if f(s)=s−s3 and p=1, then
F(s)=(1−s2)24,W(s,t)=F(s)+F(t)+Λs2t22; |
thus, condition (1.5) is satisfied if and only if
infs∈[0,1]W(s,s)=W(1√1+Λ,1√1+Λ)=Λ2(1+Λ)>14=F(0), |
that is, if and only if Λ>1. Notice that, if (1.5) is violated, we have have non-existence of non-constant solutions [9,13], and hence (1.5) is sharp in this case.
The proof of Theorem 1.1 consists in a 2 steps procedure. At first, we construct a solution to (1.3) in a ball BR with the desired symmetry properties, combining variational methods with an auxiliary parabolic problem. In a second step, we pass to the limit as R→+∞, obtaining convergence to an entire solution of (1.3). Assumption (1.5) enters in this second step in order to rule out the possibility that the limit profile (u,v) is a pair with v=u, with u possibly a constant. Roughly speaking, (1.5) makes the coexistence of u and v in the same region unfavorable with respect to the segregation, from the variational point of view.
This kind of construction is inspired by [6,11,12], where an analogue strategy was used to prove existence of solutions to
Δu=uv2,Δv=u2v,u,v>0 in RN. | (1.6) |
With respect to [6,12,11], however, the method has to be substantially modified. Solutions to (1.6) "look like" harmonic function in the same way as solutions to (1.3) "look like" solutions to the Allen-Cahn equation. Therefore, tools related with harmonic functions such as monotonicity formulae and blow-up analysis, which were crucially used in [6,12,11], are not available in our context, and have to be replaced by a direct inspection of the variational background. In such an inspection it emerges the role of the competition parameter Λ, which is not present in (1.6) (of course, Λ could be added in front of the coupling on the right hand side in (1.6); but it could be absorbed with a scaling, and hence it would not play any role).
Remark 1.3. Let us consider the scalar equation Δw+f(w)=0. The existence of a saddle-type (or pizza) solution wk with the properties
(i) −M<wk<M in R2;
(ii) wk∘Ti=−wk for every i=1,…,k, and wk(x1,−x2)=wk(x1,x2) in R2;
(iii) wk>0 in Sk.
was established by Alessio, Calamai and Montecchiari in [5], under slightly stronger assumption on F with respect to those considered here. In some sense, uk−vk looks like wk, since they share the same symmetry properties, and for this reason we can call (uk,vk) saddle-type (or pizza) solution.
Our method for Theorem 1.1 can be easily adapted also in the scalar case, giving an alternative proof for the existence result in [5]. For the sake of completeness, we present the details in the appendix of this paper. The main advantage is that our construction easily gives the following energy estimate
∫BR(12|∇wk|2+F(wk))≤CR, | (1.7) |
with C>0 depending only on k, but not on R. Such estimate seems to be unknown, expect for the case k=2, where it was proved in [7].
Theorem 1.1 establishes the existence of infinitely many positive solutions to (1.3) in the plane. These can be regarded as solutions also in higher dimension N≥3, but it is natural to ask whether there exist solutions to (1.3) in RN not coming from solutions in RN−1. We can give a positive answer to this question in any even dimension. Let N=2m, and let us consider the Simons cone
C={x∈R2m: x21+⋯+x2m=x2m+1+⋯+x22m}. |
We define two radial variables s and t by
s=√x21+⋯+x2m≥0,t=√x2m+1+⋯+x22m≥0. | (1.8) |
Theorem 1.4 (Saddle solutions in R2m). Let m≥2 be a positive integer, p≥1, f∈C0,1(R) be odd, and suppose that its primitive F satisfies (1.4). Suppose moreover that (1.5) holds. Then, for every positive integer m, there exists a positive solution (u,v) to system (1.3) in R2m having the following properties:
(i) 0<u,v<M in R2m;
(ii) v(s,t)=u(t,s);
(iii) u−v>0 in O={s>t}.
Notice that {u−v=0}=C, and that (u,v) looks like the saddle solution of the scalar Allen-Cahn equation found in [7]. The strategy of the proof is the same as the one of Theorem 1.4. However, the proof of Theorem 1.4 is a bit simpler, since we can take advantage of an energy estimate like (1.7), which is known to hold for saddle solutions in R2m (see formula (1.15) in [7]) but, as already observed, was unknown for saddle-type solutions in the plane.
Structure of the paper. In Section 2 we prove Theorem 1.1. In Section 3 we prove Theorem 1.4. In the appendix, we give an alternative proof with respect to [5] of the existence of saddle-type solutions for the scalar equation in the plane, yielding to the energy estimate (1.7).
In this section we prove Theorem 1.1. The existence of a solution in the whole plane R2 will be obtained by approximation with solutions in BR.
Throughout this section, the positive integer k (index of symmetry) will always be fixed, and hence the dependence of the quantities with respect to k will often be omitted.
In the sector S=Sk, we define
wk=min{M,αx1−|x2|√2}, |
where α=αk=tan(π/(2k)). Notice that wk>0 in Sk and wk=0 on ∂Sk. Thus, we can extend wk in the whole of R2 by iterated odd reflections with respect to the lines ℓi. In this way, we obtain a function, still denoted by wk, defined in R2, with
(i) −M≤wk≤M in R2;
(ii) wk∘Ti=−wk for every i=1,…,k, and wk(x1,−x2)=wk(x1,x2) in R2;
(iii) wk>0 in Sk,
that is, wk has the same symmetry properties of the saddle-type solutions in [5].
Now, for any Ω⊂R2 open, and for every (u,v)∈H1(Ω,R2), we introduce the functional
J((u,v),Ω):=∫Ω(12|∇u|2+12|∇v|2+W(u,v)). | (2.1) |
Moreover, for R>0, we let SR=Sk∩BR and consider the set
AR:={(u,v)∈H1(BR,R2)|(u,v)=(w+k,w−k) on ∂BR, 0≤u,v≤Min BRv=u∘Ti for every i=1,…,k,u(x1,−x2)=u(x1,x2) in BR, u≥vin SR, }. |
Notice that AR≠∅, since for instance (w+k,w−k)∈AR.
Lemma 2.1. For every R>0, there exists a solution (uR,vR)∈AR to
{−Δu=f(u)−Λ|u|p−1u|v|p+1inBR−Δv=f(v)−Λ|u|p+1|v|p−1vinBRu=w+k,v=w−kon ∂BR. | (2.2) |
Proof. The proof of the lemma is inspired by [6,Theorem 4.1]. Since the weak convergence in H1 implies the almost everywhere convergence, up to a subsequence, the set AR is weakly closed in H1. Moreover, the functional J(⋅,BR) is clearly bounded from below and weakly lower semi-continuous. Therefore, there exists a minimizer (uR,vR) of J(⋅,BR) in AR. To show that such a minimizer is a solution to (3.2), we consider the auxiliary parabolic problem
{∂tU−ΔU=˜f(U)−Λ|U|p−1U|V|p+1in (0,+∞)×BR∂tV−ΔV=˜f(V)−Λ|U|p+1|V|p−1Vin (0,+∞)×BRU=w+k,V=w−kon (0,+∞)×∂BR(U(0,⋅),V(0,⋅))∈AR, | (2.3) |
where ˜f:R→R is a globally Lipschitz continuous odd function such that ˜f(s)=f(s) for s∈[−M−1,M+1]. The existence and uniqueness of a local solution, defined on a maximal time interval [0,T), follow by standard parabolic theory. Notice that
∂tU−ΔU=c1(t,x)U, |
for
c1(t,x)={−Λ|U(t,x)|p−1|V(t,x)|p+1+˜f(U(t,x))U(t,x)if U(t,x)≠00if U(t,x)=0. |
Since ˜f(0)=0, we have that c1 is bounded from above by the Lipschitz constant L of ˜f, and it is not difficult to check that U(t,⋅)≥0 in BR for every t∈[0,T): indeed, taking into account the boundary conditions,
ddt(12∫BR(U−)2)=−∫BRU−(∂tU)=−∫BRU−(ΔU+c1(t,x)U)≤−∫BR|∇U−|2+L∫BR(U−)2≤L∫BR(U−)2, |
whence it follows that
ddt(e−2Lt∫BR(U−)2)≤0. |
Therefore, the non-negativity of U(t,⋅) for t∈(0,T) follows from the non-negativity of U(0,⋅). The same argument also shows that V(t,⋅)≥0 for every such t. Using the positivity of U, it is not difficult to prove that U is also uniformly bounded from above: since ˜f(M)=0, we have
∂t(M−U)−Δ(M−U)≥c2(t,x)(M−U), |
where c2 is the bounded function
c2(t,x)={˜f(U(t,x))−˜f(M)U(t,x)−Mif U(t,x)≠00if U(t,x)=M, |
and the same argument used above implies that 0≤U≤M on (0,T)×BR. Similarly, 0≤V≤M. As a consequence, the solution (U,V) can be globally continued in time on (0,+∞). Furthermore, in (2.3) we can replace ˜f with f, since they coincide on [−M−1,M+1].
We also observe that, since U is constant in time on ∂BR, the energy of the solution is non-increasing:
ddtJ((U(t,⋅),V(t,⋅));BR)=∫BR∇U⋅∇Ut+∇V⋅∇Vt+∂1W(U,V)Ut+∂2W(U,V)Vt=∫BR(−ΔU+∂1W(U,V))Ut+(−ΔV+∂2W(U,V))Vt=−∫BRU2t+V2t≤0. | (2.4) |
As in [6], we can now show that AR is positively invariant under the parabolic flow. Let (U,V) be a solution with initial datum in AR. By the symmetry of (2.3), we have that (V(t,Tix),U(t,Tix)) is still a solution. By the symmetry of initial and boundary data, and by uniqueness, such solution must coincide with (U(t,⋅),V(t,⋅)). This means in particular that V(t,x)=U(t,Tix). Likewise, U(t,x1,−x2)=U(t,x1,x2). Notice that the symmetries imply that U−V=0 on ∂Sk. Thus, recalling that SR=Sk∩BR, we have
{∂t(U−V)−Δ(U−V)=c(t,x)(U−V)in (0,+∞)×SRU−V≥0on (0,+∞)×∂SRU−V≥0on {0}×SR, | (2.5) |
where the bounded function c is defined by
c(t,x)={f(U(t,x))−f(V(t,x))U(t,x)−V(t,x)+ΛUp(t,x)Vp(t,x)if U(t,x)≠V(t,x)ΛUp(t,x)Vp(t,x)if U(t,x)=V(t,x). |
The parabolic maximum principle implies that U≥V in SR globally in time, and, in turn, this gives the invariance of AR.
At this point we consider the solution (UR,VR) to (2.3) with initial datum (uR,vR), minimizer of J(⋅,BR) in AR. By minimality in AR and by (2.4), we have
J((uR,vR);BR)≤J((UR(t,⋅),VR(t,⋅));BR)≤J((uR,vR);BR)⟹U2t+V2t≡0, |
and hence UR≡uR and VR≡vR. But then (uR(x),vR(x)) is a (stationary) solution of the parabolic problem (2.3), that is, it solves the stationary problem (2.2), and in addition (uR,vR)∈AR. This completes the proof.
We are ready to complete the:
Proof of Theorem 1.1. First, of all, we discuss the convergence of {(uR,vR):R>1}. Let ρ>1. Since 0≤uR,vR≤M, we have that
|ΔuR(x)|≤maxs∈[0,M]|f(s)|+ΛM2p+1,|ΔvR(x)|≤maxs∈[0,M]|f(s)|+ΛM2p+1. |
Thus interior Lp estimates (see e.g. [10,Chapter 9]), applied in balls of radius 2 centered in points of ¯Bρ with p>N, and the Morrey embedding theorem, imply that there exists C>0 depending only on M and Λ (but independent of R and ρ) such that
‖uR‖C1,α(¯Bρ)+‖vR‖C1,α(¯Bρ)≤Cin Bρ, for all R>ρ+2 | (2.6) |
(for every 0<α<1). By the Ascoli-Arzelà theorem, up to a subsequence {(uR,vR)} converges in C1,α(¯Bρ) to a solution in Bρ, for every 0<α<1. Taking a sequence ρ→+∞, a diagonal selection finally gives (uR,vR)→(u,v) in C1,αloc(R2), up to a subsequence, with (u,v) solution to
{−Δu=f(u)−Λupvp+1in R2−Δv=f(v)−Λup+1vpin R20≤u,v≤Min R2. |
Notice that, by convergence, (u,v) satisfies the symmetry property (ii) in Theorem 1.1, and moreover u−v≥0 in Sk. As in (2.5), for any ρ>0
{−Δ(u−v)=c(x)(u−v)in Sρu−v≥0in Sρ, |
for a bounded function c. Thus, the strong maximum principle implies that either u>v in Sρ, of u≡v in Sρ. Since ρ>0 is arbitrarily chosen, we have that either u>v in S, of u≡v in S. We claim that the latter alternative cannot take place. To prove this claim, we use a comparison argument similar as the one by Cabré and Terra in [7] for the construction of the saddle solution for scalar bystable equations. First of all, we observe that, by symmetry, any function in AR is determined only by its restriction on Sk. Thus the minimality of (uR,vR) can be read as
J((uR,vR),SR)≤J((u,v),SR)∀(u,v)∈AR. |
Let 1<ρ<R−2, and let ξ be a radial smooth cut-off function with ξ≡1 in Bρ−1, ξ≡0 in Bcρ, 0≤ξ≤1. We define
φR(x)=ξ(x)w+k(x)+(1−ξ(x))uR(x)=ξ(x)min{M,αx1−|x2|√2}+(1−ξ(x))uR(x), |
and
ψR(x)=ξ(x)w−k(x)+(1−ξ(x))vR(x)=(1−ξ(x))vR(x), |
where we recall that α=tan(π/(2k)). It is immediate to verify that (φR,ψR) is an admissible competitor for (uR,vR) on SR. Moreover, by (2.6), there exists C>0 such that
‖φR‖W1,∞(Bρ)+‖ψR‖W1,∞(Bρ)≤C∀R>ρ+2. | (2.7) |
By minimality
J((uR,vR),SR)≤J((φR,ψR),SR), |
and since (φR,ψR)=(uR,vR) in SR∖Sρ we deduce that
J((uR,vR),Sρ)≤J((φR,ψR),Sρ)≤J((φR,0),Sρ−1)+C|Sρ∖Sρ−1| | (2.8) |
where we used the global boundedness of {(φR,ψR)} in W1,∞(Bρ), see (2.7). The last term can be easily computed as
|Sρ∖Sρ−1|=πk(ρ2−(ρ−1)2)≤2πkρ. |
For the first term, recalling that F(M)=0, ξ≡1 in Bρ−1, and wk>0 in Sk, we have
∫Sρ−1(12|∇φR|2+F(φR)+F(0))=∫Sρ−1∩{αx1−|x2|<√2M}(12|∇wk|2+F(wk))+∫Sρ−1F(0)≤C|Sρ−1∩{αx1−|x2|<√2M}|+F(0)|Sρ−1|. |
The set
Sk∩{αx1−|x2|<√2M} |
is contained in the (non-disjoint) union of the two strips
{αx1−√2M<x2<αx1, x1>0}∪{−αx1<x2<2√M−αx1, x1>0}=S1∪S2. |
Therefore,
|Sρ−1∩{αx1−|x2|<√2M}|≤|S1∩{0<x1<ρ}|+|S2∩{0<x1<ρ}|=2∫ρ0(∫αx1−√2M+αx11dx2)dx1=2√2Mρ. |
Coming back to (2.8), we conclude that there exists a constant C>0 such that, for every ρ>1 and R>ρ+2,
J((uR,vR),Sρ)≤Cρ+F(0)|Sρ−1| |
for every 1<ρ<R−2, where C>0 is a positive constant independent of both ρ and R. Passing to the limit as R→+∞, we infer by C1loc-convergence that
J((u,v),Sρ)≤Cρ+F(0)|Sρ−1| | (2.9) |
for every ρ>1. Notice that, in this estimate, the leading term as ρ→+∞ is
F(0)|Sρ−1|∼πkF(0)ρ2. |
Suppose now by contradiction that u≡v in Sk. Recalling that 0≤u,v≤M, we have that
J((u,v),Sρ)=∫Sρ|∇u|2+W(u,u)≥∫Sρmins∈[0,M]W(s,s)=mins∈[0,M]W(s,s)|Sρ|∼πkmins∈[0,M]W(s,s)ρ2 |
as ρ→+∞. Comparing with (2.9), we obtain a contradiction for large ρ, thanks to assumption (1.5). Therefore, u>v in Sk. Since u=v on ∂Sk, we also infer that both u and v cannot be constant. The maximum principle implies then that u,v>0 in R2, and from this it is not difficult to deduce that u,v<M: indeed, if u(x0)=M, then x0 is a strict maximum point for u with
Δu(x0)=−f(M)+ΛMpv(x0)p+1=ΛMpv(x0)p+1>0, |
which is not possible. This completes the proof.
The proof of Theorem 1.4 follows the same strategy as the one of Theorem 1.1, being actually a bit simpler. Let m≥2 be a positive integer. By [7,Theorem 1.3]*, under our assumption (1.4) on F the Allen-Cahn equation Δw+f(w)=0 in R2m admits a saddle solution wm, that is a solution satisfying:
*For the existence and the energy estimate in the theorem, it is sufficient that f is locally Lipschitz, rather than C1
(i) w depends only on the variables s and t defined in (1.8);
(ii) wm(s,t)=−wm(t,s);
(iii) wm>0 in O={s>t}.
In addition, |wm|<M in R2m, and
∫BR12|∇wm|2+F(wm)≤CR2m−1for all R>1, | (3.1) |
Now, as in Section 2, we consider the energy functional J((u,v),Ω) defined in (2.1) (in this section Ω⊂R2m), and the set
AR:={(u,v)∈H1(BR,R2)|(u,v)=(w+m,w−m) on ∂BR,v(s,t)=u(t,s),u≥vin OR, 0≤u,v≤Min BR}, |
where OR=O∩BR.
Lemma 3.1. For every R>0, there exists a solution (uR,vR)∈AR to
{−Δu=f(u)−Λ|u|p−1u|v|p+1inBR−Δv=f(v)−Λ|u|p+1|v|p−1vinBRu=w+m,v=w−mon ∂BR. | (3.2) |
The proof is analogue to the one of Lemma 2.1, and is omitted.
Proof of Theorem 1.4. As in the 2-dimensional case, we can prove that up to a subsequence (uR,vR)→(u,v) in C1,αloc(R2) as R→+∞, with (u,v) solution to
{−Δu=f(u)−Λupvp+1−Δv=f(v)−Λup+1vp0≤u,v≤Min R2m. |
By convergence, v(s,t)=u(t,s), u−v≥0 in O, and 0≤u,v≤M in R2m. Also, for every ρ>0
{−Δ(u−v)=c(x)(u−v)in Oρu−v≥0in Oρ, |
for a bounded function c. Thus, the strong maximum principle implies that either u>v in O, of u≡v in O. We claim that the latter alternative cannot take place. Let 1<ρ<R−2, and let ξ be a radial smooth cut-off function with ξ≡1 in Bρ−1, ξ≡0 in Bcρ, 0≤ξ≤1. We define
φR=ξw+m+(1−ξ)uR,ψR=ξw−m+(1−ξ)vR. |
This is an admissible competitor in AR, which coincides with (uR,vR) on BR∖Bρ. Therefore, by minimality and recalling (3.1), we have
J((uR,vR),Bρ)≤J((w+m,w−m),Bρ−1)+C|Bρ∖Bρ−1|≤E(wm,Bρ−1)+∫Bρ−1F(0)+Cρ2m−1≤Cρ2m−1+F(0)|Bρ−1|. | (3.3) |
If, by contradiction, u≡v in O, then we have that
J((u,v),Bρ)=∫Bρ|∇u|2+W(u,u)≥∫Bρminσ∈[0,M]W(σ,σ)=minσ∈[0,M]W(σ,σ)|Bρ|. |
Comparing with (2.9), we obtain a contradiction for large ρ, thanks to assumption (1.5). Thus, u>v in O, and the conclusion of the proof is straightforward.
In this appendix we consider the scalar Allen-Cahn equation
−Δw=f(w)in R2, | (A.1) |
and we prove the following result:
Theorem A.1. Let f∈C0,α(R) be odd, and suppose that its anti-primitive F satisfies (1.4). Then, for every k∈N, there exists a solution wk having the following properties:
(i) −M<wk<M in R2;
(ii) wk∘Ti=−wk for every i=1,…,k, and wk(x1,−x2)=wk(x1,x2) in R2;
(iii) wk>0 in Sk.
Moreover, there exists a constant C>0 (possibly depending on k) such that
∫BR(12|∇w|2+F(w))≤CRfor every R>0. | (A.2) |
Remark A.2. The existence of a solution wk with the properties (i)–(iii) was established by Alessio, Calamai and Montecchiari in [5]. In [5] the authors also obtained a more precise description of the asymptotic behavior of wk at infinity. On the other hand, the validity of the estimate (A.2) was unknown.
In order to show that wk fulfills (A.2), we provide an alternative existence proof with respect to the one in [5]. It is tempting to conjecture that the solutions given by Theorem A.1, and those found in [5], coincide.
Our alternative proof is strongly inspired by [7], where Cabré and Terra proved existence of solutions in R2m to (A.1) vanishing on the Simon's cone (when restricted to the case m=1 - i.e., when we consider (A.1) in the plane - their result establishes the existence of the solution w2). We first prove the existence of a solution wR=wk,R to (A.1) in the ball BR, for every R>0, by variational argument. Passing in a suitable way to the limit as R→+∞, we shall obtain a solution in the whole plane R2 having the desired energy estimate.
The main simplification with respect to the proof of Theorem 1.1 stays in the fact that, dealing with a single equation, we will not need an auxiliary parabolic problem, but we will be able to prove the existence of a solution in BR with the desired symmetry properties directly by variational methods.
The proof of Theorem A.1 takes the rest of this appendix. Let us fix k. For any Ω⊂R2 open, and for every w∈H1(Ω), we define
E(w,Ω):=∫Ω(12|∇w|2+F(w)). |
For R>0, we consider SR:=BR∩Sk and the set
HR:={w∈H10(SR):w(x1,−x2)=w(x1,x2) a.e. in SR}. |
Lemma A.3. For every R>0, problem
{−Δw=f(w)in BRw(x1,−x2)=w(x1,x2)in BRw=0on ∂BR, | (A.3) |
has a solution wR, satisfying (ii) in Theorem A.1. Moreover, −M≤wR≤M in BR, wR≥0 in SR, and
E(wR,SR)=min{E(w,SR): w∈HR}. |
Proof. At first, we search a solution to the auxiliary problem
{−Δw=f(w)in SRw∈H10(SR),w≥0in SRw(x1,x2)=w(x1,−x2)in SR, | (A.4) |
by minimizing the function E(w,SR) in H. The existence of a minimizer follows easily by the direct method of the calculus of variations. Since E(w,SR)=E(|w|,SR), it is not restrictive to suppose that wR≥0. Also, since E(min{w,M},SR)≤E(w,SR) by assumption (1.4), we can suppose that wR≤M. Clearly, wR solves the first equation in (A.4) in the set SR∖{θ=0}. The fact that wR is also a solution across SR∩{θ=0} (thus a solution in SR) follows by the principle of symmetric criticality (see e.g., [14,Theorem 1.28] for a simple proof of this result, sufficient to our purposes).
Notice that wR∈C1({0<r<R,−π/2k≤θ≤π/2k}) by standard elliptic regularity. Thus, we can reflect wR 2k times in an odd way across ℓ1,…,ℓk, obtaining a solution in BR∖{0}. To see that wR is in fact a solution in BR, we take a smooth function ηδ∈C∞(¯BR) with ηδ≡0 in Bδ, ηδ≡1 in B2δ∖Bδ, and |∇ηδ|≤C/δ in BR. Then, for every φ∈C∞c(BR), we have
∫BR∇wR⋅∇(φηδ)−∫BRf(wR)φηδ=0, |
since φηδ is an admissible test function in BR∖{0}. Passing to the limit as δ→0+, we deduce that
∫BR∇wR⋅∇φ−∫BRf(wR)φ=0∀φ∈C∞c(BR), |
that is, wR is a weak solution to (A.1) in BR. This completes the proof.
Proof of Theorem A.1. We wish to pass to the limit as R→+∞ and obtain a solution in the whole plane R2 as limit of the family {wR}. As in the previous sections, by elliptic estimates we have that, up to a subsequence wR→w in C1,αloc(R2) as R→∞, for every α∈(0,1). The limit w inherits by wR the symmetry property (ii) in Theorem A.1. Moreover, w≥0 in the sector S=Sk, and |w|≤M in the whole plane R2. Actually, the strict inequality |w|<M holds, by the strong maximum principle. To complete the proof of the theorem, it remains then to show that w satisfies estimate (A.2), and that w≢0.
As in the proof of Theorem 1.1, {wR} has a uniform gradient bound: there exists C>0 (independent of R) such that
‖∇wR‖L∞(BR−1)≤C∀R>1. | (A.5) |
For an arbitrary ρ>1, let now R>ρ+2, and let ξ∈C∞c(Bρ), with ξ≡1 in Bρ−1. We consider the following competitor for wR:
φR(x)=ξ(x)min{αx1−|x2|√2,M}+(1−ξ(x))wR(x). |
Notice that this is the same type of competitor we used in Theorem 1.1. By minimality
E(wR,SR)≤E(φR,SR), |
and since wR=φR in SR∖Sρ we deduce that
∫Sρ(12|∇wR|2+F(wR))≤∫Sρ(12|∇φR|2+F(φR))≤∫Sρ−1(12|∇φR|2+F(φR))+C|Sρ∖Sρ−1|, | (A.6) |
where we used the global boundedness of {φR} in W1,∞(Bρ), see (A.5). At this point we can proceed as in the conclusion of the proof of Theorem 1.1: the right hand side in (A.6) can be estimated by Cρ, with C independent of ρ. Thus, we conclude that there exists a constant C>0 such that, for every ρ>1 and R>ρ+2,
E(wR,Sρ)=∫Sρ(12|∇wR|2+F(wR))≤Cρ, |
Passing to the limit as R→+∞, we infer by C1loc-convergence that
E(w,Sρ)≤Cρ, |
which implies, by symmetry, the estimate (A.2).
Suppose finally that w≡0. Then the energy estimate (A.2) would give for every ρ>1
πF(0)ρ2=E(0,Bρ)≤Cρ, |
which is not possible if ρ is sufficiently large. This proves that w≢0, and completes the proof of Theorem A.1.
The author declares no conflict of interest.
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