Qualitative properties of solutions to mixed-diffusion bistable equations
QQualitative properties of solutions tomixed-diffusion bistable equations
Denis Bonheure, Juraj F¨oldes & Alberto Salda˜na ∗† Abstract
We consider a fourth-order extension of the Allen-Cahn model withmixed-diffusion and Navier boundary conditions. Using variational andbifurcation methods, we prove results on existence, uniqueness, positivity,stability, a priori estimates, and symmetry of solutions. As an application,we construct a nontrivial bounded saddle solution in the plane.
Keywords: higher-order equations, bilaplacian, extended Fisher-Kolmogorov equation
We study the following fourth-order equation with Navier boundary conditions∆ u − β ∆ u = u − u in Ω ,u = ∆ u = 0 on ∂ Ω , (1.1)where Ω ⊂ R N , N ≥ β >
0. Such boundary condi-tions are relevant in many physical contexts [23] and they permit to rewrite (1.1)as a second order elliptic system with Dirichlet boundary conditions. In our bestknowledge (1.1) was analyzed only for N = 1, see [38] and references therein.In this paper we present results on existence, uniqueness, positivity, stability, apriori estimates, regularity, and symmetries of solutions in higher-dimensionaldomains when β ≥ √
8. The case β < √ ∂ t u + γ ∆ u − ∆ u = u − u in Ω × [0 , ∞ ) , γ > , (1.2) ∗ D´epartement de Math´ematique, Universit´e libre de Bruxelles, Campus de la Plaine CP213, Bd. du Triomphe, 1050 Bruxelles, Belgium, [email protected], [email protected],[email protected]. † The authors are supported by MIS F.4508.14 (FNRS). The first author is also supportedby INRIA – Team MEPHYSTO & ARC AUWB-2012-12/17-ULB1-IAPAS a r X i v : . [ m a t h . A P ] M a r hich was first proposed in 1988 by Dee and van Saarloos [17] as a higher-ordermodel for physical, chemical, and biological systems. The right-hand side of(1.2) is of bistable type, meaning it has two constant stable states u ≡ ± u ≡
0, see [17]. The distinctive feature ofthis model is that the structure of equilibria is richer compared to its secondorder counterpart ∂ t u − ∆ u = u − u in Ω × [0 , ∞ ) , (1.3)giving rise to more complicated patterns and dynamics. The equation (1.3) isrelated to the Fisher-KPP equation ( Fisher-Kolmogorov-Petrovskii-Piscunov orsometimes simply called
Fisher-Kolmogorov equation) proposed by Fisher [20]to model the spreading of an advantageous gene and mathematically analyzedby Kolmogorov, Petrovskii, and Piscunov [31].The equilibria of (1.3) satisfy the well-known Allen-Cahn or real Ginzburg-Landau equation − ∆ u = u − u in Ω (1.4)with the associated energy functional12 (cid:90) |∇ u | dx + 14 (cid:90) ( | u | − dx, u ∈ H (Ω) , (1.5)where H n (Ω) = W n, (Ω) denotes the usual Sobolev space. This functional isused to describe the pattern and the separation of the (stable) phases ± u describes the point-wise state of the material or the fluid. The constant equilibria correspondingto the global minimum points ± ( | u | − are called thepure phases, whereas other configurations u represent mixed states, and orbitsconnecting ± (cid:90) [( ∇ u ) + g ( u ) |∇ u | + W ( u )] dx, u ∈ H (Ω) , where ∇ u denotes the Hessian matrix of u . It appears as a simplificationof a nonlocal model [28] analyzed in dimension one in [9, 14, 35, 37] and inhigher dimensions in [13, 22, 27]. In [27], the Hessian ∇ u is replaced by ∆ u asa simplification of the model and it was also proposed as model for phase-fieldtheory of edges in anisotropic crystals in [44]. Finally, we also mention the study in the original model, the nonlinearity u is replaced by u
2f amphiphilic films [34] and the description of the phase separations in ternarymixtures containing oil, water, and amphiphile, see [25], where the scalar orderparameter u is related to the local difference of concentrations of water and oil.These models motivate the study of the stationary solutions of (1.2). Afterscaling, equilibria of (1.2) in the whole space solve∆ u − β ∆ u = u − u in R N . (1.6)We refer to (1.6) as the Extended-Fisher-Kolmogorov equation (EFK) for β >
Swift-Hohenberg equation for β <
0. This fourth-order model hasbeen mostly investigated for N = 1, i.e., u (cid:48)(cid:48)(cid:48)(cid:48) − βu (cid:48)(cid:48) + u − u = 0 in R . (1.7)When β ∈ [ √ , ∞ ), there is a full classification of bounded solutions of (1.7),which mirrors that of the second order equation. Specifically, each boundedsolution is either constant, a unique kink (up to translations and reflection), ora periodic solution indexed by the first integral, whereas there are no pulses.For β ∈ [0 , √ √ u = ±
1. The proof of these results rely on purely one-dimensionaltechniques, for instance, stretching arguments, phase space analysis, shootingmethods, first integrals, etc. For more details on the one-dimensional EFK werefer to [8, 38] and the references therein.For N ≥ Paneitz-Branson operator , see forinstance [10, 2], and as a special case of some elliptic systems, see e.g. [40].In the context of Schr¨odinger equations in nonlinear optics, the fourth orderoperator in (1.6) is used to model a mixed dispersion, see e.g. [19, 7].To introduce our main results denote H := H (Ω) ∩ H (Ω) (1.8)the Sobolev space associated with Navier boundary conditions (see [23] for asurvey on Navier and other boundary conditions) and let J β : H → R be theenergy functional given by J β ( u ) := (cid:90) Ω (cid:18) | ∆ u | β |∇ u | u − u (cid:19) dx for u ∈ H. (1.9)Any critical point u of J β is a weak solution of (1.1), that is, u satisfies (cid:90) Ω ∆ u ∆ v + β ∇ u ∇ v + ( u − u ) v dx = 0 for all v ∈ H. u is stable if J (cid:48)(cid:48) β ( u )[ v, v ] = (cid:90) Ω | ∆ v | + β |∇ v | + (3 u − v dx ≥ v ∈ H and strictly stable if the inequality is strict for any v (cid:54)≡ u + := max { u, } , absolute value, or rearrangements of functions sincethey do not belong to H (Ω) in general. Furthermore, the validity of maximumprinciples (or, more generally, positivity preserving properties) is a delicate issuein fourth-order problems and does not hold in general.For the rest of the paper, λ (Ω) = λ > − ∆ in Ω and a hyperrectangle is a product of N bounded nonemptyopen intervals.The following is our main existence and uniqueness result, for the second-order counterpart, we refer to [3]. Theorem 1.1.
Let β > and Ω ⊂ R N with N ≥ be a smooth bounded domainor a hyperrectangle. If λ + βλ ≥ , then u ≡ is the unique weak solution of (1.1) . If λ + βλ < , (1.10) then1) there is ε > such that (1.1) admits a positive classical solution for each β ∈ ( ¯ β − ε, ¯ β ) , where ¯ β = − λ λ .
2) for each β ≥ √ √ − / there is a positive classical solution u of (1.1) such that (cid:107) u (cid:107) L ∞ (Ω) ≤ β ( β ) and ∆ u < β u in Ω .3) for every β ≥ √ there exists a unique positive solution u of (1.1) . More-over, this solution is strictly stable and satisfies (cid:107) u (cid:107) L ∞ (Ω) ≤ . The smoothness assumptions on Ω are needed for higher-order elliptic regu-larity results. We single out hyperrectangles to use them in the construction ofsaddle solutions and patterns. Indeed, by reflexion, positive solutions of (1.1)in regular polygons that tile the plane give rise to periodic planar patterns.The quantities involved in Theorem 1.1 2) are of a technical nature. Observethat (1.10) holds for all big enough domains. As mentioned above, the threshold √ u = ±
1. For β ≥ √ β < √ u being boundedby 1. Such oscillations around one can be proved for radial global minimizersarguing as in [5, proof of Theorem 6]. Intuitively, for β ≥ √ β ∈ (0 , √
8) the bilaplacian increases its influenceresulting in a much richer and complex set of solutions. We present numericalapproximations of positive solutions using minimization techniques in Figure1 below.Figure 1: Numerical approximation of the global minimizer of (1.9) for Ω = [0 , with β = 0 . β = 4 (right). The dotted lines represent the level set { u = 1 } . Note that Theorem 1.1 1) holds for any β >
0, but only for appropriatevalues of λ .Theorem 1.1 follows directly from Theorem 5.2 and Theorem 11.1. Theproof is based on variational and bifurcation techniques. For the variational part(Theorem 5.2), we minimize an auxiliary problem for which we can guaranteethe sign and L ∞ bounds of global minimizers. Next, we prove that globalminimizers of the auxiliary problem are solutions to (1.9). The uniquenessis proved using stability, maximum principles, and bifurcation from a simpleeigenvalue (Theorem 11.1). After the paper was accepted Tobias Weth suggestedus an alternative proof for uniqueness, see Remark 6.5.We depict a numerical approximation of the bifurcation branch in Figure 2.This branch can be continued even for β <
0, see Section 13 for an example ofsuch a branch and we refer again to [38] for a survey on (1.7) for β <
0. See alsoRemark 11.2 for a discussion on the explicit values of the bifurcation points.We now use the solution given by Theorem 1.1 to construct a saddle solutionfor (1.1). We call u a saddle solution if u (cid:54)≡ u ( x, y ) xy ≥ x, y ) ∈ R . See Figure 3 below. Theorem 1.2.
For β ≥ (cid:113) √ − the problem ∆ u − β ∆ u = u − u in R hasa saddle solution. We refer to [11, 16] for more information on saddle solutions for second orderbistable equations. Computed with FreeFem++ [26] and Mathematica 10.0, Wolfram Research Inc., 2014. Computed with AUTO-07P [18].
Numerical approximation of the bifurcation branch and some radial solutions.Here Ω is a ball in R of radius 240 . Figure 3:
Saddle solutions for β < √ β ≥ √ { u = 1 } . In the following we explore properties of positive classical solutions with aspecial focus on stability and symmetry properties.
Theorem 1.3.
Let Ω be a ball or an annulus and let u be a stable solution of (1.1) with β > √ − λ such that (cid:107) u (cid:107) L ∞ (Ω) ≤ . Then u is a radial function. Note that Theorem 1.3 does not assume positivity of solutions. We believethat the restriction on β is of a technical nature, but it is needed in our approach,see also Remark 7.1.More generally, for reflectionally symmetric domains we have the following.We say that a domain is convex and symmetric in the e − direction if for every x = ( x , x , . . . , x N ) ∈ Ω we have { ( tx , x , . . . , x N ) : t ∈ [ − , } ⊂ Ω . Proposition 1.4.
Let β ≥ √ and let Ω ⊂ R N be a hyperrectangle or a boundedsmooth domain which is convex and symmetric in the e − direction. Then, anypositive solution of (1.1) satisfies u ( x , x , . . . , x N ) = u ( − x , x , . . . , x N ) for all x = ( x , . . . , x N ) ∈ Ω ,∂ x u ( x ) < for all x ∈ Ω such that x > . β ∈ (0 , √
8) thesolution oscillates when close to 1 in big enough domains, in particular it is notmonotone, although it may still be symmetric. In balls , Proposition 1.4 impliesTheorem 1.3 with the stability assumption replaced by positivity of the solution.In the following we focus on properties of particular solutions, namely, globalminimizers. The next theorem states positivity of global radial minimizers, thatis, functions u ∈ H r := { v ∈ H : v is radial in Ω } such that J β ( u ) ≤ J β ( v ) forall v ∈ H r . Theorem 1.5.
Let Ω be a ball or an annulus, β > , (1.10) hold, and let u ∈ H r be a global radial minimizer of (1.9) with (cid:107) u (cid:107) L ∞ (Ω) ≤ . Then ∂ r u does notchange sign if Ω is a ball and ∂ r u changes sign exactly once if Ω is an annulus. Note that global minimizers satisfy the required bound for β ≥ √
8, seeProposition 4.1 below. The proof of Theorem 1.5 relies on a new flippingtechnique that preserves differentiability while diminishing the energy and it istherefore well suited for variational fourth-order problems. Theorem 1.5 clearlyimplies that global radial minimizers do not change sign and we conjecture thatthis property holds in general, even for β < √
8. For β large we can relax theassumptions on the solution, as stated in the following. Corollary 1.6.
Let Ω be a ball or an annulus, β > √ − λ , (1.10) hold, andlet u be a global minimizer of (1.9) in H . Then u is radial and does not changesign in Ω . Moreover, ∂ r u does not change sign if Ω is a ball while ∂ r u changessign exactly once if Ω is an annulus.Proof. From β > √ − λ and (1.10) follows β ≥ √ (cid:107) u (cid:107) L ∞ (Ω) ≤
1. Theorem 1.3 therefore implies that u isradial. Then u is in particular the global minimizer in H r and the corollaryfollows from Theorem 1.5.As last theorem we present a uniqueness and continuity result for γ ∆ u − ∆ u = u − u in Ω ,u = ∆ u = 0 on ∂ Ω , (1.11)when γ → Theorem 1.7.
Let Ω ⊂ R N be a smooth bounded domain with the first Dirichleteigenvalue λ < . Let γ ≥ and u γ be a global minimizer in H of (cid:90) Ω (cid:18) γ | ∆ u | |∇ u | u − u (cid:19) dx. (1.12) There is δ (Ω) = δ > such that, for all γ ∈ [0 , δ ] , u γ is the unique globalminimizer in H and u γ > in Ω . Moreover, the function [ − δ, δ ] → C (Ω); γ (cid:55)→ u γ is continuous and u is the global minimizer of (1.12) in H (Ω) with γ = 0 . γ , see Lemma 2.3.The paper is organized as follows. In Sections 2, 3, and 4 we prove crucialauxiliary lemmas for obtaining a priori estimates of solutions. Section 5 containsthe proof of the variational part of Theorem 1.1. Our study of the stability ofsolutions is contained in Section 6 and 7. In Section 8 we present our flippingmethod and prove Theorem 1.5. The proof of Proposition 1.4 can be found inSection 9 and the saddle solution is constructed in Section 10. The bifurcationresult involved in the proof of Theorem 1.1 is contained in Section 11 and thecontinuity result, Theorem 1.7, is proved in Section 12. Finally, in Section 13we include two numerical approximations of bifurcation branches.To close this Introduction, let us mention that the case β < √ Acknowledgements:
We thank Guido Sweers and Pavol Quitter for very help-ful discussions and suggestions related to the paper. We also want to thankChristophe Troestler for introducing us to FreeFem++.
We prove a very helpful lemma that allows us to obtain a priori bounds onclassical solutions. These arguments were used simultaneously in [6, Lemma3.1.] to obtain similar a priori estimates in R N .For the rest of the paper we denote by C ( ¯Ω) the space of continuous func-tions in Ω vanishing on ∂ Ω. Lemma 2.1.
Let Ω ⊂ R N be a bounded domain, β > , f : R → R a continuousfunction satisfying f (0) = 0 , and let u ∈ C (Ω) ∩ C ( ¯Ω) be a solution of ∆ u − β ∆ u = f ( u ) in Ω such that ∆ u ∈ C ( ¯Ω) . Set u := max Ω u, u := min Ω u, and g : R → R given by g ( s ) := β f ( s ) + s . Then u ≤ max [ u,u ] g and u ≥ min [ u,u ] g. (2.1) Moreover,1. If u ≤ max [0 ,u ] g and f ≤ in (1 , ∞ ) , then u ≤ max [0 , g.
2. If u ≥ min [ u, g and f ≥ in ( −∞ , − , then u ≥ min [ − , g. Proof.
Let w ∈ C (Ω) ∩ C ( ¯Ω) be given by w ( x ) := − ∆ u ( x ) + β u ( x ). Weprove only the second inequality in (2.1) as the first one follows similarly. Fix x , ξ ∈ Ω such that u ( x ) = u and w ( ξ ) = min Ω w. If x ∈ ∂ Ω, then u = u ( x ) = 0 = g ( u ( x )) ≥ min [ u,u ] g ξ ∈ ∂ Ω, then − ∆ u + β u ≥ u = 0 and the second inequality in(2.1) follows. If x , ξ ∈ Ω, then w ( ξ ) ≤ w ( x ) = − ∆ u ( x ) + β u ( x ) ≤ β u ( x ).Since − ∆ w ( ξ ) ≤ f ( u ( ξ )) + β u ( ξ ) = ∆ u ( ξ ) − β ∆ u ( ξ ) + β u ( ξ )= − ∆ w ( ξ ) + β w ( ξ ) ≤ β u ( x ) , that implies u ≥ min [ u,u ] g .We now prove claim 2. only as claim 1. is analogous. Assume u ≤ − f ≥ −∞ , − g ( s ) ≥ s in ( −∞ , − [ u, − g ≥ min [ u, − s = u . Thus, if u ≥ min [ u, g then u ≥ min [ − , g as claimed. In this section we prove two regularity results. The first one is a rather standardapplication of known arguments in the fourth order setting and we includedetails for reader’s convenience. The second lemma is more subtle and thecrucial point is the dependencies of the constants involved.Recall the definition of the space H given in (1.8). Lemma 2.2.
Let Ω ⊂ R N , N ≥ be a smooth bounded domain or hyperrect-angle and fix β, γ > , and f ∈ L ∞ (Ω) . Let u ∈ H be a weak solution of γ ∆ u − β ∆ u = f , that is, (cid:90) Ω γ ∆ u ∆ φ + β ∇ u ∇ φ − f φ dx = 0 ∀ φ ∈ H. (2.2) Then for each p > one has u ∈ W ,p (Ω) ∩ C (Ω) with u = ∆ u = 0 on ∂ Ω and (cid:107) u (cid:107) W ,p (Ω) ≤ C (cid:107) f (cid:107) L ∞ (Ω) , where C = C ( β, γ, p, Ω) . In addition, if f ∈ C α (Ω) for some α ∈ (0 , , then u ∈ C ,α (Ω) with ∆ u ∈ C (Ω) and (cid:107) u (cid:107) C ,α (Ω) ≤ ˜ C (cid:107) f (cid:107) C α (Ω) , (2.3) where ˜ C = ˜ C ( β, γ, p, Ω) .Proof. Throughout the proof
C > β , γ , p , and Ω. 9ssume first that Ω ⊂ R N , N ≥ , is a smooth bounded domain. Bythe Riesz representation theorem there are weak solutions ¯ u, ¯ v ∈ H (Ω) of theequations − ∆¯ v + βγ ¯ v = f and − γ ∆¯ u = ¯ v in Ω . (2.4)Fix p >
1. Then, by [24, Lemma 9.17] and [33, Ch 9 Sec 2 Thm 3], (cid:107) ¯ u (cid:107) W ,p (Ω) ≤ C (cid:107) ¯ v (cid:107) W ,p (Ω) ≤ C (cid:107) f (cid:107) L p (Ω) , which gives the first estimate in thestatement.By embedding theorems, we have ¯ u ∈ C (Ω) and ¯ u = 0 = ¯ v = − γ ∆¯ u on ∂ Ω. Let u ∈ H be as in the statement. By integration by parts, ¯ u satisfies (2.2)and u ≡ ¯ u in Ω, since weak solutions of (2.2) are unique. Finally, if f ∈ C α (Ω),by Schauder estimates [32, Thm. 6.3.2], ¯ v ∈ C ,α (Ω) , u ∈ C ,α (Ω) for some α ∈ (0 , , and (2.3) holds.Now without loss of generality assume that Ω is a hyperrectangle of theform Ω = (cid:81) Ni =1 [0 , l i ] for some l i > , i = 1 , . . . , N . Let ¯ u, ¯ v ∈ H (Ω) be weaksolutions of (2.4). Using odd reflections and the Dirichlet boundary conditionswe extend ¯ u and ¯ v to ˜Ω = (cid:81) Ni =1 [ − l i , l i ] and obtain weak solutions of (2.4)(with the odd extension of f as well) defined in ˜Ω. Then interior regularity [33,Theorem 1 Sec 4 Ch 9] implies that for any Ω ⊂ Ω ⊂ ˜Ω (cid:107) ¯ u (cid:107) W ,p (Ω ) ≤ C ( (cid:107) f (cid:107) L ∞ (Ω ) + (cid:107) ¯ u (cid:107) L (Ω ) ) . (2.5)Note that we replaced L p by L on the right hand side, which can be doneusing Sobolev embeddings and iteration in (2.5). Also, (cid:107) ¯ u (cid:107) L (˜Ω) ≤ C (cid:107) f (cid:107) L (˜Ω) by testing (2.4) by ¯ v and ¯ u respectively and using standard estimates.As before, u ≡ ¯ u in Ω by integration by parts and uniqueness of weaksolutions. Finally, (2.3) follows analogously from interior Schauder estimates[32, Thm. 7.11]. Lemma 2.3.
Let Ω be a smooth domain, β, γ > , and let u ∈ H ∩ L ∞ (Ω) be a weak solution of γ ∆ u − β ∆ u = u − u in H . For every p > there is γ = γ ( β, p, Ω) > and C = C ( β, p, Ω) such that if γ ∈ (0 , γ ) then (cid:107) u (cid:107) W ,p (Ω) ≤ C (1 + (cid:107) u (cid:107) L ∞ (Ω) ) . (2.6) Proof.
In this proof, C denotes different positive constants which depend onΩ , β , and p , but are independent of γ .Since u ∈ L ∞ (Ω) we have by bootstrap and Lemma 2.2 that u ∈ C ,α (Ω)with ∆ u ∈ C (Ω). Then ( u, v ) with v = − γ ∆ u and f = u − u solves (2.4) inthe classical sense. By [33, Ch. 8. Sec. 5. Thm 6], there is γ = γ ( β, p, Ω) > γ ∈ (0 , γ ), (cid:107) v (cid:107) W ,p (Ω) ≤ C (1 + (cid:107) u (cid:107) L ∞ (Ω) ) . On the other hand, − β ∆ u = u − u − γ ∆ u = u − u + ∆ v and by [24, Lemma9.17] for every p ∈ (0 , ∞ ) we have (cid:107) u (cid:107) W ,p (Ω) ≤ C (cid:107) u − u + ∆ v (cid:107) L p (Ω) ≤ C (1 + (cid:107) u (cid:107) L ∞ (Ω) ) . (2.7)10n particular, (cid:107) ∆ u (cid:107) L p (Ω) ≤ C (1 + (cid:107) u (cid:107) L ∞ (Ω) ) . Set w = ∆ u. Since u ∈ C ,α (Ω) , we have that w is a weak solution of γ ∆ w − β ∆ w = f := ∆( u − u ) = w − u |∇ u | − u w in Ωwith w = 0 on ∂ Ω. Moreover, ∆ w = ∆ u = γ ( u − u + β ∆ u ) = 0 on ∂ Ω, since u ∈ C ,α (Ω) and u = ∆ u = 0. Observe that (cid:107) f (cid:107) L p (Ω) ≤ C (1 + (cid:107) u (cid:107) L ∞ (Ω) ) . Then we can repeat the argument for w and f instead of u and f respectivelyto obtain that w ∈ C ,α (Ω) and (cid:107) w (cid:107) W ,p (Ω) ≤ C (1 + (cid:107) u (cid:107) L ∞ (Ω) ) . (2.8)Then ξ := ∆ w ∈ C ,α (Ω) , (cid:107) ξ (cid:107) L p (Ω) ≤ C (1 + (cid:107) u (cid:107) L ∞ (Ω) ) , and ξ is a weak solutionof γ ∆ ξ − β ∆ ξ = f := ∆ f and ξ = 0 on ∂ Ω. Additionally, ∆ ξ = ∆ w = γ ( f + β ∆ w ) = 0 on ∂ Ω , where we used u = ∆ u = w = ∆ w on ∂ Ω. Herewe are fundamentally using that f vanishes on ∂ Ω. Note that (cid:107) f (cid:107) L p (Ω) ≤ C (1 + (cid:107) u (cid:107) L ∞ (Ω) ). Thus we can iterate the above procedure one more time with u and f replaced by ξ and f respectively, to obtain (cid:107) ξ (cid:107) W ,p (Ω) ≤ C (1 + (cid:107) u (cid:107) L ∞ (Ω) ) . (2.9)Note that we cannot iterate anymore since f is not vanishing on ∂ Ω, andtherefore we cannot obtain boundary conditions for higher derivatives. Finally,(2.6) follows by (2.7), (2.8), (2.9), and [33, Ch.9 Sec. 2 Thm 3].
For any β > C β := (cid:114) β , M β := 1 β (cid:18) β (cid:19) , K := (cid:115) √ − ≈ . . (3.1) Remark 3.1.
We now state some facts that will be useful later in the proofs.i) C β > is the unique positive root of h ( s ) = β ( s − s ) + s (cf. Lemma2.1). In particular, h > in (0 , C β ) . ii) M β ≥ is the maximum value of h in (0 , ∞ ) iii) If β ≥ √ then h is increasing on (0 , .iv) M β ≤ C β if β ≥ K . Lemma 3.2.
Let Ω ⊂ R N be a bounded domain, β > , and let u be a classicalsolution of (1.1) .i) If u is nonnegative in Ω , then (cid:107) u (cid:107) L ∞ (Ω) ≤ M β . i) If β ≥ √ and u is nonnegative in Ω , then (cid:107) u (cid:107) L ∞ (Ω) ≤ . iii) If β ≥ √ and (cid:107) u (cid:107) L ∞ (Ω) < (cid:113) β + 1 , then (cid:107) u (cid:107) L ∞ (Ω) ≤ . Proof.
We use the notation of Lemma 2.1 with f ( s ) = s − s . We prove claim iii ) first. Assume without loss of generality that (cid:107) u (cid:107) L ∞ (Ω) = u < (cid:113) β + 1 . Bydirect computation g ( s ) < − s for s ∈ ( − (cid:113) β + 1 , s ∈ [ − u, g ( s ) < max s ∈ [ − u, − s = u. (3.2)Moreover, by Lemma 2.1, u ≤ max [ − u,u ] g ≤ max [ − u,u ] g. Then, in virtue of (3.2), we get u ≤ max [0 ,u ] g. It follows from Lemma 2.1 and Remark 3.1 iii) that u ≤ max [0 , g = 1as claimed.Now, let β > u ≥
0. By Lemma 2.1 and Remark 3.1, u ≤ max [0 , ∞ ] g = M β and the first claim follows. If β ≥ √ , then g is nondecreasing in[0 ,
1] and, by Lemma 2.1, u ≤ max [0 , g = g (1) = 1 . Recall the definition of K , M β , and C β given in (3.1) and of H given in (1.8). Proposition 4.1.
Let v be a global minimizer of (1.1) in H . Then (cid:107) v (cid:107) L ∞ (Ω) ≤ C β if β ≥ K and (cid:107) v (cid:107) L ∞ (Ω) ≤ if β ≥ √ . When β ∈ (0 , √
8) we expect that global minimizers are not bounded by 1in big enough domains, see Figure 1 and Figure 2.
Proof.
Assume β ≥ K , let J β be as in (1.9), and let v denote a global minimizerin H, i.e. J β ( v ) ≤ J β ( w ) for all w ∈ H. We show first that v ∈ L ∞ (Ω). Let f : R → R , f ( s ) := (cid:40) s − s if s ∈ [ − C β , C β ] , sign( s )( C β − C β ) = − β sign( s ) C β otherwise , and let J : H → R ; J ( u ) := (cid:90) Ω | ∆ u | β |∇ u | − F ( u ) dx, (4.1)where F ( s ) := (cid:82) s f ( t ) dt. Let u be the global minimizer of J in H. By Lemma2.2, u is a classical solution of∆ u − β ∆ u = f ( u ) in Ω ,u = ∆ u = 0 on ∂ Ω . (4.2)12sing Lemma 2.1 (and its notation), since g ≥ , ∞ ), we have u ≥ min [ u,u ] g =min [ u, g , and consequently u ≥ min [ − , g ≥ − M β . If β ≥ √
8, we additionally have u ≥ min [ − , g = − u by − u and noting that β ≥ K we conclude (cid:107) u (cid:107) L ∞ (Ω) ≤ M β ≤ C β and (cid:107) u (cid:107) L ∞ (Ω) ≤ β ≥ √
8. In particular f ( u ( x )) = u ( x ) − u ( x ) for x ∈ Ω, and therefore J ( u ) = J β ( u ) ≥ J β ( v ) ≥ J ( v ) , where the last inequality is strict if (cid:107) v (cid:107) L ∞ (Ω) > C β , a contradiction to theminimality of u . Thus v ∈ L ∞ (Ω) and, by Lemma 2.2, v is a classical solutionof (4.2). If β ≥ √ (cid:107) v (cid:107) L ∞ (Ω) >
1, then, by Lemma 3.2 part iii ) , (cid:107) v (cid:107) L ∞ (Ω) ≥ (cid:113) β + 1 > C β and we obtain a contradiction as above. Recall the definition of K , M β , and C β given in (3.1). Lemma 5.1.
Let Ω ⊂ R N be a bounded domain, β > ,f : R → R ; f ( s ) := − β s if s < ,s − s if s ∈ [0 , C β ] ,C β − C β = − β C β if s > C β , (5.1) and let u be a classical solution of (4.2) for this choice of f . Then ≤ u ≤ M β . Moreover, if β ≥ √ , then u ≤ . In particular, if β ≥ K , then u is a classicalsolution of (1.1) .Proof. By Lemma 2.1 (and using the same notation) u ≥ min R g ≥ C β , that is, u ≥ . On the other hand, again by Lemma 2.1we have that u ≤ max [0 ,u ] g = max [0 , g = M β . If β ≥ √ , then g is nondecreasing in [0 , u ≤ max [0 ,u ] g implies,by Lemma 2.1, that u ≤ max [0 , g = g (1) = 1 . Finally, if β ≥ K , then M β ≤ C β , and thus f ( u ( x )) = u ( x ) − u ( x ) for all x ∈ Ω , that is, u solves (1.1). Theorem 5.2.
Let β > and Ω ⊂ R N with N ≥ be a smooth bounded domainor a hyperrectangle. If λ + βλ ≥ , then u ≡ is the unique weak solution of (1.1) . If λ + βλ < , then for β ≥ K there is a positive classical solution u of (1.1) such that (cid:107) u (cid:107) L ∞ (Ω) ≤ M β and ∆ u < β u in Ω . Additionally, if β ≥ √ then (cid:107) u (cid:107) L ∞ (Ω) ≤ . roof. Let λ + βλ ≥ u ∈ H of (1.1). Testing equation (1.1) with u yields0 = (cid:90) Ω | ∆ u | + β |∇ u | + u − u dx > ( λ + βλ − (cid:90) Ω u dx ≥ , by the Poincar´e inequality, a contradiction.Now, assume that λ + βλ <
1, let f be as in (5.1), and J , F as in (4.1)with this choice of f . Note that F ( s ) ≤ for all s ∈ R . Thus J ( u ) ≥ − | Ω | forall u ∈ H. Standard arguments show that J attains a global minimizer u in H and that u is a weak solution of ∆ u − β ∆ u = f ( u ) in Ω with Navier boundaryconditions. Observe that u (cid:54)≡
0, since λ + βλ < J ( δϕ ) < δ >
0, where ϕ is the first Dirichlet eigenfunction of theLaplacian in Ω.Arguing as in Proposition 4.1 we have that u ∈ L ∞ (Ω), and therefore u ∈ C (Ω) (cid:84) C (Ω) and ∆ u ∈ C (Ω), by Lemma 2.2. Then u is a classical solutionof (1.1) satisfying 0 ≤ u ≤ M β if β ≥ K , and 0 ≤ u ≤ β ≥ √
8, by Lemma5.1.The strict positivity of u (recall u (cid:54)≡
0) and − ∆ u + β u is a consequence of themaximum principle for second order equations and the following decompositioninto a second order system − ∆ u + β u = w, − ∆ w + β w = (1 + β u − u in Ω , u = w = 0 on ∂ Ω , where (1 + β ) u − u ≥ ≤ u ≤ M β ≤ C β for β ≥ K . In this section we prove the following result.
Theorem 6.1.
Let ∂ Ω be of class C , and β ≥ √ . Then any positive solutionof (1.1) is strictly stable. The proof of Theorem 6.1 is an easy consequence of the following.
Proposition 6.2.
Assume that β ≥ √ , and let u be a positive solution of (1.1) . Then, µ = inf v ∈ H \{ } (cid:82) Ω | ∆ v | + β |∇ v | + ( u − v dx (cid:82) Ω v dx = 0 . Indeed, assume for a moment that Proposition 6.2 holds. Then we have.
Theorem 6.1.
Let H := { v ∈ H : (cid:107) v (cid:107) L (Ω) = 1 } . By Proposition 6.2,inf v ∈H (cid:90) Ω | ∆ v | + β |∇ v | +(3 u − v dx = µ + 2 inf v ∈H (cid:90) Ω u v dx > Theorem 6.3.
Let L := (cid:18) − ∆ 00 − ∆ (cid:19) and M be a × continuous matrixsuch that1) − M is essentially positive , that is, − M ≥ and − M ≥ in Ω .2) M is fully coupled , that is, M , (cid:54)≡ and M , (cid:54)≡ in Ω .3) there is a positive strict supersolution φ of L + M , i.e., a function φ > such that ( L + M ) φ > in Ω .Then there are (cid:101) v, (cid:101) w ∈ W ,Nloc (Ω) (cid:84) C (Ω) unique (up to normalization) posi-tive functions such that ( L + M ) (cid:18) (cid:101) v (cid:101) w (cid:19) = λ (cid:18) (cid:101) v (cid:101) w (cid:19) , (6.1) where λ > is the smallest eigenvalue (smallest real part) of (6.1) .Moreover, there are positive functions v, w ∈ W ,Nloc (Ω) (cid:84) C (Ω) unique upto normalization such that ( L + M ) (cid:18) vw (cid:19) = λ B B (cid:18) vw (cid:19) , (6.2) where λ B > is the smallest eigenvalue (smallest real part) of (6.2) and B (cid:54)≡ is a matrix with B ij ∈ C ( ¯Ω) and B ij ≥ . Theorem 6.3 is a particular case of [41, Theorem 1.1] and [36, Theorem 5.1].In [36, Theorem 5.1] the result is formulated for matrices B ∈ C ( ¯Ω), but thesame proof applies for B ∈ C ( ¯Ω). Proposition 6.2.
Let L := (cid:18) − ∆ 00 − ∆ (cid:19) , (cid:102) M := (cid:32) β − u − − β β (cid:33) , B := (cid:18) (cid:19) , (6.3)and M := (cid:102) M + β B = (cid:18) β − u − β (cid:19) . Let ϕ > − ∆ ϕ = λ ϕ in Ω , ϕ = 0 on ∂ Ω . (6.4)Note that − M is fully coupled, M is essentially positive by Lemma 3.2 ii), and,for β ≥ √
8, the function φ = (cid:18) ϕ ϕ (cid:19) is a positive strict supersolution of L + M ,15ecause λ + β − > λ + u − β >
0. Then, by Theorem 6.3, thereare unique (up to normalization) positive functions v, w ∈ W ,Nloc (Ω) (cid:84) C (Ω))such that ( L + (cid:102) M ) (cid:18) vw (cid:19) = ( λ B − β B (cid:18) vw (cid:19) , where λ B > ∂ Ω is C , standard regularity arguments imply that v ∈ C (Ω) ∩ C ( ¯Ω), ∆ v ∈ C ( ¯Ω) and then∆ v − β ∆ v + ( u − v = ( λ B − β v = µ v in Ω , (6.5)where µ := λ B − β is the smallest eigenvalue of (6.5). By multiplying thisequation by u > µ (cid:90) Ω v u dx since u is a solution of (1.1). Therefore µ = 0 , because u and v are positive.This ends the proof by the variational characterization of µ . Remark 6.4.
Note that without the assumption β ≥ √ the result still holds forany solution u such that < u ≤ in Ω , λ + β − > , and λ + u − β > . Remark 6.5.
Uniqueness of positive solutions of (1.1) when β ≥ √ can beproved using Proposition 6.2 as follows. Let u and v denote two positive solutionsof (1.1) . Then w := u − v solves ∆ w − β ∆ w + ( u − w = − ( uv + v ) w in Ω and w = ∆ w = 0 on ∂ Ω . By testing with w we have (cid:90) Ω | ∆ w | + β |∇ w | + ( u − w dx = (cid:90) Ω ( − uv − v ) w dx ≤ . But then Proposition 6.2 yields that w ≡ . Theorem 1.3.
Let L and B be as in (6.3) and let (cid:101) Q := (cid:32) β − u − − β β (cid:33) . Note that Q := (cid:101) Q + (cid:18) β − (cid:19) B = (cid:18) β − u − β (cid:19)
16s a fully coupled matrix and − Q is essentially positive (see Theorem 6.3).Moreover, since β > √ − λ there is δ > / ( λ + β ) < δ < ( λ + β ) / . For such δ , if ϕ is as in (6.4), we have( L + Q ) (cid:18) δϕ ϕ (cid:19) = (cid:18) ( δλ + β δ − ϕ ( λ + 3( u − δ + β ) ϕ (cid:19) > , and therefore L + Q has a positive strict supersolution. Therefore, by Theorem6.3 there is λ B > v, w ∈ W ,Nloc (Ω) (cid:84) C (Ω) such that( L + (cid:101) Q ) (cid:18) vw (cid:19) = µ B (cid:18) vw (cid:19) , with µ = λ B − β . By standard regularity arguments, v ∈ C (Ω) (cid:84) C (Ω) and v solves∆ v − β ∆ v + (3 u − v = µv in Ω , that is, v is the first eigenfunction and µ is a simple (first) eigenvalue. Stabilityof u implies µ ≥
0. We now argue by contradiction, assume that u is notradially symmetric. Then there is a nontrivial angular derivative u θ = ∂ θ u (cid:54)≡ u θ must change sign in Ω and ∆ u θ − β ∆ u θ + (3 u − u θ = 0in Ω , since ∂ θ and − ∆ commute and u is a solution of (1.1). This implies that u θ is a sign-changing eigenfunction associated to the zero eigenvalue, but thiscontradicts the fact that µ ≥ u is aradial function. Remark 7.1.
Note that Theorem 1.3 holds true for any λ if β ≥ √ . On theother hand, if u is a nontrivial solution, then, by Theorem 5.2, we have that λ + βλ < . Combined with β > √ − λ , we obtain that the infimum of β ’s satisfying these inequalities is √ with corresponding λ = √ − √ . Before we prove Theorem 1.5, we introduce some notation. Let I := (cid:26) [0 , R ) if Ω = B R , ( R , R ) if Ω = B R \ B R , for some R > R >
0, where B r denotes the open ball of radius r centered atthe origin. To simplify the presentation we abuse a little bit the notation andalso denote u ( r ) = u ( | x | ) . Recall that H r = { u ∈ H : u is radially symmetric } . Theorem 1.5.
Suppose first that Ω = B R (0) and for a contradiction, assumethat u changes sign. Then either u or − u has a positive local maximum in170 , R ). Without loss of generality, assume there is η ∈ (0 , R ) such that 1 ≥ M := u ( η ) = max [0 ,R ] u > . Let v : I → R be given by v := (cid:26) − M M ( M − u ) + M in [0 , η ] ,u in ( η, R ] . Note that v is just a rescaled reflection of u with respect to u = M in [0 , η ].Since v (cid:48) ( η ) = u (cid:48) ( η ) = v (cid:48) (0) = u (cid:48) (0) = 0 we have that v ∈ C ( I ). Also | v (cid:48) | ≤ | u (cid:48) | in (0 , R ) and | v (cid:48)(cid:48) | ≤ | u (cid:48)(cid:48) | in (0 , R ) \ { η } , with strict inequalities in (0 , η ), whenever u (cid:48) (cid:54) = 0 and u (cid:48)(cid:48) (cid:54) = 0. (8.1)Furthermore in (0 , η ) one has 0 ≤ v ≤ − M M ( M + 1) + M = 1 and v − u = (cid:18) − M M + 1 (cid:19) ( M − u ) ≥ ,v + u = (cid:18) − − M M + 1 (cid:19) u + (cid:18) − M M + 1 (cid:19) M = (cid:18) M M (cid:19) ( u + 1) ≥ . Then v − u = ( v − u )( v + u ) ≥ , thus | − v | = 1 − v ≤ − u = | − u | in [0 , R ] , and then (cid:90) I ( v − dx ≤ (cid:90) I ( u − dx. (8.2)By (8.1) and (8.2) one has J β ( v ) < J β ( u ), a contradiction to the minimality of u and thus u does not change sign in Ω . The proof for the annulus Ω = B R (0) \ B R (0) for some R > R > u changes sign, then there are η, µ ∈ I such that M := u ( η ) = max [0 ,R ] u > m := − u ( µ ) = − min [0 ,R ] u > . Without loss of generality, assume that η < µ.
Let v : I → R be given by v := u in [ R , η ] , m − Mm + M ( M − u ) + M in ( η, µ ) , − u in [ µ, R ] . Then in ( η, µ ) v ≤ m − Mm + M ( M + m ) + M = m ≤ m > M,v = M − mm + M ( u − M ) + M ≤ M ≤ m ≤ M,v − u = (cid:18) M − mm + M − (cid:19) u + (cid:18) m − Mm + M + 1 (cid:19) M = 2 mm + M ( − u + M ) ≥ ,v + u = (cid:18) M − mm + M + 1 (cid:19) u + (cid:18) m − Mm + M + 1 (cid:19) M = 2 Mm + M ( u + m ) ≥ . η, µ )) and (8.2) also hold in this case.Then J β ( v ) < J β ( u ) , which contradicts the minimality of u and thus u does notchange sign in Ω.It only remains to prove the assertions about ∂ r u . We prove the case whenΩ is an annulus, since the case when Ω is a ball can be treated similarly.Without loss of generality assume u ≥ . Now, suppose by contradictionthat u (cid:48) has more than one change of sign. Then u (cid:48) has at least three changesof sign: two local maxima and one local minimum. Let η ∈ I be such that u ( η ) = max I u =: M and ˜ η ∈ I another local maximum. We only provethe case η < ˜ η, the other one is analogous. Let µ ∈ ( η , ˜ η ) be such that u ( µ ) = min [ η , ˜ η ] u =: m and η ∈ ( µ, ˜ η ] such that u ( η ) = max [ µ, ˜ η ] u =: M . Clearly m < M ≤ M . Define v : I → [0 ,
1] by v := u in [ R , η ] ∪ [ η , R ] , M − M M − m ( u − M ) + M in ( η , µ ) ,M in [ µ, η ) , By similar calculations as above it is easy to see that v ∈ C ( I ) , and that J β ( v ) < J β ( u ) , which contradicts the minimality of u and thus u (cid:48) only changessign once in I . Proposition 1.4.
We can write (1.1) as the following system. − ∆ u + β u = w in Ω , − ∆ w + β w = u − u + β u in Ω ,u = w = 0 on ∂ Ω . (9.1)By Lemma 3.2 and β ≥ √ (cid:107) u (cid:107) L ∞ (Ω) ≤
10 Saddle solution
Recall the definition of K , M β , and C β given in (3.1). Theorem 1.2.
Let β ≥ K . By odd reflection, it suffices to find a positive19 ∈ C ( R ) solving ∆ u − β ∆ u = u − u in R , ∆ u = u = 0 on ∂ R , (10.1)where R = { ( x , x ) ∈ R : x > , x > } . Indeed, denote again by u theextension to R by odd reflections. Since u = 0 on H := { x : x = 0 } onehas u x = u x x = u x x x = 0 on H . Also ∆ u = 0 on H implies u x x = 0on H , and consequently u x x x = 0 on H . Since u x , u x x , u x x , u x x x , u x x x are odd with respect to H they are continuous on H . Moreover, allother partial derivatives (up to third order) are continuous on H as well, sincethey are even functions. The same procedure applies to H := { x : x = 0 } andthus u ∈ C ( R ).From the equation (10.1) we also obtain continuity of ∆ u and ∆ u = 0 on ∂ R . By a similar reasoning as above we obtain that the extension is of class C ( R ) and it is a classical solution of (1.1) with Ω = R , and consequently itis a saddle solution.We find a positive solution of (10.1) by a limiting procedure using Theorem5.2 with Ω R := (0 , R ) and letting R → ∞ . Note that for R big enough, thesolution u R given by Theorem 5.2 satisfies that 0 < u R ≤ M β in Ω R , andtherefore, by Lemma 2.2, there is some C > R such that (cid:107) u R (cid:107) C ,α (Ω R ) < C for all R > . (10.2)By the Arzela-Ascoli theorem there is a sequence R N → ∞ such that u R N → u in C ( R ) for some u satisfying (10.1). We now prove that u (cid:54)≡
0. Indeed, fix r > β ) (cid:101) C + 160 , (10.3)where (cid:101) C > R specified below. We show that (cid:107) u R (cid:107) L ∞ ([0 ,r +2] ) ≥ √ R > r + 3 . (10.4)Assume by contradiction that there is R > r + 3 such that (cid:107) u R (cid:107) L ∞ ([0 ,r +2] ) ≤ √ . (10.5)We define the following sets ω := { x ∈ Ω R : dist( x, ∂ Ω R ) ≤ , x ≤ r + 1 , x ≤ r + 1 } ,ω := { x ∈ Ω R : dist( x, ∂ Ω R ) ≥ , x ≤ r + 1 , x ≤ r + 1 } ,ω := { x ∈ Ω R : r + 1 ≤ max { x , x } ≤ r + 2 } ,ω := { x ∈ Ω R : x ≥ r + 2 , or x ≥ r + 2 } . R = (cid:83) i =1 ω i . Now, let φ ∈ C (Ω R ) (cid:84) C (Ω R ) , φ ∈ C (Ω R ) suchthat 0 ≤ φ i ≤ , (cid:107) φ i (cid:107) C (Ω R ) ≤ K for i = 1 , K > R, and φ ≡ ω , φ ≡ ω , φ ≡ ω , φ ≡ ω . Further, let ψ ∈ C (Ω R ) (cid:84) C (Ω R ) be given by ψ := φ + φ u R . Then ψ ≡ ω , ψ ≡ u R in ω , and there is some (cid:101) C > , depending only on C (from (10.2))and K , such that (cid:107) ψ (cid:107) C (Ω R ) ≤ (cid:101) C. For i = 1 , . . . , J i ( v ) := (cid:90) ω i | ∆ v | β |∇ v | v − dx for v ∈ H (Ω R ) (cid:92) H (Ω R ) . Note that (cid:80) i =1 J i ( v ) = J β ( v ) + | Ω R | for v ∈ C (Ω R ) , here J β is as in (1.9) forΩ = Ω R . Then (cid:80) i =1 J i ( ψ ) ≤ [( + β ) (cid:101) C + 1]( | ω | + | ω | ) + J ( u R ) , and by (10.5), (cid:80) i =1 J i ( u R ) ≥ | ω | + J ( u R ) . Therefore J β ( u R ) − J β ( ψ ) ≥ r (cid:16) r − β ) (cid:101) C + 2) (cid:17) > , by (10.3), a contradiction to the minimality of u R . Therefore (10.4) holds andthe maximum principle yields that u > R is a solution of (10.1).
11 Bifurcation from a simple eigenvalue
Theorem 11.1.
Let Ω ⊂ R N , N ≥ be a smooth bounded domain or a hyper-rectangle. If the first Dirichlet eigenvalue λ < , then there is ε > such that (1.1) admits a positive solution u β ∈ C ,α (Ω) for all β ∈ ( ¯ β − ε, ¯ β ) , where ¯ β = 1 − λ λ . (11.1) Additionally, if ¯ β > √ and Ω is smooth, then (1.1) admits a unique positivesolution u β such that (cid:107) u β (cid:107) L ∞ (Ω) ≤ for all β ∈ [ √ , ¯ β ) .Proof. Let X = { u ∈ C ,α (Ω) ∩ C (Ω) : u = ∆ u = 0 on ∂ Ω } and Y = C ,α (Ω) . Consider the operator G : R × X → Y ; G ( β, u ) := ∆ u − β ∆ u − u + u . Then we have G ( β,
0) = 0 for all β . Moreover u ∈ X solves (1.1) if and only if G ( β, u ) = 0 . We consider the partial derivative ∂ u G : (0 , ∞ ) × X → L ( X, Y ) , ∂ u G ( γ, u )[ v ] = ∆ v − β ∆ v − v + 3 u v. β > A β := ∂ u G ( β, A β v = ∆ v − β ∆ v − v . Let N ( A β ) and R ( A β ) denote the kernel and the range of A β respectively. Note that v ∈ N ( A β )if and only if ∆ v − β ∆ v = v in Ω . Let ϕ be the first eigenfunction of theLaplacian in Ω, see (6.4) with (cid:107) ϕ (cid:107) L (Ω) = 1.By the definition of ¯ β >
0, one has ϕ ∈ N ( A ¯ β ) . Moreover, by the Krein-Rutman Theorem N ( A ¯ β ) = { αϕ : α ∈ R } . Further, since A ¯ β is self adjoint,by the Fredholm Theory R ( A ¯ β ) = { v ∈ Y : (cid:82) Ω ϕ v dx = 0 } . In particular, ddβ A β ϕ | β = ¯ β = − ∆ ϕ = λ ϕ (cid:54)∈ R ( A ¯ γ ) . Hence, by [15, Lemma 1.1] there are ε > C − functions β : ( − ε, ε ) → (0 , ∞ ) and u : ( − ε, ε ) → X such that β (0) = ¯ β and G ( β ( t ) , u ( t )) = 0 for all t ∈ ( − ε, ε ), moreover G − ( { } ) near( ¯ β,
0) consists precisely of the curves u ≡ β ( t ) , u ( t )), t ∈ ( − ε, ε ). Since ∂ uu G ( ¯ β, ϕ , ϕ ] = 0 and (cid:82) Ω ϕ ∂ uuu G ( ¯ β, ϕ , ϕ , ϕ ] dx = 6 (cid:82) Ω ϕ dx > u = ± c ( ¯ β − β ) ϕ + o ( t ) for someconstant c > t ∈ (0 , ε ). This proves the first claim.For the second claim, assume ¯ β > √ , T ) be the maximal interval of existence in [ √ , ¯ β ] for the curve u with T ∈ (0 , ∞ ]. By this we mean that β ( t ) ≥ √ t ∈ (0 , T ), that γ ( t ) can beuniquely extended for each t ∈ (0 , T ), and, if β ( T ) > √
8, then γ cannot beuniquely extended at T . In particular, the curve ceases to exist if it intersectsanother curve e.g. ( β,
0) or if it bifurcates.We show first that u ( t ) > t ∈ (0 , T ). By the C − continuity ofthe curve C := { u ( t ) : t ∈ (0 , T ) } we have that 0 < u β < t ∈ (0 , T )sufficiently close to zero. By Lemma 3.2 iii) and the continuity of C it followsthat (cid:107) u ( t ) (cid:107) L ∞ (Ω) ≤ t ∈ (0 , T ) . To show that u ( t ) > t ∈ (0 , T ) we argue by contradiction. Assume that ¯ t = sup { t ∈ (0 , T ) : u ( s ) > s ∈ (0 , t ] } < T . Let ¯ u = u (¯ t ) ∈ C ,α (Ω) . Note that ¯ u satisfies thesystem − ∆¯ u + β ¯ u = w in Ω , − ∆ w = ¯ u − ¯ u in Ω , w = u = 0 on ∂ Ω , for some β > , and ¯ u − ¯ u ≥ ≤ ¯ u ≤ . Since
T > ¯ t we have that¯ u (cid:54)≡
0. Then, by the maximum principle and the Hopf Lemma, ¯ u > ∂ ν ¯ u < ∂ Ω, where ν denotes the exterior normal vector to ∂ Ω. But thiscontradicts the C − continuity of C and the definition of ¯ u . Therefore u ( t ) > t ∈ (0 , T ) . Then, by Lemma 3.2, (cid:107) u ( t ) (cid:107) L ∞ (Ω) ≤ t ∈ (0 , T ), and standardelliptic regularity theory yields that (cid:107) u ( t ) (cid:107) C ,α (Ω) ≤ C for all t ∈ (0 , T ) andfor some C >
0. Moreover, since the positivity is preserved along the curve, u ( T ) (cid:54)≡
0. Indeed, ( β ( t ) , u ( t )) cannot return to a neighborhood of ( ¯ β,
0) byuniqueness close to ( ¯ β, β,
0) as any other branchbifurcating from ( β,
0) ( β < ¯ β ) consists locally of sign changing solutions becausethe corresponding eigenfunction directions are sign changing (perpendicular tothe principal eigenfunction). By the first part of Theorem 5.2, we know that β ( t ) < ¯ β for all t ∈ (0 , T ) . This implies that necessarily ( √ , ¯ β ) ⊂ { β ( t ) : t ∈ (0 , T ) } . This proves the existence of solutions for all β ∈ [ √ , ¯ β ).We now show that u β is the unique positive solution of (1.1) for β ∈ [ √ , ¯ β ).22ndeed, let v denote a positive solution of (1.1) for some β ∈ [ √ , ¯ β ). ByTheorem 6.1, v is a strictly stable solution, and therefore D u G ( v, β ) is aninvertible operator. Then, by the implicit function theorem there exists ε > γ : ( β − ε, β + ε ) → X such that G ( β, γ ( β )) = 0 and forany solution of G ( β, u ) = 0 sufficiently close to ( β , v ) one has u ∈ γ .Arguing as before, we can extend γ to a maximal interval ( β , β ) with γ containing only positive solutions. Then the strict stability in Theorem 6.1guarantees that γ does not have bifurcation or turning points. Since the onlysolution for β ≥ ¯ β is zero, by the first part of Theorem 5.2, we have that β ≤ ¯ β and γ ( β ) is a non-negative function. Arguing as before, one obtainsthat necessarily γ ( β ) ≡ β = ¯ β . Here, as above, we have used thatall other bifurcation points of the form ( β,
0) must correspond locally to signchanging solutions. The uniqueness of the branch close to the bifurcation point( ¯ β,
0) yields that necessarily v = u β , as desired.If Ω is a hyperrectangle, one proves the positivity along the curve usingSerrin’s boundary point Lemma [39, Lemma 1] at corners and the rest of theproof remains unchanged. Remark 11.2.
For balls of radius
R > we can explicitly write the relationshipbetween R and the bifurcation point ¯ β . Indeed, in this case, R := √ j N/ − , (cid:113) − β + √ β +4 ,where N is the dimension and j N/ − , is the first positive zero of the Besselfunction J N/ − , see for instance [29, Section 4.1]. For example, for β = √ ,the bifurcation occurs at balls of radius R N := j N/ − , √ √ −√ , for instance, R : ≈ . , R : ≈ . , R : ≈ . , R : ≈ . , etc...
12 Continuity result
Theorem 1.7.
Fix p > N , β = 1, let γ ( β, p, Ω) = γ > < γ < min { γ , } , u γ be the global minimizer of (1.12),and µ = γ − . Note that w : µ Ω → R given by w ( x ) := u γ ( µ − x ) is a weaksolution in H of ∆ w − µ ∆ w = w − w in µ Ω . Also note that µ ≥ √ γ ≤ . By Proposition 4.1 and Lemma 2.3 we have that (cid:107) u γ (cid:107) L ∞ (Ω) = (cid:107) v (cid:107) L ∞ (Ω) ≤ (cid:107) u γ (cid:107) C ,α (Ω) ≤ C for some C > γ. Let u ∗ ∈ H (Ω) be a global minimizer of (1.12) in H (Ω) with γ = 0. It iswell known that u ∗ is a unique global minimizer, smooth, strictly stable, and itdoes not change sign (see, for example, [3]).Now, since u γ is bounded in C ,α independently of γ it is easy to see that u γ → u ∗ in C as γ →
0, by the uniqueness of global minimizers of the limitproblem (1.11) with γ = 0.Let G ∈ C ( R × C (Ω)) be given by G ( γ, u ) = γ ∆ u − ∆ u − u + u . Noticethat ∂ u G ( γ, u ) ∈ L ( C (Ω) , R ) and ∂ u G (0 , u ∗ ) has trivial kernel by the strictstability of u ∗ . Therefore, by the implicit function theorem (see for example[30, Theorem I.1.1]), there is a neighborhood I × V ⊂ R × C (Ω) of (0 , u ∗ ) and23 continuous function λ : I → V with λ (0) = u ∗ such that G ( γ, λ ( γ )) = 0 for all γ ∈ I and every solution of G ( γ, u ) = 0 in I × V is of the form ( γ, λ ( γ )) for some γ ∈ I. Since u γ → u ∗ in C as γ → + and u γ is an arbitrary global minimizer,we obtain that u γ is the unique global minimizer for all γ ∈ I . Finally, if thefirst Dirichlet eigenvalue λ (Ω) < , then u ∗ (cid:54)≡ u ∗ > , by the Hopf Lemma ∂ ν u ∗ < ∂ Ω, where ν denotes the exterior normal vector, and therefore u γ > γ ∈ I , by making I a smaller neighborhood of 0 if necessary. Remark 12.1.
Note that the proof of Theorem 1.7 also shows the existence ofsolutions in C ,α (Ω) for equation (1.11) with γ ∈ [ − γ , .
13 Numerical approximations
In this Section we present some numerical approximations of a bifurcationbranch with respect to β from ¯ β = − λ λ for the problem (1.1) (cf. Theo-rem 11.1). These approximations were computed with the software AUTO-07P[18].Figure 4: Here Ω = (0 , π ). In this picture we present approximations of solutions of (1.1)along the branch. Note that for β < β > Here Ω = (0 , π ). In this picture we show numerical evidence that positivesolutions are not bounded by one and present oscillations. Also that the bifurcation branchmay return to positive values of β (although the positivity is lost). On the right we have anapproximation of two solutions of (1.1) along the branch corresponding to β = 0 . References [1] S. Allen and J.W. Cahn. A microscopic theory for antiphase boundarymotion and its application to antiphase domain coarsening.
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