The semiflow of a reaction diffusion equation with a singular potential
aa r X i v : . [ m a t h . A P ] F e b The semiflow of a reaction diffusion equation with a singular potential
Nikos. I. Karachalios,
Department of Mathematics , University of the Aegean , Karlovassi, 83200 Samos, Greece
Nikolaos B. Zographopoulos,
Department of Sciences , Division of Mathematics , Technical University of Crete , Abstract
We study the semiflow S ( t ) defined by a semilinear parabolic equation with a singular square potential V ( x ) = µ | x | . It is known that the Hardy-Poincar´e inequality and its improved versions, have a prominent roleon the definition of the natural phase space. Our study concerns the case 0 < µ ≤ µ ∗ , where µ ∗ is the optimalconstant for the Hardy-Poincar´e inequality. On a bounded domain of R N , we justify the global bifurcation ofnontrivial equilibrium solutions for a reaction term f ( s ) = λs − | s | γ s , with λ as a bifurcation parameter. Theglobal bifurcation result is used to show that any solution φ ( t ) = S ( t ) φ , initiating form initial data φ ≥ φ ≤ φ
0, tends to the unique nonnegative (nonpositive) equilibrium.
Keywords:AMS Subject Classification (2000): 35K57, 35B40, 35B41, 37L30, 46E35
Fundamental issues of the linear heat equation with a singular potential ∂ t φ − ∆ φ − µ | x | φ = 0 , x ∈ Ω , t > ,φ ( x,
0) = φ ( x ) , x ∈ Ω , (1.1) φ | ∂ Ω = 0 , t > , where Ω is in general an open set of R N , have been analyzed in the works [3, 10, 24]. The behavior of the solutionsdepends heavily on the critical value of the parameter µ (denoted by µ ∗ ), which is the best constant of the Hardy’sinequality. The first fundamental result was that of [3] for the Cauchy-Dirichlet problem in an open set of R N :If φ ( x ) ≥ φ ( x ) = 0, there exists a global solution if 0 < µ ≤ µ ∗ . Not even a local solution exists if µ > µ ∗ (complete instantaneous blowup). The importance of Hardy’s inequality for the result of [3] was shown in [10].Further fundamental results ranging from the removal of the sign condition on the initial data, the uniquenessof solutions in an appropriate functional space, the possible decay of solutions and its rate when µ < µ ∗ , to thedescription of the behavior of solutions at the critical value µ ∗ as well as analysis of the Cauchy problem, havebeen addressed in [24]. The legitimate analysis at the transition µ = µ ∗ and beyond, for the aforementionedquestions is related to the Hardy inequality and to its improved versions [24], in bounded as well in unboundeddomains.Concerning the bounded domain case, the situation regarding the behavior of solutions of (1.1), can bedescribed in summary as follows: When 0 < µ < µ ∗ the sign-condition on the initial data can be removed1 eaction diffusion equations with a singular potential φ ∈ L (Ω) there exist a unique, global in time solution φ ∈ C ([0 , ∞ ); L (Ω)) ∩ L ([0 , ∞ ); H (Ω))which decays at exponential rate. In the critical transition value µ = µ ∗ solutions which still exist globally in L (Ω), blow up instantaneously in H (Ω) but exist globally in the generalized Sobolev space H µ (Ω). The Hilbertspace H µ (Ω) is defined for any fixed 0 < µ ≤ µ ∗ , as the completion of the C ∞ (Ω) functions under the norm || φ || µ = R Ω |∇ φ | dx − µ R Ω φ | x | dx . The H µ (Ω)-solution, φ ( x, t ) ∼ O ( e − λt ) with the rate λ > < µ ≤ µ ∗ (e.g.singular behavior at the origin with prescribed rate, even for the solutions φ ≥ µ > µ ∗ , there exist initial data ofoscillating type for which the solution exists globally in time. Extensions of the results of the bounded domainwhen 0 < µ ≤ µ ∗ (but with major differences e.g on the rate of decay) have been made on appropriate weightedspaces based on weighted improvements of the Hardy’s inequality.Strongly motivated by the results of [24], for (1.1) on the bounded domain case, we shall discuss the dynamicsof a semilinear analogue of (1.1) ∂ t φ − ∆ φ − µ | x | φ = λ φ − | φ | γ φ, x ∈ Ω , t > ,φ ( x,
0) = φ ( x ) , x ∈ Ω , (1.2) φ | ∂ Ω = 0 , t > . with our attention restricted in this work, up to the critical case µ = µ ∗ .We start with the analysis of the set of equilibrium solutions of (1.2). The equilibrium solutions in this casesatisfy the semilinear elliptic equation − ∆ u − µ | x | u = λu − | u | γ u, (1.3) u | ∂ Ω = 0 , The results of Section 2, concern the bifurcation of equilibrium solutions with respect to the parameter λ ∈ R .Considering this type of nonlinear term with λ as a varying parameter, is of importance, having in mind theGinzburg-Landau nonlinearity. Hardy’s inequality implies for the subcritical case < µ < µ ∗ , the equivalence H (Ω) ≡ H µ (Ω). In this case, the operator L = − ∆ − µ | x | defines an unbounded self-adjoint operator in L (Ω)with compact inverse. Thus, a global branch of nonnegative solutions of (1.3) bifurcating from the trivial solutionat ( λ , λ is the positive principal eigenvalue of the linear eigenvalue problem − ∆ u − µ | x | u = λu, (1.4) u | ∂ Ω = 0 , is naturally expected.The analysis carried out in [24] for the critical case µ = µ ∗ , suggests that we cannot expect H -solutions forthe eigenvalue problem (1.3). Instead, the main result of Section 2, is stated in the following THEOREM 1.1
Let Ω ⊂ R N , N ≥ , be a bounded domain. Assume that < µ ≤ µ ∗ , and that < γ ≤ N q − N + 2 q N − q ) := γ ∗ , for any NN + 2 < q < . (1.5) Then, the principal eigenvalue λ ,µ of (1.4) considered in H µ (Ω) , is a bifurcating point of the problem (1.3) (inthe sense of Rabinowitz) and C λ ,µ is a global branch of nonnegative H µ (Ω) - solutions of (1.3). For comparison results and properties of the linear eigenvalue problem (1.4), we will refer to [13].The global bifurcation results of Section 2 are of a twofold meaning. On the one hand, they establish theexistence of a global branch C λ ,µ , of nonnegative solutions in the critical value µ = µ ∗ . The global branch hasthe properties proved in PROPOSITION 1.2
Let Ω ⊂ R N , N ≥ , be a bounded domain. Assume that < µ ≤ µ ∗ and that (1.5)holds. Then (i) The global branch C λ ,µ bends to the right of λ ,µ (supercritical bifurcation) and it is bounded for λ bounded.(ii) Every solution u ∈ C λ ,µ is the unique nonnegative solution for the problem (1.3). eaction diffusion equations with a singular potential λ λ µ∗µ∗µ∗ ||u|| H λ λ µ∗µ∗ ||u|| H Figure 1: (a) Possible bifurcation diagrams in (a) H µ ∗ (Ω) and (b) in H (Ω).On the other hand, it is well known that qualitative properties of a dynamical system may not depend continuouslyon the variation of the parameters. A first question of this nature can be addressed, regarding the behavior ofglobal branches of nonnegative solutions possessed by (1.3) in domains not containing the origin, Ω r = Ω \ B r (0).We may consider r > C λ ,µ,r in R × H (Ω r ), for any 0 < µ ≤ µ ∗ . How these branches behave as r →
0? A simple but carefulanalysis on the asymptotics of the eigenpairs ( λ, u λ,r ) ∈ C λ ,µ,r as r → H µ (Ω) and the regularity results in H µ (Ω r ), combined with the Whyburn’s Theorem is used to prove THEOREM 1.3
Let Ω ⊂ R N , N ≥ , be a bounded domain. Assume that < µ ≤ µ ∗ and that (1.5) holds.Then C λ ,µ,r → C λ ,µ in R × H µ (Ω) , as r ↓ . It seems even more interesting to discuss how the branches C λ ,µ for µ < µ ∗ behave as µ ↑ µ ∗ . Regarding thebehavior of the global branch C λ ,µ as the parameter µ varies to the transition value µ ∗ , the answer is given inthe following theorem, showing that the situation in H (Ω) and in H µ (Ω) is qualitatively totally different (figure1 demonstrates a possible configuration). THEOREM 1.4
Let Ω ⊂ R N be a bounded domain. We assume that < γ ≤ q − − q ) := γ ∗ , for any < q < , if N = 3 , (1.6) and when N ≥ , we assume condition (1.5).A. Let µ n ↑ µ ∗ , as n → ∞ . Assume that ( λ n , u n ) ∈ C λ ,µn , be such that λ n is bounded, i.e. | λ n | < L . Then, u n must be bounded too, in H µ ∗ (Ω) . Moreover, ( λ n , u n ) → ( λ ∗ , u ∗ ) in R × H µ ∗ (Ω) , with ( λ ∗ , u ∗ ) ∈ C λ ,µ ∗ .B. Let µ n ↑ µ ∗ , as n → ∞ . Assume that ( λ n , u n ) ∈ C λ ,µn , be such that λ n → λ ,µ ∗ . Then, u n must be unboundedin H (Ω) . Observe that the condition (1.5) is slightly modified, distinguishing between the cases N = 3 and N ≥ λ ≤ λ ,µ and the asymptotic stability of the uniquenonnegative equilibrium when λ > λ ,µ , for 0 < µ ≤ µ ∗ . However the setting of [24], enables for a stronger result:Following closely the semiflow theory [2, 17, 23], we define a gradient semiflow in H µ (Ω), for any 0 < µ ≤ µ ∗ .This is one of the basic results proved in Section 3, stated in PROPOSITION 1.5
Let Ω ⊂ R N , N ≥ , be a bounded domain, < µ ≤ µ ∗ and condition (1.5) be fulfilled.The semiflow (3.49), possesses a global attractor A in H µ (Ω) . Let E denote the bounded set of equilibrium pointsof S ( t ) . For each complete orbit φ lying in A , the limit sets α ( φ ) and ω ( φ ) are connected subsets of E on which theLyapunov functional J associated to S ( t ) , is constant. If E is totally disconnected (in particular if E is countable),the limit z − = lim t →−∞ φ ( t ) , z + = lim t → + ∞ φ ( t ) , eaction diffusion equations with a singular potential λ µ∗ ||u|| H µ∗ Figure 2: Supercritical pitchfork bifurcation for the semiflow defined by (1.2) in H µ ∗ (Ω). exist and are equilibrium points. Furthermore, any solution of (1.2), tends to an equilibrium point as t → ∞ . Armed with the fact, that the limit set ω ( φ ) for each positive orbit φ lying in the global attractor A , is aconnected subset of the bounded set E of the equilibrium solutions, the global bifurcation result of Theorem1.1 will be crucial: It actually shows that E = { } when λ ≤ λ ,µ , and is totally disconnected when λ > λ ,µ , E = { u − , , u } , u − = − u in H µ (Ω), for any 0 < µ ≤ µ ∗ . The trivial solution is unstable when λ > λ ,µ , thus thelimit set ω ( φ ) for every φ ∈ H µ (Ω) of definite sign, is described for all 0 < µ ≤ µ ∗ by THEOREM 1.6
Let Ω ⊂ R N , N ≥ , be a bounded domain. Assume that < µ ≤ µ ∗ and that (1.5) isfulfilled. Let φ ∈ H µ (Ω) , φ . If λ ≤ λ ,µ , then A = { } . If λ > λ ,µ , then ω ( φ ) = { u } when φ ≥ and ω ( φ ) = { u − } when φ ≤ . The above result is a rigorous verification that (1.2) which undergoes a pitchfork bifurcation of supercriticaltype for any µ < µ ∗ in H (Ω) preserves this behavior up to the transition µ = µ ∗ in the H µ ∗ (Ω)-phase space(see figure 2). We remark that in the case λ > λ ,µ Proposition 1.5, clearly implies that for any φ any solution φ ( t ) = S ( t ) φ converges to one of the equilibrium solutions u or u − , possibly through an heteroclinicorbit connecting them.However, Theorem 1.4 B. combined with Theorem 1.6 indicate for the “explosive” behavior of the attractor A in H (Ω) when µ → µ ∗ . Theorem 1.6 could also be viewed as the analogue of [24, Theorem 4.1, pg. 123] for (1.2),with the exponential decay, replaced by the convergence to the unique nonnegative or the unique nonpositiveequilibrium, for any λ > λ ,µ , according to the sign of the initial data φ .At this point, we also remark [12] for bifurcation results on H (Ω) with µ as a bifurcation parameter, regardingthe semilinear elliptic problem − ∆ u − µ | x | u = u q , u > , u | ∂ Ω = 0 . For bifurcation results on the degenerate elliptic problem −| x | ∆ u = λf ( u ) , u > , u | ∂ Ω = 0 , related to the Hardy inequality, we refer to [14]. We also point out [7], on recent bifurcation results for the ellipticproblem − ∆ u = λm ( x ) u + b ( x ) u γ , ∂u∂n | ∂ Ω = 0 , where the functions m, b : Ω → R are this time, smooth functions but of changing sign. For a brief reference toexisting results on the issue of convergence of solutions of global solutions of evolution equations to steady states,we refer to [9] (see also [18, pg. 366]). For improvements related to second order Hardy-type inequalities, we referto the recent work [22]. eaction diffusion equations with a singular potential This section is devoted to the proof of the existence of bifurcation branches for the equilibrium solutions of (1.2)given by the semilinear elliptic equation (1.3). Here Ω will be an open bounded and connected subset of R N , N ≥ < µ ≤ µ ∗ , where µ ∗ := (cid:18) N − (cid:19) , is the best constant of Hardy’s inequality Z Ω |∇ u | dx > (cid:18) N − (cid:19) Z Ω u | x | dx. (2.1)In subsection 2.1 we recall the basic properties of the delicate functional framework developed in [24, Section 4, pg.121-123], and we present some auxiliary results regarding the nonlinear maps defined in this setting. Subsection2.3 refers to the proof of Theorem 1.1, while subsection 2.3 is devoted to the approximation of the global branchby the associated branches of systems considered in domains not containing the origin. In subsection 2.4 wediscuss the proof of Theorem 1.4. It well known that the constant µ ∗ is optimal and it is not attained in H (Ω). In [6] it was given the followingimproved version of (2.1) Z Ω |∇ u | dx ≥ (cid:18) N − (cid:19) Z Ω u | x | dx + λ Ω Z Ω u dx, (2.2)where λ Ω = z ω N N | Ω | − N , where ω N and | Ω | denote the volume of the unit ball and Ω respectively, and z =2 . . . . denotes the first zero of the Bessel function J ( z ). This constant is optimal when Ω is a ball, but it isalso not achieved in H (Ω). In [15] was proved that inequality (2.1) admits an infinite series of correction terms.The analysis of [24], recovered that the natural phase space for the study of linear equation (1.1) system (1.2)is the Hilbert space H µ (Ω), defined for any fixed 0 < µ ≤ µ ∗ , as the completion of the C ∞ (Ω) functions underthe norm || φ || µ = Z Ω |∇ φ | dx − µ Z Ω φ | x | dx, (2.3)and endowed with the scalar product( φ, ψ ) µ = Z Ω ∇ φ ∇ ψ dx − µ Z Ω φ ψ | x | dx. Consequently, this is also the case for the semilinear analogue (1.2). Friedrich’s extension theory is applicable dueto the inequality (2.2): is the main ingredient which can be used to consider the operator L = − ∆ − V ( x ) as apositive and self adjoint operator with domain of definition D ( L ) = (cid:8) φ ∈ H µ (Ω) : L φ ∈ L (Ω) (cid:9) . (2.4)The improved Hardy-Poincar´e inequalities Z Ω (cid:20) |∇ φ − µ ∗ φ | x | (cid:21) dx ≥ C ( q, Ω) || φ || W ,q (Ω) , ≤ q < , (2.5) Z Ω (cid:20) |∇ φ − µ ∗ φ | x | (cid:21) dx ≥ C ( s, r, Ω) || φ || W s,r (Ω) , ≤ s < , ≤ r < r ∗ = 2 NN − − s ) , (2.6)for all φ ∈ C ∞ (Ω), imply the continuous embeddings, H µ (Ω) ֒ → W ,q (Ω) , H µ (Ω) ֒ → H s (Ω) , ≤ q < , ≤ s < . (2.7) eaction diffusion equations with a singular potential ≤ q < ≤ s <
1. Furthermore, since W ,q (Ω) is compactly embedded in H s for suitable q = q ( s ) closeenough to 2, and H s (Ω) is compactly embedded in L (Ω), we infer the compact embeddings H µ (Ω) ֒ → ֒ → L (Ω) , H µ (Ω) ֒ → ֒ → H s (Ω) , ≤ s < . (2.8)In the subcritical case 0 < µ < µ ∗ we have the following property of H µ (Ω). LEMMA 2.1 ([24]) Let < µ < µ ∗ . Then H µ (Ω) ≡ H (Ω) . Proof:
Clearly from (2.3), || u || µ ≤ Z Ω |∇ u | dx = || u || H (Ω) . (2.9)On the other hand, Hardy’s inequality (2.1), implies that || u || µ ≥ (cid:20) − (cid:18) N − (cid:19) − µ (cid:21) || u || H (Ω) . (2.10)Thus, from inequalities (2.9) and (2.10) we conclude that c || u || H (Ω) ≤ || u || H µ (Ω) ≤ C || u || H (Ω) , for c = 1 − (cid:18) N − (cid:19) − µ > µ < µ ∗ , and C = 1. (cid:4) A remarkable property was shown in [24] concerning the critical case µ = µ ∗ : H µ ∗ (Ω) is larger than H (Ω), sinceit contains singularities of the form f ∼ | x | ( N − / , and it is smaller than ∩ q< W ,q (Ω).With the continuous embeddings (2.7) at hand, we can handle the nonlinearity of (1.2). LEMMA 2.2
Let condition (1.5) be satisfied and assume that µ ≤ µ ∗ . The function g ( s ) = | s | γ s, s ∈ R , definesa sequentially weakly continuous map g : H µ (Ω) → L (Ω) . Let G ( φ ) := R φ g ( s ) ds . The functional G : H µ (Ω) → R defined by G ( φ ) = R Ω G ( φ ) dx , is C and sequentially weakly continuous. Proof:
Starting by the standard Sobolev embeddings, we recall that W ,q (Ω) ֒ → L p (Ω) for any 1 ≤ p ≤ qNN − q , q < N. (2.11)We consider the critical exponent p ∗ := qNN − q for any 1 ≤ q < . (2.12)Thus, as an immediate consequence of the embedding (2.7) we infer that H µ (Ω) ֒ → L p (Ω) , for any 1 ≤ p ≤ p ∗ . (2.13)Using (2.13) it can be easily deduced that the functional g is well defined, under the restriction (1.5). Furthermore,it follows from (2.13), that G is well defined if0 < γ ≤ N q − N + 2 q ( N − q ) := γ ∗ , for any 2 NN + 2 < q < . noting that γ ∗ < γ ∗ .That both functional are sequentially weakly continuous, can be verified by using the compact embeddings(2.8) and repeating the arguments of [2, Lemma 3.3, pg. 38 & Theorem 3.6, pg. 40]. We will check that G is a C -functional, and its derivative is G ′ ( φ )( z ) = h g ( φ ) , z i , for every φ ∈ H µ (Ω) , z ∈ H − µ (Ω) . (2.14) eaction diffusion equations with a singular potential φ, ψ ∈ H µ (Ω), the quantity G ( φ + sψ ) − G ( φ ) s = 1 s Z Ω Z ddθ G ( φ + θsψ ) dθdx = Z Ω Z g ( φ + sθψ ) ψdθdx. (2.15)We set σ = qNN ( q − q , σ − + p ∗− = 1, and we get (cid:12)(cid:12)(cid:12)(cid:12)Z Ω g ( φ + θsψ ) ψdx (cid:12)(cid:12)(cid:12)(cid:12) ≤ c (cid:18)Z Ω ( | φ | (2 γ +1) + | ψ | (2 γ +1) ) σ dx (cid:19) σ (cid:18)Z Ω | ψ | p ∗ dx (cid:19) p ∗ . (2.16)To apply the continuous embedding (2.13) we need the requirement(2 γ + 1) σ ≤ p ∗ . This requirement produces the restriction (1.5). Letting s →
0, and using the dominated convergence theorem,we infer that G is differentiable with the derivative (2.14).For the continuity, we consider a sequence { φ n } n ∈ N of H µ (Ω) such that φ n → φ in H µ (Ω) as n → ∞ . We notefirst, that hG ′ ( φ n ) − G ′ ( φ ) , z i ≤ || g ( φ n ) − g ( φ ) || L σ || z || L p ∗ . (2.17)Setting then p = p ∗ σ , the requirement for p >
1, produces again the restriction for NN +2 < q <
2. Now for p = N ( q −
1) + qN ( q − − N + 2 q , p − + p − = 1 , we get the inequality || g ( φ n ) − g ( φ ) || σL σ ≤ c (cid:18)Z Ω ( | φ n | γ + | φ | γ ) σp dx (cid:19) p (cid:18)Z Ω | φ n − φ | p ∗ dx (cid:19) p . The embedding (2.13) is applicable if 2 γσp < p ∗ , giving (1.5). Under this condition and aslim n →∞ Z Ω | φ n − φ | p ∗ dx = 0 , we conclude from (2.17), the continuity of G ′ . (cid:4) < µ ≤ µ ∗ The existence of a global branch of nonnegative solutions will be proved via the classical Rabinowitz’s theorem:
THEOREM 2.3
Assume that X is a Banach space with norm || · || and consider G ( λ, · ) = λL · + H ( λ, · ) , where L is a compact linear map on X and H ( λ, · ) is compact and satisfies lim || u ||→ || H ( λ, u ) |||| u || = 0 . (2.18) If λ is a simple eigenvalue of L then the closure of the set C = { ( λ, u ) ∈ R × X : ( λ, u ) solves u = G ( λ, u ) , u } , possesses a maximal continuum (i.e. connected branch) of solutions, C λ , such that ( λ, ∈ C λ and C λ either:(i) meets infinity in R × X or,(ii) meets ( λ ∗ , , where λ ∗ = λ is also an eigenvalue of L . eaction diffusion equations with a singular potential λ ,µ of the problem (1.4), for any µ ≤ µ ∗ . LEMMA 2.4
Assume that < µ ≤ µ ∗ . Problem (1.4), admits a positive principal eigenvalue λ ,µ , given by λ ,µ = inf φ ∈ H µ (Ω) φ R Ω |∇ φ | dx − µ R Ω φ | x | dx R Ω | φ | dx . (2.19) with the following properties:(i) λ ,µ is simple with a positive associated eigenfunction u ,µ , which belongs at least to C ,ζloc (Ω \{ } ) , for some ζ ∈ (0 , ,(ii) λ ,µ is the only eigenvalue of (1.4) with nonnegative associated eigenfunction. Proof:
The existence and the variational characterization (2.19) of the principal eigenvalue follows from thecompactness of the embeddings (2.8) implying that L = − ∆ − µ | x | for 0 < µ ≤ µ ∗ , has an orthonormal basis ofeigenfunctions in H µ (Ω) with an eigenvalue sequence0 < λ ≤ λ ≤ · · · ≤ λ n ≤ · · · → ∞ , (2.20)(cf. [24, pg. 122]) The regularity results (cf. [16, Theorem 8.22]) imply that if u is a weak solution of the problem(1.4), then u ∈ C ,ζloc (Ω \{ } ), for some ζ ∈ (0 , u ,µ follows from [13, Lemma 2.2]-we also referto the weak maximum principle of [4]. The simplicity and the uniqueness up to positive eigenfunctions of λ ,µ can be verified, by using Picone’s identity [18]. (cid:4) For some further properties of the principal eigenvalue and the corresponding eigenfunction, we refer to [13].We remark [15], where the weighted space Hilbert space W , (Ω; | x | − ( N − was used, defined as the completionof C ∞ -functions under the norm || u || W , (Ω; | x | − ( N − ) = Z Ω | x | − ( N − |∇ w | dx + Z Ω | x | − ( N − w dx and endowed with the inner product < u, v > W , (Ω; | x | − ( N − ) = Z Ω | x | − ( N − ∇ f ∇ g dx + Z Ω | x | − ( N − f g dx. In [15], the space W , (Ω; | x | − ( N − was considered for the proof of the existence of principal eigenvalues for theeigenvalue problem − ∆ u − µ u | x | = λ V ( x ) u (2.21) u | ∂ Ω = 0 , Furthermore, it was assumed that V ( x ) ≥ V ( x ) ∈ L p (Ω), p = N/
2. A comparison of the spaces H µ ∗ (Ω) and W , (Ω; | x | − ( N − ) implies that u ∈ H µ ∗ (Ω) if and only if | x | ( N − / u ∈ W , (Ω; | x | − ( N − ).Proceeding to the proof of the global bifurcation result, we discuss first the behavior of λ ,µ , 0 < µ < µ ∗ as µ ↑ µ ∗ . Next lemma demonstrates the qualitative differences in H (Ω) for the solutions of the linear eigenvalueproblem (1.4) as µ converges to the transition value µ ∗ . PROPOSITION 2.5
Let µ ↑ µ ∗ . Then,(i) λ ,µ is a decreasing sequence, and there exists λ ∗ > such that λ ,µ ↓ λ ∗ .(ii) The corresponding normalized eigenfunctions u ,µ are converging weakly to 0, in H (Ω) . Proof: (i) Let µ < µ . Then the variational characterization of the principal eigenvalue λ ,µ (2.19) implies that λ ,µ > λ ,µ . Thus λ ,µ is decreasing. Applying next the improved Hardy’s inequality (2.2) we infer that λ ,µ isbounded from below by λ Ω . Thus, there exists λ ∗ >
0, such that λ ,µ ↓ λ ∗ . eaction diffusion equations with a singular potential u ,µ should satisfy the weak formula Z Ω ∇ u ,µ ∇ φ dx − µ Z Ω u ,µ φ | x | dx = λ ,µ Z Ω u ,µ φ dx, (2.22)for any φ ∈ C ∞ (Ω). We still denote by u ,µ the sequence of normalized eigenfunctions, forming a boundedsequence in H (Ω). We deduce that there exists some u ∈ H (Ω) such that up to a subsequence (not relabelled), u ,µ ⇀ u in H (Ω) and u ,µ → u in L q (Ω), for any 1 < q < NN − . For some fixed ε >
0, small enough and any φ ∈ C ∞ (Ω), we have that Z Ω ( u ,µ − u ) φ | x | dx ≤ || φ || L ∞ (Ω) (cid:18)Z Ω | u ,µ − u | N − εN − − ε (cid:19) N − − εN − ε (cid:18)Z Ω | x | − N + ε (cid:19) / ( N − ε ) → , thus Z Ω u ,µ φ | x | dx → Z Ω u φ | x | dx, as µ ↑ µ ∗ . Let us now assume by contradiction that u
0. Passing to the limit in (2.22), we get that u must satisfy Z Ω ∇ u ∇ φ dx − µ ∗ Z Ω u φ | x | dx = λ ∗ Z Ω u φ dx, for any φ ∈ C ∞ (Ω), or equivalently that u must be a nontrivial solution of the problem − ∆ u − µ ∗ u | x | = λ ∗ u, u ∈ H (Ω) . (2.23)However, since µ ∗ is the optimal constant of (2.2) which is not achieved in H (Ω), (2.23) implies that u ≡ (cid:4) Proof of Theorem 1.1:
For the justification of Theorem 2.3, the improved Hardy’s inequality (2.2), will allowus to employ the method developed in [8]: On the account of (2.19), we define a bilinear form in C ∞ (Ω) by < u, v > X = Z Ω ∇ u ∇ v dx − µ Z Ω u v | x | dx − c Z Ω u v dx, for all u, v ∈ C ∞ (Ω) , c = λ ,µ . (2.24)We define next the space X , as the completion of C ∞ (Ω) with respect to the norm induced by (2.24), || u || X = X : Then due to the improved Hardy’s inequality (2.2), we deduce the equivalence of norms12 || u || H µ (Ω) ≤ || u || X ≤ || u || H µ (Ω) , for all u, v ∈ C ∞ (Ω) , Since C ∞ (Ω) is dense both in X and H µ (Ω), it follows that X = H µ (Ω). Henceforth we may suppose that thenorm in X coincides with the norm in H µ (Ω) and that the inner product in X is given by < u, v > X = < u, v > H µ (Ω) .Let us note that the identification principle [25, Identification Principle 21.18, pg. 254]) implies that if < · , · > X,X ∗ denotes the duality pairing on X , then < · , · > X,X ∗ = < · , · > . To proceed further we note that the bilinear form a ( u, v ) = Z Ω u v dx, for all u, v ∈ X, is clearly continuous in X . The Riesz representation theorem implies that we can define a bounded linear operator L such that a ( u, v ) = < L u, v >, for all u, v ∈ X. (2.25)The operator L is self adjoint and compact and its largest eigenvalue ν is characterized by ν = sup u ∈ X < L u, u >< u, u > = sup u ∈ X R Ω u dx R Ω |∇ u | dx − µ R Ω u | x | dx . Then, by Lemma 2.4 it readily follows that the positive eigenfunction u of (1.4) corresponding to λ ,µ is a positiveeigenfunction of L corresponding to ν = 1 /λ ,µ . eaction diffusion equations with a singular potential N ( λ, · ) : R × X → X ∗ as < N ( λ, u ) , v > = Z Ω ∇ u ∇ v dx − µ Z Ω u v | x | dx − λ Z Ω u v dx + Z Ω | u | γ u v dx, (2.26)for all v ∈ X . Since the functional S : X → R defined by S ( v ) = Z Ω ∇ u ∇ v dx − µ Z Ω u v | x | dx − λ Z Ω u v dx + Z Ω | u | γ u v dx, v ∈ X, is a bounded linear functional we have that N ( λ, u ) is well defined from (2.26). Moreover by using the fact that X = H µ (Ω) and relation (2.25), we can rewrite N ( λ, u ) in the form N ( λ, u ) = u − G ( λ, u ) where G ( λ, u ) := λ L u − H ( u ), < H ( u ) , v > = Z Ω | u | γ uv dx for all v ∈ X. Under condition (1.5), the embedding H µ (Ω) ֒ → L γ +2 (Ω) is compact, implying that the map H is compact. Tocheck condition (2.18) of Theorem 2.3, we derive first the inequality1 || u || X | < H ( u ) , v > | ≤ || u || X || u || γL γ +2 || u || L γ +2 || v || L γ +2 ≤ c || u || γX || v || X . (2.27)Then, we get from (2.27) thatlim || u || X → || H ( u ) || X ∗ || u || X = lim || u || X → sup || v || X ≤ || u || X | < H ( u ) , v > | = 0 . It remains to prove that C λ ,µ is global. We proceed in two steps , adapting the arguments of [18].(a) We shall prove first that all solutions ( λ, u ) ∈ C λ ,µ close to ( λ ,µ ,
0) are positive for all x ∈ Ω. More preciselywe shall prove that there exists ǫ >
0, such that any ( λ, u ( x )) ∈ C λ ,µ ∩ B ǫ (( λ ,µ , u ( x ) >
0, for any x ∈ Ω. Here B ǫ (( λ ,µ , C λ ,µ of center ( λ ,µ ,
0) and radius ǫ )We argue by contradiction, assuming that ( λ n , u n ) is a sequence of solutions of (1.3), such that ( λ n , u n ) → ( λ ,µ ,
0) and that u n are changing sign in Ω. Let u − n := min { , u n } and U − n =: { x ∈ Ω : u n ( x ) < } . Since u n = u + n − u − n is a solution of the problem (1.3) it can be easily seen that u − n , satisfies (in the weak sense) theequation − ∆ u − n − µ u − | x | − λ n u − n + | u n | γ u − n = 0 , (2.28) u − n | ∂ Ω = 0 . Then, multiplying (2.28) with u − n and integrating over Ω we have that Z U − n |∇ u − n | dx − µ Z U − n | u − n | | x | dx − λ n Z U − n | u − n | dx + Z U − n | u n | γ | u − n | dx = 0 . (2.29)Since λ n is a bounded sequence, it follows from (2.29) and H¨older’s inequality that || u − n || H µ ( U − n ) ≤ λ n Z U − n | u − n | dx ≤ C |U − n | qN − N +2 qqN (cid:18)Z U − n | u n | p ∗ (cid:19) p ∗ (2.30) ≤ C |U − n | qN − N +2 qqN || u − n || H µ ( U − n ) . eaction diffusion equations with a singular potential p ∗ is the critical exponent defined in (2.12), for any q ∈ [1 , M ≤ |U − n | , for all n, (2.31)with the constant M being independent of n . We denote now by ˜ u n , = u n / || u n || the normalization of u n . Thenthere exists a subsequence of ˜ u n (not relabelled) converging weakly in H µ (Ω) to some function ˜ u . It can be seenthat ˜ u = u ,µ . Moreover, ˜ u n → u ,µ > L (Ω). Passing to a further subsequence if necessary, by Egorov’sTheorem, ˜ u n → u ,µ uniformly on Ω with the exception of a set of arbitrary small measure. This contradicts(2.31) and we conclude the functions u n cannot change sign.(b) We shall exclude next that for some solution ( λ, u ) ∈ C λ ,µ , there exists a point ξ ∈ Ω, such that u ( ξ ) < C λ ,µ is connected (see Theorem 2.3) and the C ,ζloc (Ω \{ } )- regularity ofsolutions, we deduce that there exists ( λ , u ) ∈ C λ ,µ , such that u ( x ) ≥
0, for all x ∈ Ω, except possibly somepoint x ∈ Ω, such that u ( x ) = 0. Then, the maximum principle (see [5, 13]) and the fact that the solutions aresingular at the origin imply that u ≡ { ( λ n , u n ) } ⊆ C λ ,µ , such that u n ( x ) >
0, for all n and x ∈ Ω, u n → H µ (Ω), and λ n → λ . However, this is true only for λ = λ ,µ . As aconsequence, we have that C λ ,µ cannot cross ( λ,
0) for some λ = λ , and every function which belongs to C λ ,µ is strictly positive. (cid:4) In this subsection we prove Theorem 1.3. The proof is also an alternative approach, to show Theorem 1.1,approximating (1.3) by the family of problems,( A ) r (cid:26) − ∆ u − µ u | x | = λ u − | u | γ u, in Ω r = Ω \ B r (0) ,u | ∂ Ω r = 0 , for some r > u is a weak solution of the problem(( A ) r ), for some r > u belongs at least in C ,ζloc (Ω r ), for some ζ ∈ (0 , AL ) r (cid:26) − ∆ u − µ u | x | = λ u, in Ω r ,u | ∂ Ω r = 0 , admit for any r >
0, a positive principal eigenvalue λ ,µ,r , characterized by λ ,µ,r = inf φ ∈ H (Ω r ) φ R Ω r |∇ φ | dx − µ R Ω r | φ | | x | R Ω r | φ | dx . with the following properties: λ ,µ,r , is simple with a positive associated eigenfunction u ,µ,r and λ ,µ,r is theonly eigenvalue of ( P L ) r , with positive associated eigenfunction. Furthermore, we have the following LEMMA 2.6
Let < µ ≤ µ ∗ , and λ ,µ and λ ,µ,r , be the positive principal eigenvalues of the problems (1.4)and ( AL ) r , respectively. Then(i) u ,µ,r ( x ) ≤ u ,µ ( x ) , for any x ∈ ¯Ω r , and any r > .(ii) u ,µ,r → u ,µ in H µ (Ω) ∩ L ∞ loc (Ω \ { } ) , and λ ,µ,r ↓ λ ,µ , as r ↓ . Proof: (i) Having in mind, that both u λ,r and u λ are sufficiently smooth and positive functions on ¯Ω r , theassertion follows from the comparison principle (cf. [20, Theorem 10.5]).(ii) We extend u ,µ,r on Ω as ˆ u ,r ( x ) =: (cid:26) u ,µ,r ( x ) , x ∈ Ω r , , x ∈ B r , eaction diffusion equations with a singular potential r >
0, using in the sequel for convenience, the same notation u ,µ,r ≡ ˆ u ,µ,r . We notefirst that λ ,µ,r = R Ω r |∇ u ,µ,r | dx − µ R Ω r | u ,µ,r | | x | R Ω r | u ,µ,r | dx = R Ω |∇ u ,µ,r | dx − µ R Ω | u ,µ,r | | x | R Ω | u ,µ,r | dx ≥ λ ,µ . Since Ω r ⊂ Ω r , for any r > r , we deduce that λ ,µ,r is an decreasing sequence as r →
0. Moreover, u ,µ,r forms a bounded sequence in H µ (Ω), thus u ,µ,r ⇀ u ∗ in H µ (Ω) (up to a subsequence), and λ ,µ,r → λ ∗ in R .Then, by the compact embedding H µ (Ω) ֒ → L (Ω) we get that λ ,r Z Ω | u ,r | dx → λ ∗ Z Ω | u ∗ | dx, as r →
0. Therefore, || u ,µ,r || H µ (Ω) → || u ∗ || H µ (Ω) . Hence ( λ ∗ , u ∗ ) must be an eigenpair of (1.4) and from Lemma 2.4 (ii), we infer that ( λ ∗ , u ∗ ) ≡ ( λ , u ). Finally,we consider the difference ψ = u − u λ,r . Standard regularity results imply that || ψ || W , loc (Ω \{ } ) ≤ C || ψ || W , (Ω) + O ( r ) , as r → , for some positive constant C independent from r . By a bootstrap argument, we conclude that u ,µ,r → u ,µ in L ∞ loc (Ω \ { } ). (cid:4) Rabinowitz’s Theorem 2.3, is applicable for the approximating problems ( A ) r , by following closely the argu-ments used in proof of Theorem 1.1. LEMMA 2.7
Assume that < µ ≤ µ ∗ . The principal eigenvalue λ ,µ,r of ( P L ) r is a bifurcating point of theproblem ( P ) r (in the sense of Rabinowitz) and C λ ,µ,r is a global branch of nonnegative solutions , which ”bends”to the right of λ ,µ . For any fixed λ > λ ,µ these solutions are unique. The properties of the global branch C λ ,µ,r can be proved as in Proposition 1.2 (see Subsection 2.4). The nonlinearanalogue of Lemma 2.6 is stated in PROPOSITION 2.8
Assume that < µ ≤ µ ∗ , and let λ be a fixed number, such that ( λ, u λ,r ) ∈ C λ ,µ,r . Then,(i) u λ,r → u λ in H µ (Ω) , with ( λ, u λ ) ∈ C λ ,µ ,(ii) u λ,r ( x ) ≤ u λ ( x ) , for any x ∈ ¯Ω r , and any r ↓ ,(iii) u λ,r → u λ in L ∞ loc (Ω \ { } ) , as r ↓ . Proof: (i) We shall prove first that u λ,r is a bounded sequence in H µ (Ω). We argue by contradiction, assumingthat || u λ,r || H µ (Ω) → ∞ as r ↓ . (2.32)From the weak formulation of the problems ( A ) r we get that u λ,r satisfies the equation Z Ω r |∇ u λ,r | dx − µ Z Ω r | u λ,r | | x | = λ Z Ω r | u λ,r | dx − Z Ω r | u λ,r | γ +2 dx, (2.33)which implies that || u λ,r || H µ (Ω) ≤ λ || u λ,r || L (Ω) , (2.34)for any r small enough. Setting ˜ u λ,r = u λ,r || u λ,r || H µ (Ω) , we get that || ˜ u λ,r || H µ (Ω) = 1, for any r > u λ,r convergesweakly to some ˜ u ∗ in H µ (Ω), as r ↓
0, and so ˜ u λ,r → ˜ u ∗ in L (Ω) as well as in L γ +2 (Ω), as r ↓
0. In addition, itfollows from (2.34) that || ˜ u λ,r || H µ (Ω) ≤ λ || ˜ u λ,r || L (Ω) , for any r > , eaction diffusion equations with a singular potential u ∗
0. Dividing (2.33) by || u λ,r || γ +2 H µ (Ω) we get the equation that Z Ω r | ˜ u λ,r | γ +2 dx = λ || u λ,r || γH µ (Ω) Z Ω r | ˜ u λ,r | dx − || u λ,r || γH µ (Ω) (cid:18)Z Ω r |∇ ˜ u λ,r | dx − µ Z Ω r | ˜ u λ,r | | x | (cid:19) . (2.35)Passing to the limit to (2.35) as r ↓
0, (2.32) implies that ˜ u λ,r → L γ +2 (Ω), contradicting that ˜ u ∗ u λ,r is a bounded sequence in H µ (Ω) converging weakly to some u ∗ in H µ (Ω) as r ↓
0. Then (2.34),implies again that and u ∗
0. Passing to the limit to the weak formulation of (1.3), we deduce that u ∗ is asolution of (1.3). We set u ∗ = u λ . Claims (ii) and (iii) can be proved by similar arguments to those used inLemma 2.6. (cid:4) Proof of Theorem 1.3:
We are making use of Whyburn’s Theorem (see [14] and the references therein).For some
R > r n ↓
0, as n → ∞ , we define the sets A n as follows: A n = (cid:26) B R ( λ , ∩ C λ ,µ,rn (cid:27) ⊂ R × H µ (Ω) . For every n ∈ N , these sets are connected and closed. In addition, Lemma 2.8 implies thatlim inf n →∞ { A n } 6≡ ∅ . We will justify next, that the set S n ∈ N A n is relatively compact i.e., every sequence in A n contains a convergentsubsequence. To this end, we consider ( λ n , u n ) ∈ S n ∈ N A n . Then, the sequence ( λ n , u n ) is bounded in R × H µ (Ω).Henceforth there exists a subsequence still denoted by ( λ n , u n ), such that λ n → λ ∗ and u n ⇀ u ∗ in H µ (Ω), u n → u ∗ in L (Ω). Moreover, u n satisfies (2.34), from which it readily follows that || u n || H µ (Ω) → || u ∗ || H µ (Ω) as n → ∞ . Hence, the subsequence u n converges strongly to u ∗ in H µ (Ω), and arguing as in Proposition 2.8, we get that u ∗ λ ∗ , u ∗ ) is a solution of (1.3). From the same token we have thatlim inf n →∞ { A n } = lim sup n →∞ { A n } 6≡ ∅ . Applying similar arguments, we may let R → ∞ in order to obtain that C λ ,µ,rn → C λ ,µ , in R × H µ (Ω) for any R ∈ R + . (cid:4) C λ ,µ as µ → µ ∗ We conclude in this section, with the discussion on the properties of the global branches C λ ,µ when 0 < µ ≤ µ ∗ .We start with the proof of Proposition 1.2, which actually shows that the global bifurcation is of supercriticaltype. Proof of Proposition 1.2: (i) Assume by contradiction that C λ ,µ bends to the left of λ ,µ . Then there existsa pair ( λ, u ) ∈ R × H µ (Ω) with 0 < λ < λ ,µ , such that Z Ω |∇ u | dx − µ Z Ω u | x | dx = λ Z Ω | u | dx − Z Ω | u | γ +2 dx, (2.36)Last equation implies that || u || H µ (Ω) ≤ λ || u || L (Ω) , with λ < λ , contradicting the variational characterization of λ ,µ . Thus, C λ ,µ must bend to the right of λ ,µ . To show that C λ ,µ is bounded for λ bounded, we consider the weak formula satisfied by any u ∈ C λ ,µ , Z Ω ∇ u ∇ ψdx − Z Ω uψ | x | dx − λ Z Ω uψdx + Z Ω | u | γ uψdx = 0 , for all ψ ∈ H µ (Ω) . (2.37) eaction diffusion equations with a singular potential ψ = u in (2.37) and using the inequality2 λ Z Ω | u | dx ≤ λ | Ω | γγ +1 || u || L γ +2 ≤ || u || γ +2 L γ +2 + R , (2.38) R = (2 λ ) γ +1 γ γ γ ( γ + 1) γ +1 γ | Ω | , we get that any u ∈ C λ ,µ , satisfies the bound || u || H µ (Ω) ≤ R . (2.39)The bound (2.39), shows that any u ∈ C λ ,µ , is bounded for each fixed λ .(ii) Let u ∈ C λ ,µ , and suppose that v is a nonnegative solution of (1.3) with u v . Considering theapproximating solutions u λ,r of the problems ( A ) r , we get from Proposition 2.8 (ii) (comparison principle) that u λ,r ( x ) ≤ min x ∈ Ω { u ( x ) , v ( x ) } . (2.40)Then, by the L ∞ loc -convergence of u λ,r to u of Lemma 2.8 (iii) and (2.40), we infer that u ( x ) ≤ v ( x ) . (2.41)We apply next the weak formula (2.37) for the solutions u and v , setting ψ = v and ψ = u respectively. Subtractingthe resulting equations, we get that Z Ω ( | u | γ v − | v | γ u ) dx = 0 , contradicting (2.41), unless u ≡ v . (cid:4) Finally, we discuss the behavior of the branches C λ ,µ , as µ ↑ µ ∗ . The eigenfunction u ,µ ∗ does not belong in H (Ω), although the eigenfunctions u ,µ , 0 < µ < µ ∗ , belong in H (Ω). Therefore, the behavior of the branches C λ ,µ as µ ↑ µ ∗ should be completely different if considered in H µ ∗ (Ω) and in H (Ω) respectively. Proof of Theorem 1.4:
A. By assumption, the pair ( λ n , u n ), satisfies Z Ω |∇ u n | dx − µ n Z Ω | u n | | x | dx = λ n Z Ω | u n | dx − Z Ω | u n | γ +2 dx, (2.42)which implies that Z Ω |∇ u n | dx − µ n Z Ω | u n | | x | dx ≤ λ n Z Ω | u n | dx. (2.43)On the other hand, by the definition of the H µ ∗ (Ω)-norm and the hypothesis µ n ↑ µ ∗ , it follows that || u n || H µ ∗ (Ω) ≤ Z Ω |∇ u n | dx − µ n Z Ω | u n | | x | dx. (2.44)Combining (2.43) and (2.44) with the assumption that | λ n | ≤ L , we get the estimate || u n || H µ ∗ (Ω) ≤ λ n || u n || L (Ω) < L || u n || L (Ω) . (2.45)We employ an argument similar to the one used in the proof of Proposition 2.8, assuming by contradiction that || u n || H µ ∗ (Ω) → ∞ as n → ∞ . We consider the normalization ˆ u n of u n in H µ ∗ (Ω),ˆ u n = u n || u n || H µ ∗ (Ω) , which is a bounded sequence in H µ ∗ (Ω). Hence, we may extract a subsequence (not relabelled), convergingweakly to some ˆ u ∗ in H µ ∗ (Ω). The compact embedding H µ ∗ (Ω) ֒ → L (Ω) and inequality (2.45) imply thatˆ u ∗
0. Dividing (2.43) by || u n || H µ ∗ (Ω) , we get the inequality Z Ω |∇ ˆ u n | dx − µ n Z Ω | ˆ u n | | x | dx ≤ λ n Z Ω | ˆ u n | dx < ∞ . (2.46) eaction diffusion equations with a singular potential || u n || γ +2 H µ ∗ (Ω) , we get the equation Z Ω | ˆ u n | γ +2 dx = λ n || u n || γH µ ∗ (Ω) Z Ω | ˆ u n | dx − || u n || γH µ ∗ (Ω) (cid:18)Z Ω |∇ ˆ u n | dx − µ n Z Ω | ˆ u n | | x | dx (cid:19) . (2.47)Passing to the limit to (2.47) as n → ∞ , we deduce that u ∗ ≡
0, which is the contradiction. Thus u n must bebounded in H µ ∗ (Ω), and (up to some subsequence) converges weakly to some u ∗ in H µ ∗ (Ω).The strong convergence ( λ n , u n ) → ( λ ∗ , u ∗ ) in R × H µ ∗ (Ω), follows from the compactness of the embedding H µ ∗ (Ω) ֒ → L (Ω) and (2.45). Let us remark that if u ∗ ≡
0, the same argument implies that u n → H µ ∗ (Ω).In this case, division of (2.42) by || u n || H µ ∗ (Ω) and passage to the limit, shows that λ n → λ ,µ ∗ .It remains to prove that the limit ( λ ∗ , u ∗ ) ∈ C λ ,µ ∗ . Note that for any φ ∈ C ∞ (Ω), Z Ω ∇ u n ∇ φ dx − µ ∗ Z Ω u n φ | x | dx − ( µ n − µ ∗ ) Z Ω u n φ | x | dx = λ n Z Ω u n φ dx − Z Ω | u n | γ u n φ dx. Passing to the limit as n → ∞ , we need to show that the integral Z Ω u n φ | x | dx, remains bounded for any φ ∈ C ∞ (Ω) and any n ∈ N . This claim follows by H¨older’s inequality and the continuousembedding H µ ∗ (Ω) ֒ → L p ∗ (Ω), since (cid:12)(cid:12)(cid:12)(cid:12)Z Ω u n φ | x | dx (cid:12)(cid:12)(cid:12)(cid:12) ≤ || φ || L ∞ (Ω) || u n || L p ∗ (Ω) Z Ω | x | − NqNq − N + q dx. (2.48)The integral in the right hand side of (2.48) converges if q > NN − . Combining this requirement with (1.5), thecondition (1.6) follows for the case N = 3. When N ≥
4, the claim is valid under the condition (1.5).B. Let ( λ n , u n ) ∈ C λ ,µn , and assume that µ n ↑ µ ∗ and λ n → λ ,µ ∗ as n → ∞ . Assuming further that u n remainsbounded in H (Ω), we may extract a subsequence still denoted by u n , which converges weakly to some u ∗ in H (Ω). Passing to the limit in the weak formula as n → ∞ , it follows that u ∗ , µ ∗ , λ ,µ ∗ , satisfy Z Ω ∇ u ∗ ∇ φ dx − µ ∗ Z Ω u ∗ φ | x | dx = λ ,µ ∗ Z Ω u ∗ φ dx − Z Ω | u ∗ | γ u ∗ φ dx, for any φ ∈ C ∞ (Ω). However, the variational characterization of λ ,µ ∗ implies that this is true only if u ∗ ≡ u n ⇀
0, in H (Ω). On the other hand, arguing as in part A., it can be seen from (2.42) that thenormalization ¯ u n = u n / || u n || H (Ω) converges (up to a subsequence) weakly to u ,µ ∗ in H (Ω) which is impossible.Thus, u n must be unbounded in H (Ω). (cid:4) In this section we shall define a gradient semiflow associated to the semilinear parabolic equation (1.2), S ( t ) : H µ (Ω) → H µ (Ω) , < µ ≤ µ ∗ , (3.49)with J ( φ ) := 12 Z Ω |∇ φ | dx − µ Z Ω | φ | | x | dx − λ Z Ω | φ | dx + 12 γ + 2 Z Ω | φ | γ +2 dx, < µ ≤ µ ∗ , (3.50)as a Lyapunov functional. In subsection 3.1, we discuss the stability properties of the equilibrium solutions bylinearization. In subsection 3.2, and by following closely the general semiflow theory [2, 17, 23], we present theproof of Theorem 1.5, as well as the description of the limit set ω ( φ ) for nonnegative (nonpositive) initial data φ , φ eaction diffusion equations with a singular potential Seeking for nonpositive stationary solutions u = − u − with u − ≥ u −
0, it is clear that u − satisfies (1.3).Therefore, Theorem 1.1, can be restated as COROLLARY 3.1
Let Ω ⊂ R N , N ≥ , be a bounded domain. Assume that < µ ≤ µ ∗ , and that condition(1.5) is satisfied. Then, the principal eigenvalue λ ,µ of (1.4) considered in H µ (Ω) , is a bifurcating point of theproblem (1.3) (in the sense of Rabinowitz) and C λ ,µ and C − λ ,µ are global branches of nonnegative and nonpositive H µ (Ω) - solutions respectively, which bend to the right of λ ,µ . For any fixed λ > λ ,µ , every solution u ∈ C λ ,µ and u − ∈ C − λ ,µ is the unique nonnegative and unique nonpositive solutions for the problem (1.3) and u − = − u . We first verify that solutions of (1.2) initiating from nonnegative (nonpositive) initial data remain nonnegative(nonpositive) for all times. Then, we will proceed with the asymptotic stability of the nonnegative equilibrium bylinearization. For the latter, Hardy’s inequalities and their improvements, allow for the definition of appropriateGarding forms, helping us to verify that zero is not an eigenvalue for the linearized flow around the nonnegative(nonpositive) equilibrium.
LEMMA 3.2
Assume that µ ≤ µ ∗ . The set D +( − ) := (cid:8) φ ∈ H µ (Ω) : φ ( x ) ≥ ( ≤ )0 on Ω (cid:9) , is a positively invariant set for the semiflow S ( t ) . Proof:
The argument of [11, Proposition 5.3.1] for the linear heat equation, can be repeated here (see also [18]).We assume that φ ∈ H µ (Ω), φ ≥ φ ( t ) = S ( t ) φ , the global in time solution of (1.2), initiatingfrom φ . We consider φ + := max { φ, } , φ − := − min { φ, } . Both φ + and φ − are nonnegative, and φ = φ + − φ − .It can be seen from (1.2) that φ − satisfies the equation ∂ t φ − − ∆ φ − − µ φ − | x | − λ φ − + | φ | γ φ − = 0 . (3.1)Moreover, φ − satisfies the energy equation (see Proposition 1.5),12 ddt || φ − || L + Z Ω |∇ φ − | dx − µ Z Ω | φ − | | x | dx − λ || φ − || L + Z Ω | φ | γ | φ − | dx = 0 . (3.2)From (3.2) and (2.19), we get that 12 ddt || φ − || L ≤ c || φ − || L . where c = λ ,µ − λ . Thus φ − satisfies || φ − ( t ) || L ≤ e ct || φ − || L = 0 , for every t ∈ [0 , + ∞ ) , (3.3)implying that φ ≥ t ∈ (0 , + ∞ ), a.e. in Ω. (cid:4) PROPOSITION 3.3
Let µ ≤ µ ∗ . The unique nonnegative (nonpositive) equilibrium point which exists for λ > λ ,µ is uniformly asymptotically stable. On the account of Corollary 3.1, we consider only the nonnegative equilibrium u ≥ u
0. First, we observethat the linearized semiflow around the zero solution, is defined by the Cauchy-Dirichlet problem ∂ t ψ − ∆ u − µ u | x | − λ ψ = 0 , x ∈ Ω ,ψ | ∂ Ω = 0 . We have that φ = 0 is asymptotically stable in H µ (Ω) if λ ≤ λ ,µ , and unstable in H µ (Ω) if λ > λ ,µ . The linearizedsemiflow around the nonnegative equilibrium point u of (1.2), is defined by the Cauchy-Dirichlet problem − ∆ ψ − µ ψ | x | − λ ψ + (2 γ + 1) | u | γ ψ = 0 , (3.4) ψ | ∂ Ω = 0 , eaction diffusion equations with a singular potential u , we will prove that ˜ µ = 0, is not an eigenvalue for the eigenvalue problem − ∆ ψ − µ ψ | x | − λ ψ + (2 γ + 1) | u | γ ψ = ˜ µψ, (3.5) ψ | ∂ Ω = 0 . The weak formulation of (3.5) is A µ ( ψ, ω ) := Z Ω ∇ ψ ∇ ω dx − µ Z Ω ψ ω | x | dx − λ Z Ω ψ ω dx + (2 γ + 1) Z Ω | u | γ ψ ω dx = ˜ µ Z Ω ψ ω dx, (3.6)for every ω ∈ H µ (Ω). Using the improved Hardy’s inequality and the properties of the H µ (Ω)-space for any0 < µ ≤ µ ∗ , we may consider a symmetric bilinear form A µ : H µ (Ω) × H µ (Ω) → R , which in turns, defines aGarding form [25, pg. 366]: Since A µ ( ψ, ψ ) ≥ || ψ || H µ (Ω) − λ || ψ || (Ω) , Garding’s inequality is satisfied. Then, it follows from [25, Theorem 22.G pg. 369-370] and (2.8), that the problem(3.5) has infinitely many eigenvalues of finite multiplicity. Counting the eigenvalues according to their multiplicity,we derive the sequence − λ < ˜ µ ≤ ˜ µ ≤ · · · , and ˜ µ j → ∞ as j → ∞ . (3.7)The smallest eigenvalue can be characterized by the minimization problem˜ µ = min A µ ( ψ, ψ ) , ψ ∈ H µ (Ω) , || ψ || L = 1 . (3.8)The j -th eigenvalue, can be characterized by the minimum-maximum principle˜ µ j = min M ∈L j max ψ ∈ M A µ ( ψ, ψ ) . (3.9)where M = { ψ ∈ H µ (Ω) : || ψ || L = 1 } and L j denotes the class of all sets M ∩ L with L an arbitrary j -dimensionallinear subspace of H µ (Ω).By using similar arguments as for the proof of Lemma 2.4 (see also Lemma 2.6), we may see that for (3.5),the (nontrivial) eigenfunction corresponding to the principal eigenvalue ˜ µ is nonnegative, i.e ψ ≥ µ , ψ satisfy (3.6) we get by setting ω = u that Z Ω ∇ ψ ∇ u dx − µ Z Ω ψ u | x | dx − λ Z Ω ψ u dx + (2 γ + 1) Z Ω | u | γ ψ u dx = ˜ µ Z Ω ψ u dx. On the other hand, by setting ψ = ψ to the weak formula (2.37) we get Z Ω ∇ ψ ∇ u dx − µ Z Ω ψ u | x | dx − λ Z Ω ψ u dx + Z Ω | u | γ ψ u dx = 0 . Subtracting these equations, we obtain that2 γ Z Ω | u | γ u ψ dx = ˜ µ Z Ω u ψ dx, (3.10)which implies that ˜ µ >
0. Thus ˜ µ = 0 is not an eigenvalue, and u is uniformly asymptotically stable. (cid:4) eaction diffusion equations with a singular potential H µ (Ω) for any < µ ≤ µ ∗ The proof of Proposition 1.5 is based on the analogue of [2, Theorem 3.6, pg. 40], this time for the parabolicequation (1.2).
Proof of Proposition 1.5:
It follows from [24], that the operator L = − ∆ − µ | x | with domain of definition (2.4)is a generator of a strongly continuous semigroup T ( t ), for any 0 < µ ≤ µ ∗ , while the function f ( s ) = | s | γ s − λs ,defines a locally Lipschitz map f : H µ (Ω) → L (Ω) as it can be easily deduced by Lemma 2.2. Thus for any0 < µ ≤ µ ∗ and any φ ∈ H µ (Ω), there exists a unique solution φ of (1.2), defined on a maximal interval [0 , T max )and in the class C([0 , T ]; H µ (Ω) ∩ C ([0 , T ]; L (Ω)). The solution satisfies the variation of constants formula φ ( t ) = T ( t ) φ + Z t T ( t − s ) f ( φ ( s )) ds. (3.11)Lemma 2.2 implies also that the functional J ∈ C ( R , H µ (Ω)), for any 0 < µ ≤ µ ∗ . Moreover, for all φ ∈ D ( L )and any t ∈ [0 , T ], T < T max , (cid:28) ∆ φ + µ φ | x | + f ( φ ) , J ′ ( φ ) (cid:29) = − Z Ω (cid:12)(cid:12)(cid:12)(cid:12) ∆ φ + µ φ | x | + f ( φ ) (cid:12)(cid:12)(cid:12)(cid:12) dx = − Z Ω | ∂ t φ | dx ≤ . (3.12)Setting h ( t ) = f ( φ ( t )), we consider the sequence h n ( t ) ∈ C ([0 , T ]; H µ (Ω)) and φ n ∈ D ( L ) such that h n → h, in C ([0 , T ]; H µ (Ω)) ,φ n → φ , in H µ (Ω) . We define φ n ( t ) = T ( t ) φ n + R t T ( t − s ) h n ( s ) ds , and it follows from [19, Corrolary 2.5, p107] that φ n ( t ) ∈ D ( L ), φ n ∈ C ([0 , T ]; H µ (Ω)) and that they satisfy ∂ t φ n − ∆ φ n − µ φ n | x | + f ( φ n ) = 0 . (3.13)Moreover, from [1, Lemma 5.5, pg. 246-247] or [2, Theorem 3.6, pg. 41], we deduce that φ n → φ, in H µ (Ω) . Finally, by using the continuity of J and (3.12), and passing to the limit to the equation J ( φ n ( t )) − J ( φ n ) = Z t (cid:28) J ′ ( φ n ( s )) , ∆ φ n ( s ) + µ φ n ( s ) | x | + h n ( s ) (cid:29) ds = − Z t || ∂ t φ n ( s ) || L ds + Z t hJ ′ ( φ n ( s )) , h n ( s ) − f ( φ n ( s )) i ds, we derive ddt J ( φ ( t )) = − Z Ω | ∂ t φ | dx, for all 0 < µ ≤ µ ∗ and t ∈ [0 , T ] , T < T max . (3.14)From (3.14) we infer that the unique solution φ , satisfies the energy equation ddt || φ || L + Z Ω |∇ φ | dx − µ Z Ω | φ | | x | − λ || φ || L + Z Ω | φ | γ +2 dx = 0 , for all 0 < µ ≤ µ ∗ . (3.15)When λ ≤ λ ,µ , we observe by using (2.19), that lim sup t →∞ || φ ( t ) || L = 0. For the case λ > λ ,µ , we insert (2.38)to (3.15), to get the estimate12 ddt || φ || L + 12 || φ || H µ (Ω) + λ || φ || L + 12 || φ || γ +2 L γ +2 dx ≤ R . eaction diffusion equations with a singular potential || φ ( t ) || L ≤ || φ (0) || L exp( − λt ) + R λ (1 − exp( − λt )) . (3.16)Letting t → ∞ , from (3.16) we obtain thatlim sup t →∞ || φ ( t ) || L ≤ ρ , ρ = R /λ. (3.17)Now assume that φ is in a bounded set B of H µ (Ω). Then (3.17) implies that for any ρ > ρ , there exists t ( B , ρ ), such that || φ ( t ) || L ≤ ρ , for any t ≥ t ( B , ρ ) . (3.18)By the definition of the Lyapunov functional J and (3.18), we have the inequality J ( φ ( t )) ≥ Z Ω |∇ φ | dx − µ Z Ω | φ | | x | dx − λ Z Ω | φ | dx ≥ Z Ω |∇ φ | dx − µ Z Ω | φ | | x | dx − λ ρ , t ≥ t . (3.19)Since J is nonincreasing in t , we conclude with the bound || φ ( t ) || H µ (Ω) ≤ J ( φ ) + λρ , t ≥ t . (3.20)establishing that solutions are globally defined in H µ (Ω), for any 0 < µ ≤ µ ∗ and λ > λ ,µ . In addition, (3.20)implies that the semiflow S ( t ) is eventually bounded and since the operator L has compact resolvent, S ( t ) isasymptotically compact (cf. [2, Proposition 2.3, pg. 36], [17, 23]). Thus, the positive orbit γ + ( φ ) for any φ ∈ H µ (Ω) is precompact and has a nonempty compact and connected invariant ω -limit set ω ( φ ). Moreover(3.14) implies that ω ( φ ) ∈ E . Equilibria of S ( t ) are extreme points of J , satisfying the weak formula (2.37).From (2.39), we have that E is bounded for any fixed λ . Hence S ( t ) is point dissipative. (cid:4) Proof of Theorem 1.6:
Lemma 3.2 and Proposition 1.5, imply that the solution φ ( t ) = S ( t ) φ , initiating frominitial data φ ≥ φ ≤ φ t → ∞ , in H µ (Ω), for any 0 < µ ≤ µ ∗ . In fact, it follows from Theorem 1.1 that the set of equilibriumsolutions E = { u − , , u } , when λ > λ ,µ , the trivial solution being unstable by Proposition 3.3. Thus for anynonnegative (nonpositive) initial condition φ , ω ( φ ) = { u } ( ω ( φ ) = { u − } ). While in the case λ < λ ,µ , Theorem1.1 combined with Propositions 3.3 and 1.5, imply that dist( S ( t ) B , { } ) → t → ∞ , for every bounded set B ⊂ H µ (Ω). Thus, when λ < λ ,µ the global attractor A = { } . (cid:4) Acknowledgements.
We would like to thank Prof. Anargyros Delis for stimulating discussions.
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