Mathematics in Applied Sciences and Engineering https://ojs.lib.uwo.ca/mase Volume 1, Number 4, December 2020, pp.275-285 https://doi.org/10.5206/mase/9314 A SOLUTION TO A FRACTIONAL ORDER SEMILINEAR EQUATION USING VARIATIONAL METHOD RAMESH KARKI AND YOUNG HWAN YOU Abstract. We will discuss how we obtain a solution to a semilinear pseudo-differential equation involving fractional power of Laplacian by using a method analogous to the direct method of calculus of variations. More precisely, we will discuss the existence of a weak form of solutions to this equation as a minimizer of a suitable energy-type functional whose Euler-Lagrange equation is the semilinear equation, and also discuss the possibility of regularity of such a weak solution so that it will be a solution to the semilinear equation. 1. Introduction We consider N as a fixed positive integer greater than 1 throughout this paper. We call a function u : Rd → R a NZd-periodic function if and only if u(x + Nej) = u(x) for all j = 1, 2, . . . , d, where each ej = (0, 0, . . . , 1, . . . , 0) is the j th standard unit vector in Rd. We assume that f : Rd ×R → R is an NZd+1-periodic continuous function such that∫ y+N 0 f(x,z) dz = ∫ y 0 f(x,z) dz (1.1) for all x ∈ Rd, y ∈ R. Under these assumptions on f, our main goal is to study the existence of a weak form of solutions to a semilinear pseudo-differential equation (ΨDE) of the form −(−∆)αu(x) = f(x,u(x)), x ∈ Rd, (1.2) where ∆ is a d-dimensional Laplacian and α is a fixed real number with 0 < α < 1. Because of nonlocal behaviors of the fractional power of Laplacian, we will consider the problem under the periodic boundary setting. This will allow us to reduce the problem into a variational type problem, or more precisely, a problem of finding a minimizer of an energy type functional I associated with Equation (1.2). Here, the main idea is to use a method that is analogous to a widely used classical method, the direct method of calculus of variations, for studying solutions to nonlinear partial differential equations (see [6, 9, 12] etc.). Problems similar to the one we mentioned above have been widely studied. Under a stronger assump- tion that f mentioned above is also infinitely smooth, R. de la Llave and E. Valdinnoci have studied the existence of a special (Birkhoff) class of solutions to a problem similar to ours by using the steepest descent method in [8]. In this method, they consider u as a real-valued function of a spacial variable x ∈ Rd and a forward time parameter t > 0, then consider a steepest descent equation corresponding to Received by the editors 19 January 2020; firstly revised 23 May 2020, secondly revised 1 September 2020; accepted 28 October 2020; published online 09 November 2020. 2000 Mathematics Subject Classification. Primary 35S15. Key words and phrases. energy-type functional, minimizer, weak solution, lower semicontinuity, minimizing sequence, regularity. 275 276 R. KARKI AND Y. H. YOU the problem, reduce this equation to the abstract Cauchy problem (a type of evolutionary problem in an infinite dimensional space), and find a solution (in the Birkhoff class) to the problem as an equilibrium solution. In [3], T. Blass, R. de la Llave and E. Valdinnoci have studied the existence of this type of solution to the semilinear elliptic equation Au + f(x,u) = 0, x ∈ Rd (1.3) involving the positive definite uniformly elliptic self-adjoint operator A = − ∑d i,j=1(a ij(x)uxi)xj of order 2 with smooth symmetric coefficients aij by using the Sobolev gradient descent method which is analogous to the steepest descent method but is more specific than the latter one. In [14], R. Karki has generalized this method to the semilinear pseudo-differential equation Aαu + f(x,u) = 0, x ∈ Rd (1.4) involving the fractional power of the operator A. The advantage of studying these semilinear equations by using the steepest or (Sobolev) gradient descent method is that the solution to the corresponding steepest or gradient descent equation converges to an equilibrium solution at a much faster rate, which can be seen via numerical simulation (see [2]). Other advantages could be exploring the the dynamics of the gradient descent equations corresponding to the problems, which can be applicable in different fields such as mathematical physics, math finance, engineering, mechanics, and Geology etc. Despite having these advantages, we use neither of these methods to solve our problem. Using the variational approach, J. Moser has thoroughly discussed the existence, the regularity and many other properties of solutions to a problem analogous to ours in [15, 16, 17], especially the case when (−∆)α is replaced by −∆. Now we use this variational approach to find a weak solution to our semilinear fractional Poisson type equation (1.2). By obtaining a weak solution to Equation (1.2), we will devise a powerful tool that is needed to obtain a classical solution to Equation (1.2). Fractional Poisson type equations (special cases of Equation (1.2)) are studied when exploring anoma- lous diffusion in the context of geophysical electromagnetics (see [24]) and near-surface geotechical engineering as non-local electromagnetic effects occur due to fractures and stratigraphic layering (see [11, 23]). For the case λ = 1 2 , Equation (1.2) and the problem similar to ours appear in many fields, and have been widely studied for last several years. For instance, they appear in the theory of water waves when approximating the Dirichlet problem to the Neumann problem (see [7, 18]). Their applications to the ultrarelativistic limit of quantum mechanics (see [10]) and recently, to phase transition problems involving fractional powers of Laplacian have appeared (see [1, 4, 13]). The operator (−∆) 1 2 also plays a vital role in the thin obstacle problem (see [5, 22]). We will prove the existence of a weak solution to Equation (1.2) in two main steps. First, we will show that a regular enough minimizer (i.e., a member of a suitable Sobolev space) of the energy-type functional I is a weak solution to Equation (1.2), and then show that the functional I does indeed have a minimizer. Schematically, we will first discuss some function spaces with respective norms, and some operators such as Laplacian and fractional Laplacian defined on these spaces in Section 2.1. In Section 2.2, we will introduce a few key terms such as an energy-type functional I whose Euler-Lagrange equation is Equation (1.2), a minimizer of I and a weak form of solution to Equation (1.2), and show that a minimizer with sufficient regularity is indeed such a solution. In Section 2.3, we will prove the existence of weak solutions to Equation (1.2) in the form of minimizer of I. Finally, we will comment on possibilities of proving regularity results related to a weak solution to Equation (1.2), and also briefly discuss other stronger forms of solutions to Equation (1.2) in Section 3. A SOLUTION VIA VARIATIONAL METHOD 277 2. Existence of weak solutions in the form of minimizers 2.1. Some preliminaries. In this subsection, we will characterize some function spaces and their norms using the ideas adopted in [14]. Such characterizations will provide us some tools which will be really handy while working with these spaces and their norms later. We use the notation NTd to denote quotient space Rd/NZd, which is the same as the d-dimensional square [0,N]d after identifying its opposite sides. In other words, NTd is a d-dimensional torus. We use C∞(NTd) to denote the space of all smooth functions u : Rd → R that are NZd-periodic and equip C∞(NTd) with the uniform norm. Any u in C∞(NTd) has a Fourier series representation u(x) = ∑ j∈Zd e 2π N i〈x,j〉ûj, x ∈ Rd (2.1) where each ûj is the j th Fourier coefficient of u and is given by ûj = 1 Nd ∫ NTd e− 2π N i〈x,j〉u(x) dx. (2.2) We use L2(NTd) to denote the space of all square integrable functions u : Rd → R that are NZd- periodic. Then L2(NTd) is the completion of C∞(NTd) with respect to the norm ‖u‖L2(NTd) = ∑ j∈Zd |ûj|2. (2.3) Using (2.1) together with the convergence in L2(NTd), each u in L2(NTd) can be expressed as u(x) = ∑ j∈Zd e 2π N i〈x,j〉ûj, x ∈ Rd (2.4) where the equality needs to be understood in almost everywhere sense. We consider the Sobolev space Hs(NTd) with s > 0 as the completion of C∞(NTd) in L2(NTd) under the norm ‖u‖Hs(NTd) = ∥∥∥(I + (−∆)s) 12 u∥∥∥ L2(NTd) . (2.5) Throughout the rest of this subsection, we will be referring to s > 0. As we can express the operator −∆ : D(−∆) ⊆ L2(NTd) → L2(NTd) as (−∆)u(x) = ( 2π N )2 ∑ j∈Zd |j|2ûje 2π N i〈x,j〉, (2.6) we can use the spectral integral from the spectral theory of unbounded self-adjoint operators on Hilbert spaces (see [19, 20, 21]) to express the operator (−∆)s on L2(NTd) as (−∆)su(x) = ( 2π N )2s ∑ j∈Zd |j|2sûje 2π N i〈x,j〉. (2.7) Therefore, by the virtue of Equations (2.3) - (2.7), we have ‖u‖2Hs(NTd) = ∑ j∈Zd [ 1 + ( 2π N )2s |j|2s ] |ûj|2. (2.8) Also, u ∈ D((−∆)s) if and only if u ∈ L2(NTd) and( 2π N )2s ∑ j∈Zd |j|2s|ûj|2 < ∞, 278 R. KARKI AND Y. H. YOU which is true if and only if ‖u‖2Hs(NTd) < ∞, meaning u ∈ H s(NTd). Moreover, for each u ∈ D((−∆)s) we can define ‖u‖Ḣs(NTd) =  (2π N )2s ∑ j∈Zd |j|2s|ûj|2   1 2 , (2.9) to obtain a seminorm (not a norm) ‖·‖Ḣs(NTd) on D((−∆) s). Actually, the mean û0 = 1 Nd ∫ NTd u(x) dx for any u ∈ D((−∆)s) over NTd has no contribution in ‖u‖Ḣs(NTd) even if u is nonzero. 2.2. Minimizers and weak solutions. In order to study solutions to the semilinear ΨDE (1.2), we first consider the energy-type functional I(u) = ∫ NTd { 1 2 [(−∆) α 2 u(x)]2 + F(x,u(x)) } dx, (2.10) defined on a subspace of L2(NTd), where F(x,y) = ∫ y 0 f(x,z) dz, x ∈ Rd, y ∈ R, (2.11) and then study the critical values of I. Among those critical values, we are basically interested on minimizers of I as discussed in Section 1. We observe that I is naturally defined for all u ∈ Hα(NTd). Taking this into account, we introduce a minimizer of I and a weak solution to Equation (1.2). Definition 2.1 (Minimizer). A u ∈ Hα(NTd) is a minimizer of I if I(u + φ) ≥ I(u) for all φ ∈ C∞(NTd). It follows from Definition 2.1 that if u ∈ Hα(NTd) is a minimizer of I, then d dt I(u + tφ)|t=0 = 0 (2.12) for all φ ∈ C∞(NTd). Definition 2.2 (Weak solution). A u ∈ Hα(NTd) is a weak solution to Equation (1.2) if u satisfies 〈(−∆) α 2 u, (−∆) α 2 φ〉 + 〈f(.,u),φ〉 = 0 (2.13) for all φ ∈ C∞(NTd). Now we establish a fundamental result that relates a minimizer of I to a weak solution to Equation (1.2). Theorem 2.1. Let f : Rd × R → R be an NZd+1-periodic continuous function such that Equation (1.1) holds. If u ∈ Hα(NTd) is a minimizer of I given by Equation (2.10), then u is a weak solution to Equation (1.2). Proof. Let u ∈ Hα(NTd) be a minimizer of I and let φ ∈ C∞(NTd). Then I(u + tφ) − I(u) = ∫ NTd { 1 2 [ (−∆) α 2 (u + tφ) ]2 + F(.,u + tφ) } − ∫ NTd { 1 2 [ (−∆) α 2 u ]2 + F(.,u) } = ∫ NTd 1 2 { 2t(−∆) α 2 u(−∆) α 2 φ + t2 [ (−∆) α 2 φ ]2} + ∫ NTd {F(.,u + tφ) −F(.,u)} , A SOLUTION VIA VARIATIONAL METHOD 279 so we have I(u + tφ) − I(u) t = ∫ NTd (−∆) α 2 u(−∆) α 2 φ + t 2 ∫ NTd [ (−∆) α 2 φ ]2 (2.14) + ∫ NTd F(.,u + tφ) −F(.,u) t Since u ∈ Hα(NTd) and φ ∈ C∞(NTd), we have (−∆) α 2 u, (−∆) α 2 φ ∈ L2(NTd), so (−∆) α 2 u(−∆) α 2 φ,[ (−∆) α 2 φ ]2 ∈ L1(NTd). Thus the first two integrals on the right side of Equation (2.14) are finite. Moreover, since f is NZd+1-periodic on Rd × R, it is bounded there and, therefore, there exists a real number M > 0 such that |Fy(x,y)| = |f(x,y)| ≤ M for all (x,y) ∈ Rd ×R. Thus we have∣∣∣∣F(.,u + tφ) −F(.,u)t ∣∣∣∣ ≤1t |F(.,u + tφ) −F(.,u)| = 1 t ∣∣∣∣ ∫ t 0 d ds F(.,u + sφ) ds ∣∣∣∣ = 1 t ∣∣∣∣ ∫ t 0 Fy(.,u + sφ)φds ∣∣∣∣ ≤ 1 t ∫ t 0 |Fy(.,u + sφ)||φ|ds ≤ M|φ| for all t > 0. Since M|φ| ∈ L1(NTd), the Dominated Convergence Theorem implies that lim t→0 ∫ NTd F(.,u + tφ) −F(.,u) t exists and equals ∫ NTd Fy(.,u)φ = ∫ NTd f(.,u)φ. Therefore, letting t → 0 on the both sides of Equation (2.14), we obtain d dt I(u + tφ)|t=0 = 〈(−∆) α 2 u, (−∆) α 2 φ〉 + 〈f(.,u),φ〉. (2.15) Also, Equation (2.12) holds true for a minimizer u ∈ Hα(NTd) of I. From Equation (2.12) and Equation (2.15), we obtain Equation (2.13). Hence u is a weak solutions to Equation (1.2). � Theorem 2.1 guarantees that in order to find weak solutions to Equation (1.2), it suffices to prove the existence of a minimizer of I in Hα(NTd). Theorem 2.2. Let f : Rd×R → R be an NZd+1-periodic continuous function such that Equation (1.1) holds. Then there exists u ∈ Hα(NTd) such that u is a minimizer of I given by Equation (2.10), and hence a weak solution to Equation (1.2). 2.3. Proof of Theorem 2.2. We will complete the proof of Theorem 2.2 by subsequently proving a few results. The first of them is related to a coercive condition satisfied by I in Hα(NTd). Proposition 2.3 (Coercivity). Suppose I is given by Equation (2.10). Then there exists a positive constant Λ depending on f, N and d such that I(u) ≥ 1 2 ( 1 −N−d )[ 1 + ( 2π N )2α]−1 ‖u‖2Hα(NTd) − Λ (2.16) for all u ∈ Hα(NTd). 280 R. KARKI AND Y. H. YOU Proof. Notice that F given by Equation (2.11) is continuous and NTd+1-periodic on Rd × R. Let Λ0 be a positive real number such that |F(x,y)| ≤ Λ0 for all (x,y) ∈ Rd ×R. Then the integrand L((−∆) α 2 u(x),x,u(x)) := 1 2 [ (−∆) α 2 u(x) ]2 + F(x,u(x)) (2.17) of I satisfies the condition L(p,x,z) := 1 2 p2 + F(x,z) ≥ 1 2 p2 − Λ0 (2.18) for all (p,x,z) ∈ R×Rd ×R. Let u ∈ Hα(NTd). First, using (2.18) into (2.10), then using (2.3), (2.7), we get I(u) ≥ 1 2 ∥∥(−∆) α2 u∥∥2 L2(NTd) − Λ0N d = 1 2 ∑ j∈Zd ( 2π N )2α |j|2α|ûj|2 − Λ0Nd and next using Equation (2.9), we get I(u) ≥ 1 2 ‖u‖Ḣα(NTd) − Λ0N d. (2.19) Taking j = 0 ∈ Zd in Equation (2.2) and using the Cauchy-Schwartz Inequality, we get |û0| ≤ 1 Nd ∫ NTd |u(x)| dx ≤N−d · (Nd) 1 2 ‖u‖L2(NTd) =N− d 2 ‖u‖L2(NTd) Using the last inequality into Equation (2.8), we have ‖u‖2Hα(NTd) ≤ ∑ j∈Zd−{0} [ 1 + ( 2π N )2α |j|2α ] |ûj|2 + N−d‖u‖ 2 L2(NTd) ≤ ∑ j∈Zd−{0} |j|2α [ 1 + ( 2π N )2α] |ûj|2 + N−d‖u‖ 2 L2(NTd) ≤ [ 1 + ( 2π N )2α] ∑ j∈Zd |j|2α|ûj|2 + N−d‖u‖ 2 Hα(NTd) = [ 1 + ( 2π N )2α] ‖u‖2Ḣα(NTd) + N −d‖u‖2Hα(NTd) , from which, we obtain ‖u‖2Ḣα(NTd) ≥ ( 1 −N−d )[ 1 + ( 2π N )2α]−1 ‖u‖2Hα(NTd) . (2.20) Combining Inequality (2.19) and Inequality (2.20), we obtain I(u) ≥ 1 2 ( 1 −N−d )[ 1 + ( 2π N )2α]−1 ‖u‖2Hα(NTd) − Λ0N d and hence I(u) ≥ 1 2 ( 1 −N−d )[ 1 + ( 2π N )2α]−1 ‖u‖2Hα(NTd) − Λ A SOLUTION VIA VARIATIONAL METHOD 281 for some positive constant Λ depending on f, N and d. � The next important result required to prove Theorem 2.2 is related to weakly lower semicontinuity of I in Hα(NTd). Recall that I is weakly lower semicontinous at u ∈ Hα(NTd) if and only if for every sequence {uk} in Hα(NTd) converging weakly in Hα(NTd) to a limit u, I(u) ≤ lim inf k→∞ I(uk) and is weakly lower semicontinuous in Hα(NTd) if and only if it is weakly lower semicontinuous at each u ∈ Hα(NTd). Proposition 2.4 (Weakly lower semicontinuity). I is weakly lower semicontinuous in Hα(NTd). Proof. Consider any u ∈ Hα(NTd) and any sequence {uk} in Hα(NTd) that converges weakly in Hα(NTd) to u and set m = lim inf k→∞ I(uk). To complete the proof, we need to show that I(u) ≤ m. Since a weakly convergent sequence is bounded, we have sup k ‖uk‖Hα(NTd) < ∞. From definition of limit infimum, we may assume, by passing to a subsequence if necessary, that m = lim k→∞ I(uk). (2.21) By the Sobolev Embedding Theorem, the inclusion Hα(NTd) ↪→ L2(NTd) is compact. So, we may as- sume, by passing another subsequence of the last subsequence of our original sequence {uk} if necessary, that {uk} converges strongly in L2(NTd) to u, which then implies that uk → u a. e. in Ω, where Ω = NTd. Let � > 0 be given. Then, by Egoroff’s Theorem, there exists a measurable subset E� of Ω with |Ω −E�| ≤ � (here |Ω −E�| denotes the Lebesgue measure of Ω −E�) such that uk → u uniformly on E�. Define a subset F� of Ω by F� = { x ∈ Ω : |u(x)| + |(−∆) α 2 u(x)| ≤ 1 � } . As � → 0, |Ω −F�|→ 0. Define another subset G� of Ω by G� = E� ∩F�. As � → 0, |Ω −G�| ≤ |Ω −E�| + |Ω −F�|→ 0. Since L given by Equation (2.17) is bounded from below, we may assume without loss of generality that L ≥ 0 (otherwise, we may consider L̃ = L + Λ0 ≥ 0 where Λ0 is as in the proof of Proposition 2.3). Therefore, from I(uk) = ∫ Ω 1 2 [ (−∆) α 2 uk ]2 + F(.,uk) 282 R. KARKI AND Y. H. YOU for all k, we have I(uk) ≥ ∫ G� 1 2 [ (−∆) α 2 uk ]2 + F(.,uk) (2.22) for all k. Recall that if a map φ : R → R is convex, then φ(y) ≥ φ(x) + φ′(x)(y −x) for all y,x ∈ R. Since the map p 7→ 1 2 p2 mapping R into itself is convex, 1 2 p2k ≥ 1 2 p2 + p(pk −p) for all pk,p ∈ R. So, we have 1 2 ( (−∆) α 2 uk(x) )2 ≥ 1 2 ( (−∆) α 2 u(x) )2 + (−∆) α 2 u(x) ( (−∆) α 2 uk(x) − (−∆) α 2 u(x) ) for all x ∈ Rd and for all k, and thus∫ G� 1 2 [ (−∆) α 2 uk ]2 ≥ ∫ G� 1 2 [ (−∆) α 2 u ]2 (2.23) + ∫ G� (−∆) α 2 u ( (−∆) α 2 uk − (−∆) α 2 u ) . Since uk → u uniformly on G�, for each x ∈ Rd we have (−∆) α 2 uk(x) − (−∆) α 2 u(x) =(−∆) α 2 (uk −u)(x) = ∑ j∈Zd ( 2π N )α |j|α ̂(uk −u)je 2π N i〈x,j〉 = ∑ j∈Zd ( 2π N )α |j|α · 1 Nd ∫ Ω e− 2π N i〈y,j〉(uk(y) −u(y)) dy e 2π N i〈x,j〉 = ∑ j∈Zd 1 Nd ( 2π N )α |j|αe 2π N i〈x,j〉[ ∫ G� e− 2π N i〈y,j〉(uk(y) −u(y)) dy + ∫ Ω−G� e− 2π N i〈y,j〉(uk(y) −u(y)) dy], which approaches 0 as � → 0 and k →∞ because of the facts that G� ⊆ Ω with |Ω −G�|→ 0 as � → 0 and uk → u uniformly on G�. This limit and the Monotone Convergence Theorem then imply that∫ G� [( (−∆) α 2 uk − (−∆) α 2 u )]2 = ∫ Ω χG� [( (−∆) α 2 uk − (−∆) α 2 u )]2 (2.24) approaches 0 as � → 0 and k →∞. By the Cauchy-Schwartz’s Inequality,∣∣∣∣ ∫ G� (−∆) α 2 u ( (−∆) α 2 uk − (−∆) α 2 u )∣∣∣∣2 ≤ ∫ G� [ (−∆) α 2 u ]2 (2.25) · ∫ G� [( (−∆) α 2 uk − (−∆) α 2 u )]2 . From Limit (2.24) and Inequality (2.25), we can obtain that∫ G� (−∆) α 2 u ( (−∆) α 2 uk − (−∆) α 2 u ) → 0 (2.26) A SOLUTION VIA VARIATIONAL METHOD 283 as � → 0 and k → ∞. Letting � → 0 and k → ∞ on both sides of Inequality (2.23) and using Limit (2.26), we get lim �→0, k→∞ ∫ G� 1 2 [ (−∆) α 2 uk ]2 ≥ lim �→0 ∫ G� 1 2 [ (−∆) α 2 u ]2 . (2.27) Since |Ω − G�| → 0 as � → 0, χG�(x) → 1 for almost every x ∈ Ω as � → 0. By the Monotone Convergence Theorem, we have lim �→0 ∫ G� [ (−∆) α 2 u ]2 = lim �→0 ∫ Ω χG� [ (−∆) α 2 u ]2 = ∫ Ω [ (−∆) α 2 u ]2 . (2.28) From Inequality (2.27) and Equation (2.28), we have lim �→0, k→∞ ∫ G� 1 2 [ (−∆) α 2 uk ]2 ≥ ∫ Ω 1 2 [ (−∆) α 2 u ]2 . (2.29) Moreover, from the facts that F is continuously differentiable at the functional component y and Fy(x,y) = f(x,y), x ∈ Rd, y ∈ R, and uk → u uniformly on E� (⊇ G�), it follows that lim �→0 k→∞ ∫ G� F(.,uk) = ∫ Ω F(.,u) (2.30) Letting � → 0 and k →∞ on the both sides of (2.22), then applying (2.29) and (2.30), we get lim k→∞ I(uk) ≥ ∫ Ω 1 2 [ (−∆) α 2 u ]2 + ∫ Ω F(.,u) = ∫ Ω [ 1 2 [(−∆) α 2 u ]2 + F(.,u)] =I(u), and hence, by Limit (2.21), I(u) ≤ m as desired. � Finally, we are ready to prove the existence of minmizer of I in Hα(NTd). Proposition 2.5 (Existence of minimizer). I has a minimizer in Hα(NTd). Proof. Set m0 = inf u∈Hα(NTd) I(u). Without loss of generality, assume that m0 < ∞ (otherwise, each u ∈ Hα(NTd) becomes a minimizer of I). Let {uk} be a minimizing sequence in Hα(NTd), meaning a sequence satisfying lim k→∞ I(uk) = m0. Then we have sup k I(uk) < ∞. By Proposition 2.3, we have I(uk) ≥ 1 2 ( 1 −N−d )[ 1 + ( 2π N )2α]−1 ‖uk‖Hα(NTd) − Λ for all u ∈ Hα(NTd), where Λ is a positive constant depending on f, N and d. This implies that sup k ‖uk‖Hα(NTd) < ∞. Due to the Weak Compactness Theorem, every bounded sequence in a Hilbert space is weakly precom- pact, that is, every bounded sequence in a Hilbert has a weakly convergent subsequence. Therefore, 284 R. KARKI AND Y. H. YOU there exists a subsequence {ukj} of {uk} and an element u ∈ Hα(NTd) such that ukj ⇀ u in Hα(NTd). By lower semicontinuity of I from Proposition 2.4, we have I(u) ≤ lim inf j→∞ I(ukj ) = m0. On the other hand, it is clear that m0 ≤ I(u). Therefore, m0 = I(u). This shows that u is a minimizer of I in Hα(NTd). � With Proposition 2.5, we have completed the proof of Theorem 2.2. In this way, we have proved that under the given assumptions on f, Equation (1.2) does have a weak solution in Hα(NTd). 3. Some Remarks on Regularity A weak solution u ∈ Hα(NTd) to Equation (1.2) does not necessarily satisfy Equation (1.2) unless u is sufficiently regular or smooth. But in order to have sufficient regularity or smoothness of u we may need to add further assumptions on the nonlinear functional f. For the moment, suppose we have a suitable f so that u ∈ Hs(NTd) for s ≥ 2α. Since for the weak solution u, we have 〈(−∆) α 2 u, (−∆) α 2 φ〉 + 〈f(.,u),φ〉 = 0 for all φ ∈ C∞(NTd) and (−∆) α 2 is self-adjoint on L2(NTd) (see [8, 20, 21]), we have 〈(−∆)αu + f(.,u),φ〉 = 0 for all φ ∈ C∞(NTd) and hence (−∆)αu + f(.,u) = 0 a. e. (3.1) This leads us to ask the following. Remark 3.1. Under what assumptions on f, does a minimizer u ∈ Hα(NTd) of I belong to Hs(NTd), s ≥ 2α so that u satisfies Equation (1.2) in a pointwise almost everywhere sense (meaning u is a strong solution to Equation (1.2))? Furthermore, suppose we have even a better f so that the weak solution u ∈ Hs(NTd), s ≥ 2α is smooth enough to become both u and (−∆)αu continuous on Rd. Then Equation (3.1) reduces to (−∆)αu(x) + f(x,u(x)) = 0 ∀ x ∈ Rd. This means u solves Equation (1.2) in a classical sense and thus Equation (1.2) is the Euler-Lagrange equation of I(u). So, it is natural to expect the following. Remark 3.2. Additionally, how smooth does f need to be so that a weak solution u of Equation (1.2) will satisfy it in a pointwise (everywhere) sense (meaning u is a classical solution to Equation (1.2))? References [1] G. Alberti and G. Bellettini, A nonlocal anisotropic phase transitions I. The optimal profile problem, Math. Ann. 310(1998), 527- 560. [2] T. Blass and R. de la lLave, Perturbation and Numerical Method for Computing the Minimal Average Energy, Networks & Heterogeneous Media 6(2)(2011), 241-255. [3] T. Blass, R. de la lLave and E. 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