Application of Mountain Pass Theorem to superlinear equations with fractional Laplacian controlled by distributed parameters and boundary data

In the paper we consider a boundary value problem involving a differential equation with the fractional Laplacian $(-\Delta)^{\alpha/2}$ for $\alpha \in\left( 1,2\right) $ and some superlinear and subcritical nonlinearity $G_{z}$ provided with a nonhomogeneous Dirichlet exterior boundary condition. Some sufficient conditions under which the set of weak solutions to the boundary value problem is nonempty and depends continuously in the Painleve-Kuratowski sense on distributed parameters and exterior boundary data are stated. The proofs of the existence results rely on the Mountain Pass Theorem. The application of the continuity results to some optimal control problem is also provided.


Introduction
The problems with the fractional Laplacian attracted in recent years a lot of attention as they naturally arise in various areas of applications. The fractional Laplacian naturally appears in probabilistic framework as well as in mathematical finance as infinitesimal generators of stable Lévy processes [2,8,9,10,54]. One can find the problems involving the fractional Laplacian in mechanics and in elastostatics, to mention only, a Signorini obstacle problem originating from linear elasticity [7,20,21]. Then concerning fluid mechanics and hydrodynamics the nonlocal fractional Laplacian appears, for instance, in the quasi-geostrophic fractional Navier-Stokes equation [23] and in the hydrodynamic model of the flow in some porous media [11,41,52,53].
In the paper we investigate the differential equation with the fractional Laplacian (−∆) α/2 and some nonlinearity G z of the form (1) (−∆) α/2 z (x) = G z (x, z (x) , u (x)) in Ω with the nonhomogeneous Dirichlet exterior boundary condition where Ω ⊂ R n for n ≥ 3 is a bounded domain with a Lipschitz boundary and α ∈ (1,2) . First of all, we look for the weak solutions, in the fractional Sobolev space H α/2 (R n ) , of (1) satisfying (2) . Without going into details we examine the existence of the weak solution z of (1) − (2) such that z − T v ∈ H α/2 0 (Ω) where T v ∈ H α/2 (R n ) is an appropriate orthogonal extension of v to R n and a distributed parameter u is from L ∞ (Ω, R m ) space with m ≥ 1. The fractional Laplacian and relevant Sobolev spaces of fractional orders are presented in Section 2, whereas the weak formulation of (1) − (2) is stated in Section 3. Next, we address the stability issue for problem (1) − (2) . By stability here we mean the continuous dependence of solutions z on distributed parameters u and boundary data v. It is possible to prove that under some suitable assumptions, for an arbitrary pair (u, v) there exists a weak solution z u,v to problem (1) − (2) which is stable with respect to the distributed parameters u and the boundary data v. In other words, we prove that z u,v → z u0,v0 in H α/2 (R n ) provided that u tends to u 0 in L ∞ (Ω, R m ) and v tends to v 0 in H α/2 (R n \Ω) . The main stability result for problem (1) − (2) is a direct consequence of Theorem 5.1 presented in Section 5.
It should be noted that the weak formulation of system (1) corresponds to the Euler-Lagrange equation for the following integral functional up to some appropriate shift specified in (21) , is an appropriate extension of v to R n and u ∈ L ∞ (Ω, R m ) . Such an extension is possible due to Theorem 5.4 from [25]. The above functional is referred to as the functional of action or the energy functional. On the function G we impose, besides some technical, growth and regularity assumptions, the following superlinearity assumption (4) a < pG (x, z, u) ≤ zG z (x, z, u) which is satisfied for some a > 0, p > 2 and |z| sufficiently large. This condition guarantees that problem (1) − (2) can be referred to as a superlinear exterior boundary value problem and as illustrated in Example 5.1 the nonlinear functional (3) , in general, can be unbounded from above and below. For that reason we cannot adopt the approach to the existence and stability issue of Dirichlet problem involving the fractional Laplacian presented for example in [13] where the coercive functional bounded from below was studied, while in [27] only the linear case was treated. In general, in the theory of boundary value problems and its applications we consider, first of all, the problem of the existence of a solution and next questions of stability, uniqueness, smoothness, asymptotics etc.. The problem of existence of solutions to equation (1) with the homogenous Dirichlet boundary condition corresponding to critical point of mountain pass type was considered for example in the recent papers [46,47]. For more references on the existence results for problems involving nonlocal fractional Laplacian equation with subcritical nonlinearities, see, for example [45] as well as [6,24] for problems with critical nonlinearities. Moreover, the asymptotically linear case was investigated in [28] whereas in [40] one can find a bifurcation result in the fractional setting. We also refer the interested reader to papers [5,11,17,18,19,22,27,34,48] for other results related to the fractional Laplacian. In the present paper we apply to the functional defined in (3)the renowned Mountain Pass Theorem presented, for example, in [38,42,51] which enables us to obtain the existence result for problem related to (1) − (2) similarly as in [46,47].
As far as the continuous dependence results of solutions on parameters and boundary data for equation (1) are concerned, up to our best knowledge, the subject in fractional setting seems to have received almost no attention in the literature. Some continuous dependence results for homogenous Dirichlet boundary problem involving the fractional Laplacian one can find in [13] where coercive case is examined by the direct method of calculus of variations. Differentiable continuous dependence on parameters, or in other words robustness result are presented in [14] where the application of theorem on diffeomorphism leads to the stability result for the problem involving one-dimensional fractional Laplacian with zero initial condition. In the present paper we obtain the existence and the continuous dependence results for the exterior boundary value problem involving the equation with the fractional Laplacian by adopting the approach presented in [12] were superlinear elliptic boundary value with the nonhomogeneous Dirichlet boundary condition was examined.
The structure of the paper reads as follows. Section 2 contains some useful information on the fractional Sobolev spaces and the fractional Laplacian. The variational formulation of the problem and some standing assumptions are presented in Section 3, whereas in Section 4, our attention is focused on proving some auxiliary lemmas which are of a paramount importance to the rest of the paper. Some sufficient condition for the existence and continuous dependence of solutions to the exterior boundary value problem involving the equation with the fractional Laplacian on distributed parameters and boundary data can be found in Section 5. Finally, the application of the stability result leads to the existence of the optimal solution to the control problem described by (1) − (2) with a integral performance index expressed by some cost functional as asserted in Theorem 6.1 and Theorem 6.2 in Section 6. The proof of these theorem relies in an essential way on the continuous dependence results from Section 5.
By H α/2 (D), we shall denote the following space where D ⊂ R n for n ≥ 3 is a general, possibly unbounded, open domain in R n with suitably smooth boundary, for example Lipschitz, in our case D = Ω, D = R n \Ω or D = R n , for some bounded domain Ω with smooth boundary. In the literature, the fractional Sobolev spaces are also referred to as Aronszajn, Gagliardo or Slobodeckij spaces, associated with the names of the ones who introduced them almost simultaneously, see [3,29,50].
The space H α/2 (D) is a Hilbert space placed between L 2 (D) and H 1 (D) and endowed with the natural norm where the term is the so-called Gagliardo semi-norm of z. H α/2 0 (D) can be defined as completion of C ∞ 0 (D) with respect to the norm (6) and one can extend the functions from H α/2 0 (D) with 0 to R n as presented in [27]. It should be emphasized that for several domains with non-Lipschitz boundary or for α ∈ (0, 1] definitions of the space of the fractional order might be non-equivalent, for details see for example [11,27,41]. Spaces H α/2 (D) and H α/2 0 (D) are related to the fractional Laplacian operator (−∆) α/2 as we shall see in the sequel. First of all, let (−∆) α/2 : S → L 2 (R n ) be the fractional Laplacian operator defined on the Schwartz space of rapidly decaying C ∞ functions in R n denoted by S of the form where C (n, α) = π −(α+n/2) Γ((n+α)/2) Γ(−α/2) . Then, as stated in Lemma 3.5 in [25], for any z ∈ S we have the equivalent symmetrized definition for all x ∈ R n . Furthermore, for any z ∈ S, the definition involving the Fourier transform reads as for all ξ ∈ R n and a suitably chosen positive constant C depending only on n and α, see [25,Proposition 4.1] where F stands for the Fourier transform. This definition allows as to extend the definition of (−∆) α/2 to the space H α/2 (R n ) wider than S spaces. Alternatively, the space H α/2 (R n ) can be defined via the Fourier transform as which holds for any real α. From Proposition 4.2 in [25], it follows that the space H α/2 (R n ) defined by (5) coincides with the one defined by (10) . In addition, for any z ∈ H α/2 (R n ) , we have for a suitably chosen positive constant C depending only on n and α. Proposition 4.4 from [25] allows us to formulate the relation between the fractional Laplacian operator (−∆) α/2 and the fractional Sobolev space H α/2 (R n ). In fact, we have .
Thus it means that the equivalent norm in H α/2 (R n ) to the one from (6) has the form In this paper we shall use spectral properties of the fractional Dirichlet Laplacian defined on the bounded domain D with smooth boundary. The powers (−∆) α/2 of the positive Laplace operator (−∆) , in a bounded domain with the zero Dirichlet boundary condition are defined through the spectral decomposition using the powers of the eigenvalues of the original operator. Let (z k , ρ k ) for k ∈ N be the system of the eigenfunctions and eigenvalues of the Laplace operator (−∆) on D with the homogeneous Dirichlet condition on ∂D. Then (z k , ρ α/2 k ) for k ∈ N is the system of the eigenfunctions and eigenvalues of the fractional Laplacian (−∆) α/2 on D, also with the homogeneous boundary Dirichlet condition. By H α/2 0 (D) , we can denote the space of functions z = z (x) defined on a bounded, smooth domain D ⊂ R n , n ≥ 3, such that z = ∞ k=1 a k z k and ∞ k=1 a 2 k ρ α/2 k < ∞, with the norm defined by the formula , see [25,Proposition 4.4]. In addition, the fractional Laplacian acts on It is worth recalling that for a bounded domain with a Lipschitz boundary, the fractional Sobolev space H α/2 (D) is compactly embedded into L s (D) for s ∈ [1, 2 * α ) where 2 * α = 2n/ (n − α) for α < n and the inequality holds, cf. Corollary 7.2 in [25].
Furthermore, one can prove, like in [25], that every open set D with bounded Lipschitz boundary is an extension domain for H α/2 which means that every function u ∈ H α/2 (D) can be extended to a functioñ u ∈ H α/2 (R n ) in a continuous way. That is, there exists a functionũ ∈ H α/2 (R n ) such thatũ| D = u and where C = C (n, α, Ω). The interested reader is referred to the proof of this fact to Theorem 5.4 in [25]. Finally, it is well worth underlining that the fractional Sobolev spaces play an important role in the trace theory. In papers [26,30,39] one can find the trace theorem in which for α ∈ (1, 2) and a bounded Lipschitz domain D, there exists a continuous, surjective trace operator possessing a continuous right inverse. The space H (α−1)/2 (∂D) is said to be the space of traces or boundary values of functions from the space H α/2 (D) . Covering ∂D by coordinate patches, we define the space H (α−1)/2 (∂D) as before via such charts with the norm analogous to (6). For more intricate details on the definition of the space of traces H (α−1)/2 (∂D) we refer the reader to [1,16,33,35]. For the definitions of the fractional Laplacian (7) − (9), however we need the space of functions defined on the whole of R n . Contrary to the Dirichlet problem for local Laplacian, the Dirichlet problem for the nonlocal fractional Laplacian require the so-called boundary values of the function to be set also outside the given set where the equation holds, cf. [27] where the linear Dirichlet problem for the fractional Laplacian is treated.

Variational formulation and standing assumptions
In the paper, we shall consider a problem involving a weak formulation of the following equation with the fractional Laplacian in the form where the exterior boundary condition will be ascertained by claiming that is a bounded domain with a Lipschitz boundary. Let v 0 be a fixed element from the space H α/2 (R n \Ω) . By V we shall denote the set of all boundary data such that for l 1 > 0 while U denotes the set of distributed parameters of the form Let us notice that an open domain with a Lipschitz boundary of the form R n \Ω is an extension domain for H α/2 , and consequently there exists a continuous operator T : where C = C (n, α, Ω, T ) . Such an operator exists due to Theorem 5.4 from [25]. Besides, one can additionally assume that the mapping T is to be chosen so that if necessary its values are modified on Ω by composition with orthogonal projection P : H α/2 → H α/2 0 ⊥ and still denoted by T, abusing the notation. That is, for the mapping T, we have the following orthogonal decomposition so for any y ∈ H α/2 0 (Ω) and v ∈ H α/2 (R n \Ω) where (·, ·) H α/2 denotes the scalar product in H α/2 (Ω) defined, e.g., via the formula compare also results in [12]. For an alternative approach, see [27], allowing non-Lipschitz domains and a weak formulation of linear problem with more general nonlocal operators.
As it was announced we look for a weak solution of (16) (Ω). Let y = z − T v, then the problem in (16) can be rewritten in the following homogenized form It should be noted that y ∈ H α/2 0 (Ω) allows us for α ∈ (1, 2) to use the spectral definition given by (13) of the fractional Laplacian or alternatively the Fourier transform definition given by (9) extended to H α/2 0 (Ω) . The fractional Laplacian of T v can be calculated via the extended Fourier transform definition (9) to H α/2 (R n ) which for smooth and fast decaying function from Schwartz space can also be calculated via singular integrals given by (7) and (8) .
Next, we say that y ∈ H α/2 0 (Ω) is a weak or an energy solution of (19) if the identity (Ω) reads as for establishing existence results and examining the structure of the set of critical points.
In short, conditions (C1) − (C4), as we shall demonstrate, guarantee the existence of weak solution to problem (16) corresponding to the critical points of mountain pass type of the associated functional. If, additionally, condition (C5) is satisfied, it is feasible to prove that these solutions depend continuously on distributed parameter u and boundary data v.

Critical points of mountain pass type
In this section we focus our attention on proving some auxiliary results which are of a key importance to the rest of the paper. First of all, we recall some definitions. Let I : E → R be a functional of C 1 −class defined on a real Banach space E, in our case on H In what follows we will need some compactness properties of the functional I guaranteeing for example by the Palais-Smale condition. Now we recall what this means. A sequence {y k } ⊂ E is referred to as a Palais-Smale sequence for a functional I if for some C > 0, any k ∈ N, |I (y k )| ≤ C and I ′ (y k ) → 0 as k → ∞. We say that I satisfies the Palais-Smale condition if any Palais-Smale sequence possesses a convergent subsequence in E. For more details on the Palais-Smale condition we refer the reader to [38,Chapter 4.2].
In this section we shall use, as in [12], the following version of the Mountain Pass Theorem, cf. [38,51]. Throughout this section we shall use the following notation and definitions. First, (Ω) : y H α/2 0 < r and r > 0. Next, for k ∈ N 0 , let denote an arbitrary sequence of functionals of C 1 −class, and c k (r) be the value defined by setting  In what follows, we shall establish the properties of the upper limit of sets. By the Painlevé-Kuratowski upper limit of sets or in short the upper limit of sets, we shall understand the set of all cluster points with respect to the strong topology of E of a sequence {x k } such that x k ∈ X k for k ∈ N, where X k denotes the sequence of sets from the space E. Lim sup X k stands for the upper limit of the sets X k , k ∈ N. In particular, Lim sup Y k (r) is the upper limit of the sets Y k (r) , k ∈ N, so the set of all cluster points with respect to the strong topology of H α/2 0 (Ω) of a sequence {y k } such that y k ∈ Y k (r) for k ∈ N. For more details on the Painlevé-Kuratowski upper limits of sets we refer the reader to [4]. Now we prove that the upper limit in H α/2 0 (Ω) of the sequence of sets {Y k (r)} defined in (23) is nonempty and is a subset of Y 0 (r).
Lemma 4.2 Assume that (a) for any k ∈ N 0 , the functional I k , is of C 1 −class and the functional I 0 satisfies the Palais-Smale condition, (b) the sequences {I k } , {I ′ k } tend uniformly on the ball B r to I 0 , I ′ 0 , respectively, (c) for any k ∈ N 0 , the sets Y k (r) are nonempty. Then any sequence {y k } such that y k ∈ Y k (r) , k ∈ N is relatively compact in H α/2 0 (Ω) and Lim sup Y k (r) ⊂ Y 0 (r).
Proof. In the proof we shall follow the lines of the proof of Lemma 3.1 from [12]. First of all, one can prove that Lim sup Y k (r) is not empty. To do this, let {y k } be an arbitrary sequence such that y k ∈ Y k (r) for k ∈ N 0 . Such a sequence exist by assumption (c). Moreover, again, by (b) , 0 = lim k→∞ I ′ 0 (y k ) as I ′ k (y k ) = 0 for k ∈ N 0 . Furthermore, y k H α/2 0 (Ω) < r hence the sequence I 0 (y k ) is bounded. Since I 0 satisfies the Palais-Smale condition, the sequence {y k } is relatively compact in H α/2 0 (Ω), that is, Lim sup Y k (r) is not empty. Next, by (b) , it is easy to check that (24) lim k→∞ c k (r) = c 0 (r) and moreover for any sequence {y k } such that y k ∈ Y k (r) for k ∈ N, I 0 (y k ) − I k (y k ) → 0 as k → ∞. From the convergence in (24), we conclude that lim k→∞ I 0 (y k ) = c 0 (r). Since the set Lim sup Y k (r) is not empty, choosẽ y from this set, so thatỹ is a cluster point of some sequence {y k } such that y k ∈ Y k (r) for k ∈ N. Therefore, passing to a subsequence, if necessary, we may assume that y k →ỹ as k → ∞. Suppose thatỹ / ∈ Y 0 (r) , i.e. I 0 (ỹ) = c 0 (r) or I ′ 0 (ỹ) = 0. Let us observe that the second inequality is false. Indeed, assumption (b) and the first part of our proof imply I ′ 0 (ỹ) = lim k→∞ (I ′ 0 (y k ) − I ′ k (y k )) = 0.
What we need at this point of our considerations is to examine a specific form of the functional I k derived from the functional of action given by (21) and denoted by F k , see the formula (25) below. Let {v k } be a sequence of boundary data and {u k } be a sequence of distributed parameters such that v k ∈ V and u k ∈ U for k ∈ N 0 . Furthermore, {F k } stands for the sequence of functionals of the form for which we define the value  (Ω) . Here and throughout the paper, for any k ∈ N 0 , let Y k denote the set of critical points corresponding to the value c k , that is, the set of the form (27) Y k = y ∈ H α/2 0 (Ω) : F k (y) = c k and F ′ k (y) = 0 .
In Section 5, we shall prove that, for any k ∈ N, the set Y k is not empty and the sequence of {Y k } possesses nonempty upper limit in H α/2 0 (Ω) such that Lim sup Y k ⊂ Y 0 . In the proof of that results we need the following lemma in which the boundedness of the sequence {Y k } is claimed. (Ω) of radius ̺ > 0 centered at zero such that Y k ⊂ B ̺ .
Proof. First of all, let us observe that the set {c k : v k ∈ V, u k ∈ U} is bounded from above. Indeed, for any k ∈ N, conditions (C2), (C3) enable us to conclude and therefore c k ≤c where D, c, d,c are some constants, and the sets Ω + t , Ω − t are defined as Then, for any v k ∈ V, u k ∈ U and y ∈ Y k we have where D 1 , D 2 are some constants and Ω + , Ω − are some subsets of Ω of the form and < ·, · > is a dual pair in H α/2 0 (Ω) . Thus By condition (C3) , p − 2 > 0, and consequently there exists ̺ > 0 such that y ∈ B ̺ . Hence, Y k ⊂ B ̺ for any v k ∈ V and u k ∈ U, which completes the proof. Next, we state some sufficient conditions guaranteeing a uniform convergence on any ball from the space H  (Ω) and y ∈ B ̺ and any fixed ε > 0, we have for k sufficiently large. In fact, let us observe that T v k tends to T v 0 in H α/2 (Ω) , since v k tends to v 0 in H α/2 (R n \Ω). By conditions (C1) , (C2), (C5), the Krasnoselskii Theorem on the continuity of Niemyckii's operator (cf. [32]) implies that the right hand side of the above inequality tends to 0 for any y ∈ B ̺ . It means that the sequence {F k } tends uniformly to F 0 on a ball B ̺ . A similar reasoning holds for the case of a uniform convergence of the sequence {F ′ k } to F ′ 0 on a ball from H (Ω) such that h ∈ B 1 simple calculations lead to for k sufficiently large, and the claim of the lemma follows. Finally, after necessary shifts resulting in (19) and F k defined by (25) , we have that w 0 = 0. It is possible to demonstrate that there exist a bounded neighborhood B of w 0 in H α/2 0 (Ω) and some point w 1 / ∈ B such that the assumptions of Theorem 4.1 are fulfilled for F k . Indeed, we have the following lemma.
To finish the proof it is enough to show that for any v k ∈ V and u k ∈ U there exists w 1 / ∈ B η such that F k (w 1 ) < 0. For a fixed nonzero y ∈ H α/2 0 (Ω) and l > 0, from (C2) and (C3) we have the following estimates Since p ∈ (2, 2 * α ) and a 0 > 0 we see that lim l→∞ F k (ly) = −∞. Therefore, there exists l 0 > 0 such that for w 1 = l 0 y we have w 1 H α/2 ≥ η and F k (w 1 ) < 0 for any v k ∈ V and u k ∈ U, which proves the assertion of the lemma.

The main continuity result
In this section we employ Lemmas 4.2, 4.3, 4.4, 4.5 and Theorem 4.1 to prove, first of all, some sufficient conditions under which critical points of mountain pass type of the functional F k defined in (25) exist and depend continuously on parameters. Actually, we have the following theorem.
Theorem 5.1 Suppose that the function G satisfies conditions (C1)−(C5) and moreover the sequence {v k } ⊂ V tends to v 0 in H α/2 (R n \Ω) and the sequence {u k } ⊂ U tends to u 0 in L ∞ (Ω, R m ). Then for any k, the set of critical points Y k of the functional F k is nonempty, does not contain the zero critical point and any sequence {y k } such that y k ∈ Y k , k ∈ N, is relatively compact in H α/2 0 (Ω) and Lim sup Y k ⊂ Y 0 .
Proof. Applying the Mountain Pass Theorem, cf. Theorem 4.1, we shall prove the first part of the assertion of the theorem. So we need to prove that for any k, the set of critical points Y k of the functional F k is nonempty and does not contain the zero critical point. The functional F k , k ∈ N 0 is of C 1 −class on H α/2 0 (Ω). From Lemma 4.5 it follows that there exist the ball B η and the point w 1 ∈ H α/2 0 (Ω) , independent of the choice of v k , u k , such that w 1 / ∈ B η and inf y∈∂Bη F k (y) > 0 = max {F k (0) , F k (w 1 )} , for k ∈ N 0 . Conditions (C2), (C3) guarantee that the functional F k satisfies the Palais-Smale condition for k ∈ N 0 . Next, let k be fixed and {y i } be a sequence such that {F k (y i )} is bounded and F ′ k (y i ) → 0 as i → ∞. Thus, there exist constants C 1 , C 2 > 0 such that |F k (y i )| ≤ C 1 and F ′ k (y i ) ≤ C 2 for any i ∈ N. In the same manner as in the proof of Lemma 4.3, we obtain the following estimates where C 3 > 0, and D 1 , D 2 are some constants. Hence where C 1 , C 2 , C 3 , D 2 are described above. Therefore, the sequence {y i } is bounded in H α/2 0 (Ω) and it contains a subsequence, still denoted by {y i } , such that y i tends to y 0 weakly in H (Ω) is compactly embedding into the space L s (Ω) with s ∈ [1, 2 * α ) , i.e. we may assume after passing to a subsequence, still labelled {y i }, that y i → y 0 in L s (Ω) . Consequently, denoting < ·, · > a dual pair in H The equality and the growth condition (C2) lead to Let us notice that the right hand side of the above inequality tends as i → ∞ to 0, by the growth condition (C2) and since y i → y 0 in L s (Ω) , T v k → T v 0 in L s (Ω) and u i → u 0 in L ∞ (Ω, R m ). As a result, for any k, In that way we have demonstrated that for any k ∈ N 0 the functional F k satisfies the Palais-Smale condition. At this point it is possible to apply Theorem 4.1, with w 0 = 0 and c = c k , to deduce that for any v k and u k , the set of critical points for which a critical value of the functional F k denoted by c k is attained, is not empty, i.e.
Moreover, c k = inf g∈M max t∈[0,1] F k (g (t)) > max {F k (0) , F k (w 1 )} = 0, and therefore y = 0 does not belong to the set Y k for all k ∈ N 0 . What is left is to demonstrate that Lim sup Y k = ∅ in H α/2 0 (Ω) and Lim sup Y k ⊂ Y 0 . By invoking Lemma 4.3, we obtain that there exists a ball is given by (23) with I k = F k , k ∈ N 0 . Since F 0 satisfies the Palais-Smale condition and any set Y k (r) = Y k is nonempty for k ∈ N 0 , Lemma 4.4 implies that the sequences {F k } , {F ′ k } tend uniformly on any ball from H α/2 0 (Ω) to F 0 and F ′ 0 , respectively. Second part of the assertion of our theorem follows directly from Lemma 4.2. It means that the proof of the theorem is thus complete.
Let us notice that the critical value of the functional F u k ,v k defined in (21) denoted by c k satisfies the following relation where c k is defined in (26) as the critical value of F k . Thus the set of critical points of the functional F u k ,v k for which the critical value c k is attained has the form for k ∈ N 0 . Immediately from Theorem 5.1 we get the following corollaries characterizing, first of all, the set of critical points of mountain pass type of the functional of action without unimportant shift and then the set of some weak solutions of the problem involving the equation with the fractional Laplacian with the homogenous Dirichlet boundary data.
Corollary 5.1 If all assumptions of Theorem 5.1 are satisfied, then for any k the set Y u k ,v k is nonempty and does not contain zero critical point, Let us denote by Y u k ,v k the set of the weak solutions to problem (16) corresponding to the critical value c k .
Corollary 5.2 If all assumptions of Theorem 5.1 are satisfied, then for any k the set Y u k ,v k is nonempty and does not contain zero, Furthermore, it is easy to observe that Z u k ,v k = Y u k ,v k + T v k , k ∈ N 0 is a set of weak solutions to problem (1) corresponding to the critical value c k + T v k .

Corollary 5.3
If all assumptions of Theorem 5.1 are satisfied, then for any k the set Z u k ,v k is nonempty and does not contain T v k , Lim sup Z u k ,v k = ∅ in H α/2 (R n ) and Lim sup Z u k ,v k ⊂ Z u0,v0 .
Example 5.1 It is easy to check that the assumptions of Theorem 5.1 are satisfied by the following equation where U is a bounded interval and γ > 0 while Let us notice that the functional of action F u,v defined in (21) , up to some shift related to F k defined in (25) has the following form Putting, up to necessary shift, z k (x) = k sin x 1 sin x 2 sin x 3 and z k (x) = k 6/7 sin (kx 1 ) sin (kx 2 ) sin (kx 3 ) we obtain by the Sobolev inequality (14) or the Poincaré inequality, cf.
i.e. the functional F is unbounded from above and below. For that reason to obtain the existence results we cannot use methods applied, for example, in [13,55]. Applying Corollary 5.3 we have that for any u ∈ U and v ∈ V there exists a weak solution z u,v ∈ H 3/4 R 3 to problem (29) and the solution z u,v continuously, or upper semicontinuously, depends on boundary data v and distributed parameter u.

Optimal control problem
Following ideas employed for optimal control system in [36] by a direct application of Corollary 5.3 following from Theorem 5.1 the existence of optimal processes can be ascertained for the optimal control problem involving the weak formulation of the partial differential equation with the fractional Laplacian and some nonlinearity of the form (30) (−∆) α/2 z (x) = G z (x, z (x) , u (x)) in Ω with the fixed exterior boundary condition (31) z (x) = v (x) in R n \Ω and with the integral cost functional J defined via Φ as where Φ : Ω × R × R m → R is a given function and a control u ∈ U λ where U λ = {u : Ω → R m : u (x) ∈ U and |u (x 1 ) − u (x 2 )| < λ |x 1 − x 2 |} for a fixed λ > 0 and U a compact subset of R m . Recall that α ∈ (1, 2) , n ≥ 3 and m ≥ 1. Let A be the set of all admissible pairs; that is A= (z, u) ∈ H α/2 (R n ) × U λ : z is a weak solution of (30) satisfying (31) corresponding to the critical value c k of the functional F u,v defined in (21) for u ∈ U λ } .
It should be noted that Corollary 5.3 implies that the set of all admissible pairs A is nonempty. In this section, our aim is to find a pair (z u * , u * ) ∈ H α/2 (R n ) × U λ satisfying (33) J (z u * , u * ) = min (z,u)∈A J (z, u) .
The theorem on the existence of optimal process to problem (33) can now be formulated.
Proof. By (C6) and (C7) the cost functional is well-defined and continuous with respect to (z, u) ∈ R × U variables. Let (z k , u k ) , k ∈ N be a minimizing sequence for problem (30) − (32), i.e. u k ∈ U λ , the equation (−∆) α/2 z k (x) = G z (x, z k (x) , u k (x)) is satisfied in a weak sense cf. (20) and moreover z k = v in R n \Ω and lim k→∞ J (z k , u k ) = inf (z,u)∈A J (z, u) .
Entire class U λ is equicontinuous and uniformly bounded, so certainly {u k } is also. By Arzéla-Ascoli's Theorem, there exists a subsequence {u k } such that u k → u 0 uniformly on Ω and u 0 ∈ U λ . By Corollary 5.3 the sequence {z k } , or at least some its subsequence, tends to z 0 in H α/2 (R n ) and (z 0 , u 0 ) ∈ A, thus J (z 0 , u 0 ) = inf (z,u)∈A J (z, u) . It means that the process (z 0 , u 0 ) is optimal for (33) .
Analogous result for another class of controls U Ω(r) defined below can be proved. By Ω (r) we denote a fixed decomposition of Ω on r open subsets Ω i such that r i=1 Ω i ⊂ Ω, µ ( r i=1 Ω i ) = µ (Ω) and Ω i ∩ Ω j = ∅ for i = j, i, j = 1, ..., r with µ being the Lebesgue measure on R n . We shall say that a function u is constant on Ω (r) if u is constant on each subset from decomposition Ω (r) , i.e. u (x) = const i for x ∈ Ω i , i = 1, ..., r. Finally for the following class of controls U Ω(r) = {u ∈ L ∞ (Ω, R m ) : u (x) ∈ U and u is constant on Ω (r)} where U is a compact subset of R m , analogously to the proof of Theorem 6.1 one can prove the following theorem.