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On fractional plasma problems

In this paper we show existence and multiplicity results for a linearly perturbed elliptic problem driven by nonlocal operators, whose prototype is the fractional Laplacian. More precisely, when the perturbation parameter is close to one of the eigenvalues of the leading operator, the existence of three nontrivial solutions is proved.


Introduction
A Tokamak machine consists of a toroidal cavity containing a given mass of plasma surrounded by a vacuum layer. One of the issues of the Plasma Physics deals with the specification of the region occupied by such a plasma at equilibrium and the description of its flux through a cross section Ω ⊆ R 2 of the machine, which we assume to be a bounded domain.
Denoting by u the flux function, a possible description of such a phenomenon is given by the nonlinear eigenvalue problem (though λ is not an eigenvalue according to the usual terminology) with specific boundary conditions and λ ∈ R, see [12] and [29]. However, in [29] the domain Ω is far from the line x 1 = 0, and so the operator is uniformly elliptic. For this reason, a simplified but formally equivalent version of (1) is considered in [30], with the equation However, in most cases, the Laplace operator does not fit the problem in a realistic way. Indeed, since the papers of Einstein [6] and Smoluchowski [26], the Laplacian has been the successful tool to describe, among others, diffusion and Brownian motion. However, the diffusion of a particle (or of an individual in a Biological species) at a point x might be influenced by all other particles, and this is particularly true if one takes into account also long-range particle interactions. For these reasons, in this case the diffusion operator cannot act pointwise, and so it is natural to consider the average between the total contributions, i.e. an integral average, of the form where K is a weight which measures the influence of the particle at x + y on the one at x. Typically, such an influence is determined by singular interactions depending on the distance between the two points, and so a quantity of the form |y| α dy (2) for some α > 0 is the most natural operator which shows up, see [4,27] and [31] for more details on this replacement of local operators by nonlocal ones. To our best knowledge, the first replacement of a local operator with a nonlocal one for the plasma problem was considered in [1], where the author uses the fractional Laplacian in the eigenvalue problem with Ω ⊆ R N a bounded domain, u : Ω → R, s ∈ (0, 1) and a 0 a given constant. Here the fractional Laplacian taken into account is Note that, in spite of the nonlocal definition of A s , it is now well known that a local representation is available: if u ∈ H 2 (Ω) ∩ H 1 0 (Ω), u = a i φ i , where a i =´Ω uφ i dx ∈ R and φ i are the L 2 -orthonormalized eigenfunctions of −∆ in Ω with associated eigenvalues Λ i , we have see [2]. For such a problem, Mark Allen studied existence and regularity of solutions and the properties of the free boundary, using the fact that this fractional Laplacian can be seen as a Dirichlet-to-Neumann map, which makes the original nonlocal problem in a local one with an additional dimension, see [5] for the entire space and [3] for bounded domains. For completeness, we briefly recall this Dirichlet-to-Neumann procedure: set C = Ω × (0, ∞), and, given u, consider the solution v of    div(y 1−2s ∇v)(x, y) = 0 (x, y) ∈ C, v(x, y) = 0 x ∈ ∂Ω, y > 0, v(x, 0) = u x ∈ Ω.
Then, there exists a constant C s > 0 such that i.e. the operator mapping the Dirichlet datum u to the Neumann-type datum lim y→0 + −y 1−2s u y (x, y) is the s − root of the negative Laplacian −∆ in Ω. Also notice that when s = 1/2, we have C s = 1 and √ −∆u(x) = −u y (x, 0). For the precise value of C s , see [5]. However, it is clear from [2] that actually A s has a local representation in terms of eigenvalues and eigenfunctions of −∆ in Ω, and thus nonlocal interactions are not really considered. For this reason, in this paper we start from different nonlocal versions of problem (3), whose prototype is in agreement with the physical motivations which led to (2). From now on, Ω will be a bounded domain of R N with Lipschitz continuous boundary. Note that the boundary condition 'u = 0 on ∂Ω' is replaced by the nonlocal one 'u = 0' in R N \ Ω, see [28] and [8, theorem 4.4.3]. In this way, by definition of (−∆) s , it is clear that an actual nonlocal operator is in force, and so nonlocal effects describing interactions among particles can be considered. It is also worth mentioning that A s and (−∆) s are different operators with different eigenvectors and eigenfunctions, see [24]. However, there are also some very good properties that this operator enjoys, similarly to the Laplacian, or a uniformly elliptic operator: for instance, it admits a simple and positive principal eigenvalue with signed positive eigenfunction ( [24]) and it satisfies the maximum principle and the Harnack inequality ( [5]).
In this paper we shall consider more general problems of the form Here K is a singular potential, whose prototype is K(x) = 1/|x| N+2s , see assumption (H) for the precise setting, and u belongs to a suitable reference space X s 0 , see below. Even without the precise definitions of the main characters involved, if we minimize the energy functional we find a minimizer u, with associated multiplier λ, so that the couple (u, λ) solves (P). In this way, we immediately find the counterpart of the existence result proved in [1] for the spectral fractional Laplacian: The fact that λ > 0 is proved simply starting from the identitÿ using that a 0.
In the rest of the paper we are interested in another version of problem (P), that is where u : R N → R, γ > 0 and p > 2. Notice that moving from (P) to (P λ ) we have set a = 1 (just to fix the ideas), and replaced u by −u. Of course, this choice is completely irrelevant and an analogous result can be proved for In order to introduce all the elements we need to solve problem (P λ ), we start recalling the usual setting for L K , see [23]. Take a function K : R N \ {0} → (0, ∞) satisfying the assumption: (H). For s ∈ (0, 1) and N > 2s, we assume that Introduce the space which makes X s 0 a Hilbert space, see [23]. From now on, we will denote by · the norm induced by ·, · .
Operator L K is defined in (4) and u ∈ X s 0 is a solution of (P λ ) if our first easy result is Theorem 1.2. If p ∈ (2, 2 ), γ > 0, λ ∈ R and (H) holds, then ( P λ ) admits one nontrivial solution.
Finally, in order to give our main result, we recall that L K admits a non-decreasing and diverging sequence (λ n ) n∈N of eigenvalues with associated L 2 -orthonormalized eigenfunctions (e n ) n∈N , such that, for every u ∈ X s 0 , we have α n e n , with α n = u, e n ∈ R, for every n ∈ N, see [25]. Moreover, λ 1 > 0 is simple, e 1 > 0 is bounded in Ω and every eigenvalue has finite multiplicity. For further references, it is convenient to state the following remarks. Our main result is Theorem 1.5. Let p ∈ (2, 2 ), γ > 0 and (H) holds. If l ∈ N with l 1, then there exists δ l+1 > 0 such that, for every λ ∈ (λ l+1 − δ l+1 , λ l+1 ) problem (P λ ) admits three nontrivial solutions.
The proof of this result is obtained by using a critical point theorem of mixed nature proved in [11], already successfully applied in [13-17, 19, 20, 32, 33], also for variational inequalities, see [9].

Mathematical background
In this section we recall some results which will be used throughout the paper. (i) if Ω has a Lipschitz boundary, then the embedding X s 0 → L p (R N ) is compact, for every p ∈ [1, 2 ); (ii) the embedding X s 0 → L 2 (R N ) is continuous.
Let X be a Banach space, I ∈ C 1 (X, R). We say that I satisfies the Palais-Smale condition, (PS) for short, if every (u n ) n ⊆ X such that (I (u n )) n is bounded and I (u n ) → 0 in X admits a convergent subsequence. We say that I satisfies the Palais-Smale condition at level c ∈ R, (PS) c for short, if every (u n ) n ⊆ X such that I (u n ) → c and I (u n ) → 0 in X admits a convergent subsequence.
The following linking theorem, proved by Rabinowitz in [21], though well known, is here recalled in view of the estimate of the critical value, which will be needed for establishing the main result. As usual, S ρ is the sphere of radius ρ in X and B R is the ball of radius R. Theorem 2.3. Let X be a Banach space and I ∈ C 1 (X, R) be such that I (0) = 0. Suppose that X = X 1 ⊕ X 2 , where X 1 and X 2 are closed subspaces with dim X 1 < ∞. Assume that (ii) there exists e ∈ S 1 ∩ X 2 and R > ρ such that, setting Finally, we state another critical point theorem, which is one of the ∇theorems introduced by Marino and Saccon in [11]. The main feature of these theorems is the following condition, which essentially requires that the functional, constrained on a certain subspace, has no critical points with some uniformity.

Definition 2.4 (∇-condition).
Let X be a Hilbert space and I ∈ C 1 (X, R). Let C be a closed subspace of X and a, b ∈ R ∪ {−∞, ∞}.
We say that I verifies condition where P C : X → C denotes the orthogonal projection of X onto C.
Now we give the abstract theorem.
Theorem 2.5 (Theorem 2.10, [11]). Let X be a Hilbert space and X 1 , X 2 , X 3 be three sub- where P i : X → X i denotes the orthogonal projection of X onto the subspace X i , for every i = 1, 2, and Assume that Then, I has at least two critical points in

The superlinear problem (P λ )
In this section we shall prove theorem 1.2. A first tool is the following Proof. First of all, Now take a sequence (u n ) n ⊆ L p (Ω) such that u n → 0 in L p (Ω) and any subsequence (u n k ) k . Up to another subsequence, we can assume that u n k → 0 a.e. in Ω. For every subsubsequence (u n k j ) j we get that since u n k j → 0 a.e. in Ω. This being valid for any sub-subsequence, we get the claim. □ Recalling the spectral properties of L K described above, we have a standard decomposition we obtain the decomposition where H 0 . = {0}. By using the previous notation, we get this useful result, for whose proof see [25].
Of course, problem ( P λ ) has a variational structure, since it is the Euler-Lagrange equa- We are now ready to prove theorem 1.2.
Proof of theorem 1.2. Throughout the proof we will adopt the decomposition for some i ∈ N 0 , introduced at the beginning of the section. First, we observe that when λ = λ 1 , the family {te 1 } t − e1 ∞ defines a ray of solutions, as a simple calculation shows.
Case λ < λ 1 . In this case it is enough to choose the representation X s with c > 0; moreover, by (7) we get that for some C > 0. Hence, in this case the norm defined as is equivalent to the usual one · . This said, let us check that F λ satisfies the assumptions of the mountain pass theorem.
In a sphere of radius ρ > 0 small enough, by lemmas 3.1 and 2.1, we get that , and so 0 is a strict local minimum point for F λ . Now, by choosing u < 0 in Ω and t > 0 we have that Therefore, by the generalized Lebesgue theorem, we get that Finally, we need to prove the (PS) c -condition. Take a sequence (u n ) n ⊆ X s 0 such that On the other hand, we have Thus, since p > 2, it follows that (u n ) n is bounded in X s 0 . Then, we get that, up to a subsequence, u n u in X s 0 and by lemma 2.
Since (u n − u) n is a bounded sequence and F λ (u n ) → 0 in (X s 0 ) , we get that By (9),ˆΩ thus we immediately get that u n → u in X s 0 . Hence, by the mountain pass theorem, there exists a critical point u ∈ X s that is problem ( P λ ) admits one nontrivial solution, as well. Case λ > λ 1 . If λ > λ 1 , then there exists i 1 such that λ i λ < λ i+1 and we shall apply theorem 2.3 taking Then, in a sphere of radius ρ, by (7) and the same calculations used in the previous case, we have that (6), it follows that Moreover, taking v = u + te i+1, with u ∈ H i and t > 0, since u and e i+1 are orthogonal in L 2 (Ω) and in X s 0 (see remark 1.3), by (6), we get that Using the generalized Lebesgue theorem, we get that G M Canneori and D Mugnai Nonlinearity 31 (2018) 3251 and, since by remark To conclude the proof, let us check that F λ satisfies the (PS) c -condition.
. Suppose by contradiction that (u n ) n is unbounded. Then, up to a subsequence, the sequence u n n diverges, and by lemma 2.1 we may assume that there exists w ∈ X s 0 such that u n u n w in X s 0 , strongly in L p (Ω) and a.e. in Ω.
Let us observe that (10) implies that On the other hand, we can write where the first term of the right-hand-side goes to zero as n → ∞ and the other two terms are non-positive. Hence, since F λ (u n )u n / u n → 0 as n → ∞, we have that both the following limits exist and Therefore, by (10) we get that Moreover, if v ∈ C ∞ c (Ω), we obtain that so, in this way, by (10) Since C ∞ c (Ω) is dense in X s 0 (see [7]), (13) holds for every v in X s 0 . Therefore, w is a nontrivial eigenfuction, see (12), of L K with associated eigenvalue λ. If λ = λ i , this is a contradiction.
On the other hand, if λ = λ i , w = αe i , with α ∈ R, and a contradiction arises due to remark 1.4 and the fact that w 0, see (11).
Hence, every (PS) c -sequence is bounded. Therefore, by lemma 2.1 we can suppose that u n u in X s 0 , u n → u in L q (Ω), for any q ∈ [1, 2 ), u n → u a.e. in Ω. (14) We can observe that the sequence (u n − u) n is bounded and by (14) it follows that u n → u in X s 0 . Therefore, by theorem 2.3, there exists a nontrivial critical point u for F λ with F λ (u) > 0, and thus problem ( P λ ) admits a nontrivial solution. □

Multiplicity via ∇-theorems
The aim of this section is to produce a multiplicity result for problem (P λ ). The idea is to apply theorem 2.5 to the functional associated to (P λ ), which reads for every u ∈ X s 0 . Remark 4.1. Given q ∈ [1, 2 ), we define the inverse of the operator L K , L −1 Moreover, L −1 K : L q (Ω) → X s 0 is compact.
If l, m ∈ N, with l 1 and m l + 1, are such that λ l < λ l+1 = . . . = λ m < λ m+1, we will choose  We immediately observe that the previous definition is well-posed, since e 1 ∈ H i for every i ∈ N and e 1 > 0 in Ω.

Remark 4.4. It is clear by definition 4.3 that
for every u ∈ X i , u 0.
In the sequel, we will need the next property.
Proof. Fix i 2 and take u ∈ H i such that u 0 in Ω and ´Ω Therefore, since e 1 > 0 in Ω, we get that α 1 =´Ω ue 1 dx > 0 and thus

Geometry of the ∇-theorem
In this section we check that the geometric condition required in theorem 2.5 holds true.

Proposition 4.6 ((∇)-geometry for
Proof. First of all, let us observe that X 1 ⊕ X 2 = H m and X 2 ⊕ X 3 = H ⊥ l . Thus, the idea is to show that there exist R > ρ > 0 such that Concerning (a), consider u ∈ H ⊥ l . Fixed ε > 0, there exists ρ > 0 small enough such that, by lemma 3.1, (7) and lemma 2.1, we have that for u = ρ small enough. Now we prove (b). First, consider u ∈ H l . As a consequence of (6), we easily get that On the other hand, if u ∈ H m it can be written in the form with v ∈ H l and w ∈ span{e l+1 , . . . , e m }. Suppose by contradiction that there exists a sequence (v j + w j ) j , with v j ∈ H l and w j ∈ span{e l+1 , . . . , e m } for every j ∈ N, such that v j + w j → ∞ as j → ∞ and Since H m = H l ⊕ span{e l+1 , . . . , e m } is finite-dimensional, then, up to a subsequence, by lemma 2.1 we have that v j + w j v j + w j →v +w in X s 0 , L q (Ω) and a.e. in Ω, for every q ∈ [1, 2 ), with v ∈ H l and w ∈ span{e l+1 , . . . , e m }. In particular, we have that v +w ∈ H m and v +w = 1.

The (∇)-condition
This section is devoted to the proof of a suitable (∇)-condition for functional F λ . First of all, we need to prove two lemmas.
Proof. We proceed by contradiction, so we suppose that there exist σ > 0, (µ j ) j ⊆ R, with µ j ∈ [λ l +σ, λ m+1 −σ], and a sequence (u and Since u j ∈ H l ⊕ H ⊥ m for every j ∈ N, by (19) we have that Being this valid for every j ∈ N, as a consequence of (20) we get that Now, since u j ∈ H l ⊕ H ⊥ m for every j ∈ N, we can write with v j ∈ H l and w j ∈ H ⊥ m . Therefore, v j and w j are orthogonal in X s 0 , and so and and so, by (7), (6), (22) and (23), we get In this way, we havē On the other hand, by Hölder's inequality, theorem 2.1 and (23), we have that Thus, combining the previous inequality with (24) and the fact that u j ≡ 0, we find that with C = γCλm+1 σ .

G M Canneori and D Mugnai Nonlinearity 31 (2018) 3251
Therefore, u j → 0 as j → ∞, and thus, since H l ⊕ H m is finite-dimensional, But, by (25), lemmas 3.1 and 2.1, for every ε > 0 we have that and thus u j p−2 > 0 for every j ∈ N, which is in contradiction with (26). □ Lemma 4.8. Let l, m ∈ N, with l 1 and m l + 1, be such that λ l < λ l+1 = . . . = λ m < λ m+1 and let λ ∈ R, λ = λ 1 . Denote by P : Then, up to a subsequence, u j → ∞ as j → ∞ and so we may assume that there exists u ∈ X s 0 such that (see lemma 2.1) u j u j u in X s 0 , strongly in L p (Ω) and a.e. in Ω.
By (i) and (27), dividing F λ (u j ) by u j p , we get that with o(1) → 0 as j → ∞, and thus u 0 in Ω.
Moreover, every u j can be written in the form Observe that and thatˆΩ since Pu j ∈ span {e l+1 , . . . , e m } and span {e l+1 , . . . , e m } is orthogonal to H l and H ⊥ m also in L 2 (Ω).
Therefore, by remark 4.2, (29)- (31) and (15) we have that Observe that, by the Holder inequality and lemma 2.1, we have that On the other hand, by (32), we have that and thus, by (i)-(iii) and throwing away the non-positive terms, we get with o(1) → 0 as j → ∞. Therefore, using the Holder inequality and lemma 2.1, we get that with a 1 > 0 and o(1) → 0 as j → ∞. Now, since Pu j → 0, see (ii), and u j → ∞ as j → ∞, the previous inequality implies that there exists a 2 > 0 such that Thus, we can suppose that M(u j ) → M as j → ∞. Therefore, passing to the limit as j → ∞ in (34), we have that, for every ε > 0 which implies that M = 0. In this way, we get that and thus Then, since L −1 K : L p (Ω) → X s 0 is a compact operator, by (27) and (36), we have that Moreover, by remark 4.2, (29) and (iii) we get that and thus, by (27), (ii) and (37) with u ∈ H l ⊕ H ⊥ m and u = 1. Therefore, for every v ∈ X s 0 , considering (ii), (36), (38) and the fact that L −1 K is compact, we have that and thus, by (15) and the fact that Qu = u, since u ∈ H l ⊕ H ⊥ m , we have that for every v ∈ X s 0 . This means that u is a nontrivial ( u = 1) and non-negative (see (28)) eigenfunction in the space H l ⊕ H ⊥ m , but this is impossible since λ = λ 1 . □ Now we are ready to prove the (∇)-condition for F λ .
Proof. Let P, Q denote the orthogonal projections introduced in lemma 4.8.

G M Canneori and D Mugnai Nonlinearity 31 (2018) 3251
On the other hand, by remark 4.2, we can write Moreover, by (41) and the generalized Lebesgue theorem, and so Therefore, since L −1 K : L p (Ω) → X s 0 is a compact operator, by (42) and (41) we have that Now, by (40), we get and hence Now, again from (40), for every ϕ ∈ H l ⊕ H ⊥ m . But, on the other hand, by (44), (41) and the generalized Lebsegue theorem, for every ϕ ∈ H l ⊕ H ⊥ m , and thus u is a critical point of Fλ, constrained on H l ⊕ H ⊥ m . Then, by lemma 4.7, u ≡ 0 and hence Fλ(u) = 0.

The multiplicity result
First of all, we want to produce an existence result for problem (P λ ) using theorem 2.5. In order to achieve this result, we need to prove the following two lemmas.
Lemma 4.10. Let l, m ∈ N, with l 1 and m l + 1, be such that Proof. By contradiction, suppose that there exists a sequence (u j ) j ⊆ H m and a constant M ∈ R such that u j → ∞ as j → ∞ and Therefore, by lemma 2.1 and the fact that H m is finite-dimensional, u j u j → u in X s 0 , in L q (Ω) and a.e. in Ω, for every q ∈ [1, 2 ) and u = 1. Now, dividing both sides of (45) by u j p , by (46) we get that with o(1) → as j → ∞, and thus u 0 in Ω.
Since ε is any number smaller than ε σ and larger than sup F λ (∆) (see the notation of theorem 2.5), we get that for every i = 1, 2. □ Remark 4.13. Of course, the existence of two solutions for λ near λ l is obvious from bifurcation theory, but the application of the ∇-theorem gives us precise estimates on the associated critical values, which are fundamental to prove our main theorem below, which is a precise formulation of theorem 1.5. and thus, using the generalized Lebesgue theorem and the fact that (e m+1 ) − ≡ 0, we get that Then, the geometric condition of the classical linking theorem holds and, since we have already proved the (PS) c -condition for F λ for every λ = λ 1 (see the proof of theorem 1.2), we find another solution u 3 for problem (P λ ) with the property that Therefore, by lemma 4.11, (51) and (53), there exists δ δ such that, for every λ ∈ (λ l+1 − δ, λ l+1 ), and thus finally finding the announced third nontrivial solution. □

γ → ∞
In order to underline the dependence of the functional and of the solutions from γ, in this section we will use the following replacements: In this way, we notice that all the solutions we found in theorem 4.14 enjoy the property that for every γ > 0.
< ∞ by lemma 4.10, since λ < λ m+1. Moreover, also in theorem 1.2 we have a similar uniform estimate. Indeed, if λ < λ 1 we got the existence of a nontrivial solution u γ via the mountain pass theorem (see the proof of theorem 1.2). Hence, we have the following information: for every γ > 0, where Γ = ϕ ∈ C [0, 1] , X s 0 : ϕ(0) = 0, ϕ(1) = e . In this way, choosing e = −Re 1 for some R > 0 large enough and taking the path ϕ(t) = −tRe 1 , t ∈ [0, 1], we get that and hence On the other hand, if λ > λ 1 , and so λ i λ < λ i+1 for some i ∈ N, a nontrivial solution is obtained using the linking theorem with the decomposition X s 0 = H i ⊕ H ⊥ i . In particular, if we choose e = e i+1 and R > ρ > 0 (see theorem 2.3), every nontrivial critical point where Q = {v = u + te i+1 u ∈ H i , u R, t ∈ 0, R/ λ i+1 } and H = {h ∈ C (Q, X s 0 ) : h |∂Q = Id}. Taking h = Id Q and v ∈ Q, by the same estimates provided in the proof of theorem 1.2, we have that This being valid for every γ > 0, we get that Finally, if λ = λ 1 , we have F γ λ (te 1 ) = 0 for every γ, t > 0.
Notice that in this last case it is clear that no upper bound is available for the set of solutions. On the other hand, in our next result we will show that all the solutions found in theorem 1.2 for λ = λ 1 and in theorem 1.5 are bounded thanks to the estimates found in (55), in (56) and in (54). More precisely, we have the following a priori estimate. Theorem 5.1. If (u γ ) γ>0 is a family of solutions of (P γ λ ) with sup γ F γ λ (u γ ) < ∞ and λ = λ 1 , then (u γ ) γ>0 is bounded in X s 0 .
Proof. Case λ < λ 1 . By the Poincaré inequality we immediately get that with C > 0. Thus, by assumption, (u γ ) γ is bounded in X s 0 , as claimed. Case λ > λ 1 . Suppose by contradiction that (u γ ) γ is unbounded in X s 0 . Then, there exists a sequence (u n ) n . = (u γn ) n such that u n → ∞ as n → ∞ and u n u n w in X s 0 , strongly in L p (Ω) and a.e. in Ω.
(61) We observe that Therefore, starting from (61), we obtain that lim n→∞ (λ 1 − λ)ˆΩ u n e 1 u n dx = (λ 1 − λ)ˆΩ we 1 dx = 0 and thus, by (60), w ≡ 0, which is in contradiction with (59). Hence, (u γ ) γ>0 is bounded in X s 0 . □ We notice that, thanks to (54)-(56), all the set of solutions we found in the previous sections are equibounded, and so we may assume that any sequence of solutions (u γn ) n converges weakly in X s 0 as n → ∞. From now on, we will write (u γ ) γ>0 to denote any sequence (u γn ) n and we will write u γ u as γ → ∞ meaning that u γn u as n → ∞. Since u is a weak limit in X s 0 , we cannot consider any set defined by point-wise values of u. For instance, it would be natural to define the 'contact set' as {x ∈ Ω : u(x) = −1}, which is a closed set if u is continuous, but, at this stage, we do not have any tool which lets us say that u has such a regularity. For this reason, we introduce the following sets: where two sublevel S γ and S γ are equivalent according to ∼ if they differ for a set of measure zero. Then, we define the 'free' set F = x ∈ Ω : there exists a neighborhood U of x and γ 0 > 0 such that |U ∩ S γ | = 0 for all γ γ 0 , where |A| here stands for the Lebesgue measure of a set A. Of course, F is an open subset of Ω. Thus, C := Ω \ F is closed, where for every v ∈ C ∞ c (Ω). Finally, we prove that suppµ ⊆ C: if x 0 ∈ Ω \ C, then there exists a neighborhood U of x 0 and γ 0 > 0 such that u γ + 1 0 a.e. in U for every γ γ 0 . Now, take φ ∈ C ∞ c (U), so that¨O for every γ γ 0 . Passing to the limit as γ → ∞, we get the claim. □