Local versus nonlocal elliptic equations : short-long range eld interactions

Abstract: In this paper we study a class of one-parameter family of elliptic equations which combines local and nonlocal operators, namely the Laplacian and the fractional Laplacian. We analyze spectral properties, establish the validity of the maximum principle, prove existence, nonexistence, symmetry and regularity results for weak solutions. The asymptotic behavior of weak solutions as the coupling parameter vanishes (which turns the problem into a purely nonlocal one) or goes to in nity (reducing the problem to the classical semilinear Laplace equation) is also investigated.


Introduction and main results
Consider the following class of mixed local-nonlocal elliptic equations (−∆) s u − ϵ∆u = f (x, u) in Ω, u = in R N \ Ω, (1.1) where Integro-di erential equations of the form (1.1) arise naturally in the study of stochastic processes with jumps. The generator of an N-dimensional Lévy process has the following general structure: where ν is the Lévy measure and satis es R N min{ , |y| }dν(y) < +∞, and χ B is the usual characteristic function of the unit ball B of R N . The rst term of (1.2) on the right-hand side corresponds to the di usion, the second one to the drift, and the third one to the jump part.
In the particular case when ν = , namely when there are no jumps, L turns into a classical and extensively studied second-order di erential operator. On the other hand, the case when the process has no di usion and no drift has attracted a lot of attention recently (we refer to the book [18] for the development of the existence theory for several nonlinear nonlocal problems; see also the survey [21] for related regularity properties of solutions). In particular, one of the most prominent operators belonging to this class is the fractional Laplacian (−∆) s , which turns out to be the in nitesimal generator of isotropic s-stable Lévy processes and plays an important role in analysis and probability theory. The issue of the existence of solutions to fractional Laplacian problems, as well as the study of qualitative properties of such solutions, has been addressed by many authors and the literature devoted to the eld grows up continuously (we mention, in particular, the following recent contributions [1,4,6,7,11,17,23,25,26] and the references therein). Finally, it is well-known, e.g. from [13,19], that Markov processes with both di usion and jump components are suitable to modeling many situations in nance and control theory, see also [8][9][10]29]. For the above reasons, the study of the operator L with di usion, drift and jump components appears quite intriguing and, as far as we know, there are just a few contributions in this direction, mainly relying on a combination of probabilistic and analytic techniques. For instance in [12], under the assumption that the di usion is uniformly elliptic and suitable conditions on ν, the author established a Harnack-type inequality and a regularity result for functions that are nonnegative in R N and harmonic in some domain. In [14,15] the author derived interior Schauder estimates for both classical and weak solutions of the equation Lu = f . Here we focus on the case L = ϵ∆ − (−∆) s , obtained from (1.2) by setting a ij = ϵδ ij , ϵ > , b j = and ν given by the symmetric kernel |x − y| −N− s . We study this problem from a purely analytical point of view. Precisely, the paper is divided into two parts. Firstly, for xed ϵ > (without loss of generality, we may assume ϵ = ), we provide some basic results related to (1.1), including spectral properties, maximum principles, existence, nonexistence, symmetry and regularity of weak solutions. With these preliminary tools, in the second part we will address further issues related to the asymptotic pro le, as ϵ → as well as ϵ → +∞, of ground states of the model problem in Ω, u = in R N \ Ω.
If Ω is a bounded star-shaped domain, it was proved in [22] that (1.3) has no nontrivial bounded solutions if p ≥ * := N N− . When ϵ = , problem (1.3) possesses a positive solution for < p < * s := N N− s , N > s, and, if Ω is a bounded, C , and star-shaped domain, do not exist positive bounded solution for p = * s and there are no nontrivial bounded solution for p > * s , see [23,24]. It is natural to investigate the behavior of the ground state uϵ in (1.3), as ϵ → (corresponding to the fractional case), which may converge, blow up or vanish, depending on the range of p. On the other hand, in the case ϵ → +∞, setting vϵ(x) = ϵ − p− uϵ(x), the function vϵ satis es (1.5) By Sobolev embedding theorems and regularity theory, classical solutions to (1.5) do exist for < p < * with N > , and thus one may suspect in this case that uϵ(x) blows up almost everywhere in Ω, as ϵ → +∞. We mention that the operator L can be given a local realization by means of a Ca arelli-Silvestre-type extension [6]. However, if one proceeds as in [5] by directly extending u ∈ H s (Ω) to the cylinder C Ω := Ω × ( , +∞), i.e. considering C Ω as the domain of the extension w of u, with w = on the lateral boundary ∂ L C Ω of C Ω , then one would end up with the fractional Laplacian operator de ned by spectral decomposition, which on bounded domains does not coincide with (−∆) s . The idea is then to start from the half-space R N+ + and to require the trace of the extension to vanish a.e. outside Ω. This can be achieved as follows. De ne the space It is known from [6] that, for all u ∈ H s (R N ) there exists a unique function E(u) ∈ Y s , called the harmonic extension of u to R N+ and (−∆) s u corresponds to the Dirichlet-to-Neumann map for a suitable normalizing constant ks > . Next de ne the following subspace whose elements have traces over R N which vanish outside Ω. We say that w ∈ Y s is a weak solution to in the weak sense, then its trace over R N weakly solves Before presenting our main results, let us introduce the functional framework. and, by Poincaré's inequality, this norm in X s is also equivalent to We cliam that (X s , · s) is an Hilbert space with scalar product For the proof, see Lemma 12. Throughout this paper, we will use the symbol · p, p ∈ [ , +∞] to denote the usual L p -norm in R N , while the letters C, C i will represent generic positive constants which may change from line to line.

Preliminary results
Let us begin with some spectral properties of the operator L which, it is well known, are crucial for the solvability of linear and resonance nonlinear problems.
Theorem 1. Let s ∈ ( , ), N > , and consider the following eigenvalue problem: and a nonnegative eigenfunction e ∈ X s corresponding to λ , satisfying e = and λ = e s ; ( ) λ is simple; ( ) the set of the eigenvalues of (1.13) consists of a sequence {λ k }, k ∈ N, with < λ < λ ≤ . . . ≤ λ k ≤ . . . and λ k → +∞ as k → +∞. where M k+ = {u ∈ X s : u, e j s = , ∀j = , , · · · , k}. Moreover, for any k ∈ N, there exists e k+ ∈ M k+ that is an eigenfunction of L corresponding to λ k+ and that veri es e k+ = , λ k+ = e k+ s ; ( ) the sequence {e k } of eigenfunctions corresponding to λ k is an orthonormal basis of L (Ω) and an orthogonal basis of X s ; ( ) each eigenvalue λ k has nite multiplicity; more precisely, if λ k is such that λ k− < λ k = . . . = λ k+h < λ k+h+ , for k > and for some h ∈ N, then the set of all the eigenfunctions corresponding to λ k agrees with span{e k , . . . , e k+h }.
Theorem 1 can be viewed as the generalization of similar eigenvalue problems for the Laplacian and the fractional Laplacian operators, see the monograph [18,Chapter 3]. Next, we focus on maximum principles, both for classical and for weak supersolutions of the problem Lu = , u |R N \Ω = . De ne Theorem 2. Assume that u ∈ Ls ∩ C (Ω) is lower semicontinuous onΩ. If Theorem 3. Assume that u ∈ H (Ω), [u] s,R N < +∞, and u satis es Then u(x) ≥ a.e. in Ω.
We have used the arguments of [7] to prove Theorem 2. Basically, if we have u(x ) = min Ω u(x) < for some x ∈ Ω, then (−∆) s u(x ) ≥ (−∆) s u(x ) − ∆u(x ) ≥ and we reduce the problem to the fractional setting. On the other hand, by Stampacchia's theorem the proof of Theorem 3 relies only on local arguments. In a classical nonlinear setting, we derive the following existence result for problem (1.1), where we set for simplicity ϵ = .
Theorem 4. Assume that f : Ω × R → R satis es the following conditions: Then, the problem possesses a nonnegative solution u ∈ X s .
for all x ∈ Ω and t ∈ R.
If u is a bounded solution of (1.18), then u ≡ .
A consequence of Theorems 4 and 5 we have Corollary 6. Assume that Ω ⊂ R N is a bounded domain, s ∈ ( , ) and p ∈ ( , * ). Then, the problem The remaining results are related to symmetry properties and regularity of weak (classical) solutions.

Theorem 9.
Assume that u ∈ X s is a weak solution of the problem Then the following facts hold: ( ) if f ∈ L q (Ω) with q = N , then there exists a constant α > such that the positive weak solution of (1.21) satis es Ω e αu dx < +∞.
In particular, u ∈ L q (Ω) for every q < +∞; ( ) if f is a positive function and f ∈ L q (Ω) with N N+ ≤ q < N , then there exists a constant C > such that the positive weak solution of (1.21) satis es Note that the results in Theorem 9 are consistent with the ones known for the Laplace operator. As we will see in the proof, this is due to the choice of appropriate truncated functions which downplay the e ects of the nonlocal term.
In the second part of this paper we aim at investigating the asymptotic behavior of ground states (weak solutions) of problem (1.3) when < p < * , N > . For this purpose set Note that Sϵ can be also written in the form . Hereafter let us denote The compactness of Sobolev embeddings yields the existence of non-negative minimizers wϵ of Sϵ. Then, by Lagrange's multiplier rule, there exists some λϵ > such that By multiplying both sides of (1.22) by wϵ and then integrating, we get λϵ = Sϵ. Moreover, by maximum principles, wϵ > . The energy associated to (1.22) is Since wϵ is a solution of (1.22), we get wϵ ϵ = Sϵ wϵ p p and wϵ p = , and thus On the other hand, any nontrivial solution v of (1.22) necessarily satis es We have also which yields v p ≥ . This means that, by (1.25), v should satisfy This fact, together with (1.24), implies that wϵ is a ground state of equation (1.22) and so uϵ = S − −p ϵ wϵ is a ground state of (1.3). Note that uϵ also satis es Pϵ(uϵ) = Sϵ. Let us stress the fact that we do not know whether the mountain-pass solution obtained from Theorem 4 and the minimal solution uϵ obtained above agree. Next we will focus on the minimal solution uϵ. Before stating our main results, we introduce the following best constants

Main results
Theorem 10. The following facts hold:

there exists a minimizer u of S and uϵ → u in H s as ϵ
Moreover, there exist a minimizer uϵ such that uϵ ϵ → as ϵ → .
Finally let us make a few comments on Theorems 10 and 11. We expect the minimizers uϵ obtained in (i) to be uniformly bounded with respect to ϵ and thus that the convergence should hold in the classical sense. In this context, the existence or nonexistence of positive solutions of the problem with < p < * s plays a very important role. However, the moving plane methods can not be applied directly here because one of the most important tools -the Kelvin transform -is not available. On the other hand, the classical minimizers uϵ obtained in (ii) blow up as ϵ → , up to a subsequence if necessary. Indeed, if not one has uϵ → u in classical sense for supp(u ) ⊂ Ω, and then u is a minimizer of S, thus a contradiction. Obviously, for the case (iii), the minimizers uϵ vanishe a.e. in Ω. We stress that the constant plays a crucial role to get (iii), while to prove Theorem 11, we need the constant The rest of this paper is organized as follows. We prove Theorems 1-9 in Section 2 whence Section 3 is devoted to prove Theorems 10 and 11.

Proofs of theorems 1-4 and 7-9
Let us prove rst the following preliminary result Proof. It is straightforward to see that ·, · s is a scalar product on X s which induces the norm · s. So, it su ces to prove that (X s , · s) is complete. Let {u k } be a Cauchy sequence in X s , that is, for any ε > , there exists k > such that for k, j ≥ k , H (Ω) as k → +∞. Moreover, since u k = a.e. in C Ω, we may de ne u = a.e. in C Ω and then u k → u in H (R N ) as k → +∞. Thus, up to a subsequence, still denoted by {u k }, u k → u a.e. in R N . Fatou's lemma and (2.1) yield and hence u ∈ X s . Finally, by using again (2.1), we get and u j → u in X s as j → +∞, as desired.
Proof of Theorem 1. The proof is modeled on the one of the nonlocal operator −L K studied in [18]. However, for reader's convenience, we give the detailed proof. (1) Let {u j } be a minimizing sequence for I, that is, Then and this implies that I(u * ) = inf u∈M I(u).
Let t ∈ (− , ), v ∈ X s , C t = u * + tv and u t = (u * + tv)/C t . Then u t ∈ M and one has (2.5) By the very de nition of u * we get Now, assume that λ is attained at some e ∈ X s with e = . By (2.6), we get I(e ) = λ . To prove that e ≥ , we rst show that if e is an eigenfunction related to λ with e = , then both e and |e| realize the minimum in (1.14).
Note that |e| ∈ X s and |e| = e = . Thus, since e is a minimizer, we get I(|e|) = I(e) = I(e ) = λ . This also implies that e ≥ or e ≤ a.e. in Ω. By replacing e by |e |, we may conclude that e ≥ .
(2) Suppose by contradiction that there exists another eigenfunctionẽ ∈ X s ,ẽ ≠ e , corresponding to λ . Then, by (1) we may assume thatẽ ≥ . Setting u :=ẽ ẽ and v := e − u , we claim that v = . In fact, direct calculations show that v is also an eigenfunction related to λ , and so from (1) we deduce v ≥ or v ≤ a.e. in Ω, that is, e ≥ u or e ≤ u a.e. in Ω. However, since e − u = , we get e = u and hence v = a.e. in Ω. This also implies that v = a.e. in R N and therefore, if w ∈ X s is an eigenfunction related to λ , then w = ± w e a.e. in R N .
(3) Since M k is a weakly closed subspace of X s , arguing as in step (1), we deduce that λ k+ is attained at some e k+ ∈ X s . On the other hand, being M k+ ⊆ M k , we get < λ ≤ λ ≤ . . . ≤ λ k ≤ . . . Now, we claim that λ ≠ λ . If not, e ∈ M would be also an eigenfunction related to λ . By (2), e = ± e e a.e. in R N . Thus, we would get = e , e s = ± e e s , which yields e ≡ a.e. in R N , a contradiction. Analogously to the proof of (2.6), we have e k+ , w s = λ k+ R N e k+ wdx, ∀w ∈ M k+ . (2.7) We claim that equality (2.7) holds for any w ∈ X s , i.e., that e k+ is an eigenfunction related to λ k+ . Let us consider the decomposition X s = span{e , · · · , e k } ⊕ (span{e , · · · , e k }) ⊥ = span{e , · · · , e k } ⊕ M k+ , and write any given v ∈ X s as which proves the claim. Next we show that λ k → +∞ as k → +∞. Suppose, by contradiction, that λ k → C ∈ R + as k → +∞. Since e k s = λ k , up a subsequence, e k j → e in L (Ω) as k j → +∞. On the other hand, being e k i , e k j s = for k i ≠ k j , we get Ω e k i e k j dx = for k i ≠ k j and thus which is a contradiction, as Finally, let us show that any eigenvalue of (1.13) can be written in the form (1.15). Arguing again by contradiction, assume that there exists an eigenvalue λ ∉ {λ k } and let e be an eigenfunction corresponding to λ with e = . Then λ = e s ≥ e s = λ and this implies the existence of k ∈ N such that λ k < λ < λ k+ . Thus, e ∉ M k+ and so there exists j ∈ { , , · · · , k} such that e, e j s ≠ , a contradiction.
(4) Let us prove that {e k } is a basis of X s by a standard Fourier analysis technique. Given u ∈ X s , we de ne that is, which implies that {v j } is a Cauchy sequence in R, and thus there exists v * ∈ X s such that v j → v * in X s as j → +∞. For j ≥ i we also have v j ,ẽ i s = u,ẽ i s − v,ẽ i s = u,ẽ i s − v,ẽ s = .
Passing to the limit, we get v * ,ẽ i s = for any i ∈ N. This means that v * = and therefore u j = u − v j → u in X s as j → +∞. So, u = ∞ i= u,ẽ i sẽi, as we wanted.
In order to show that {e i } is a basis for L (Ω), too, chosen v ∈ L (Ω), we let v j ∈ C (Ω) such that v − v j L (Ω) < j . Since {e i } is a basis of X s and X s ⊂ L (Ω), there exist k j ∈ N and a function w j ∈ span{e , . . . , e k j } such that v j − w j s ≤ j . Thus, by Sobolev embeddings, which yields the claim.
(5) X s = span{e k , · · · , e k+h } ⊕ (span{e k , · · · , e k+h }) ⊥ . So, any eigenfunction u ∈ X s \ { } corresponding to λ k admits the representation u = u + u , with u = k+h i=k c i e i ∈ span{e k , · · · , e k+h } for some c i ∈ R, and u ∈ (span{e k , · · · , e k+h }) ⊥ . Obviously, u , u s = , which yields λ k u = u s = u s + u s . (2.8) By de nition, u is also an eigenfunction corresponding to λ k . Thus, Recalling that u and u are eigenfunctions corresponding to λ k , we deduce that u enjoys the same property and thus, u ∈ (span{e , · · · , e k+h }) ⊥ = M k+h+ . We claim that u ≡ . If not, we would have (2.10) Collecting (2.8), (2.9) and (2.10), we would obtain clearly a contradiction. Therefore, u = u ∈ span{e k , · · · , e k+h } and this completes the proof.
Proof of Theorem 2. Inspired by [7], assume that u(x) < in Ω. Owing to the lower semi-continuity of u onΩ, there exists some x ∈Ω such that u(x ) = min Ω u(x) < .
Moreover, since u ≥ in R N \ Ω, we deduce that x is an interior point of Ω. Taking into account the fact that ∆u(x ) ≥ , we get which contradicts the assumptions. Moreover, if u(x ) = at some point x ∈ Ω, then As a consequence of the previous result one has Corollary 13. Assume that u ∈ Ls ∩ C (Ω) is upper semi-continuous onΩ. If Using G(v) as a test function, we get and since (v(x) − v(y))(G(v(x)) − G(v(y))) ≥ , we get Ω ∇v∇G(v)dx ≤ . (2.14) Now, by Stampacchia's theorem (cf. [3]), the proof is standard. In fact, by (2.14) we get G (v)|∇v| = a.e. in Ω and hence ∇G(u) = a.e. in Ω. Since G(v) ∈ H (Ω), we deduce G(v) = a.e. in Ω and, as a consequence, v ≤ and u ≥ a.e. in Ω.
The proof of the existence result established in Theorem 4 requires some preliminary steps. Since we focus on nonnegative solutions, we consider the truncated energy functional associated with problem (1.18), namely where u + := max{u, }. Thanks to (f ), (f ) and Sobolev inequalities, F is well-de ned, Fréchet di erentiable and In order to exploit the mountain pass theorem, let us rst prove that F exhibits the suitable geometry.
Proof. For any u ∈ X s , by (f ), (f ) and Sobolev embeddings, we get By the de nition of q, we can choose ρ > su ciently small such that − C u q− s > and the result follows by setting δ = ρ − Cρ q+ .

Lemma 16.
There exists e ∈ X s satisfying e s > ρ and F (e) < .

Proof.
Fix v ∈ X s , with v s = and v > a.e. in R N ,, and let u = tv for t > . Then, by (f ) we obtain The assertion follows by taking e = tv, with t su ciently large.
A key-ingredient is the Palais-Smale condition, (PS) for short.

Lemma 17. The functional F satis es (PS).
Proof. Let c ∈ R, and let {u k } be a sequence in X s such that where C(R) > , and hence {u k } is bounded in X s . Then, up to a subsequence still denoted by {u k }, there exists u ∈ X s such that u k u in X s , u k → u in L m (Ω), < m < * , and u k → u a.e. in Ω as k → +∞ (note that, since u k = a.e. in C Ω, we may de ne u = a.e. in C Ω). By (2.18) and the boundedness of the sequence {u k − u}, we get (2.20) On the other hand, by (f ) and the compactness of the embeddings (2.21) A joint use of (2.20), (2.21) and the fact that u k u in X s , lead to u k s → u s as k → +∞ and, by classical arguments, to u k − u s → as k → +∞, as desired. and tu ∈ N i ϕ u (t) = . By direct computations, and so ϕu has a unique critical point (a local maximum), namely Moreover ϕu is increasing in ( , t(u)), decreasing in (t(u), +∞) and t(u)u ∈ N − . Let us split N as follows we easily derive thatc > . Indeed, if v = u/ u s and t(v) is as in (2.23), we get for any u ∈ N − , being cp > the best constant of the embedding X s → L p (R n ). The next step is to show thatc is attained at some function within N − . Let {u k } ⊂ N − be a minimizing sequence, i.e. such that F (u k ) →c as k → +∞. Clearly {u k } is bounded, so, up to a subsequence, u k u ∈ X s and u k → u ∈ L p (R N ) . So, which yields u ≠ . If u k ↛ u in X s , we would obtain We know that there exists a unique t ∈ ( , ) such that t u ∈ N − and that t → ϕu k (t) admits its maximum at t = , so F (γ(t)), The above considerations yieldc = c * . Moreover, if u ∈ X s \ { }, as t → +∞ and hence c ≤ c * . If X and X are the two components of X s induced by N, with ∈ X , there will exist r > such that B( , r) ⊂ X . Since ϕ u (t) = F (tu), u ≥ for all t ∈ [ , t(u)], one has F (u) ≥ for all u ∈ X . As a result, any γ ∈ Γ will have a non-empty intersection with N and thenc ≤ c. This implies thatc = c * = c and the proof is complete.

Remark 18.
It is worth pointing out that each ground state solution u of (1.19) does not change sign. Indeed, denoting as usual u + = max{u , } and u − = min{u , } the positive and the negative part of u , respectively, multiplying (1.19) by u ± and integrating, we get u ± ∈ N. As a result Before proving Theorem 8, it is necessary to introduce some notations and some preliminary results, namely an antisymmetric maximum principle holding for "narrow" regions. This is needed to apply the moving plane method directly to problem (1.20) (cf. [7]). For λ ∈ R, denote by the moving planes, by Σ λ := {x ∈ R n : x < λ} the region to the left of the plane and by the re ection of x ∈ R N with respect to T λ , and set

Lemma 19.
Let V be a bounded subset of Σ λ contained in {x ∈ R n : λ − l < x < λ}, l > su ciently small. Suppose that u ∈ Ls ∩ C (V) is lower semicontinuous onV. If c : R n → R is bounded from below in V and then u(x) ≥ in V for l su ciently small. Moreover, if u(x) = for some x ∈ V, then u(x) ≡ for a.e. x ∈ R n .
Proof. Arguing by contradiction, assume that u solves (2.24) and u < somewhere in V. Since u is lower semicontinuous, there exists x ∈ int(V) such that Since u is anti-symmetric in Σ λ , following [7] one has and, after a change of variables, it is not di cult to see that the last term diverges to −∞ as l → + . Being ∆u(x ) ≥ and c bounded below in V, we get which contradicts (2.24).
Proof of Theorem 8. Since u is a positive solution of (1.20), we get for every x ∈ Σ λ , where and |c λ (x)| ≤ L, for some L > . Applying Theorem 2.24, for λ → − + one has w λ (x) ≥ in Σ λ . Now, setting we claim that λ = . Assuming the contrary, then it should be otherwise there would exist x ∈ Σ λ : Let us estimate which together with the fact that ∆w λ (x ) ≥ obviously contradicts (2.25). So, there exists δ > such that, for any λ ∈ (λ , λ + δ) we have w λ (x) ≥ , x ∈ Σ λ , which contradicts the de nition of λ . Therefore w (x) ≥ , x ∈ Σ , or equivalently u (−x , x , . . . , xn) ≤ u(x , x , . . . , xn), Since the x -direction is arbitrary, the previous relation implies that u is radially symmetric with respect to the origin. The monotonicity follows from the fact that w λ (x) ≥ for all λ ∈ (− , ].
Finally, we deal with the regularity of solutions to the problem where Ω ⊂ R N is a bounded domain, N > . To be precise, we will study the L p regularity, including the boundedness of solution via Stampacchia's method, exponential summability and a Calderón-Zygmund type result (for the purely nonlocal case we refer to [16]). Let us recall following lemma by Stampacchia Lemma 20. Let ψ : R + → R + be a nonincreasing function such that where M > , δ > and γ > .
Proof of Theorem 9 (1) We follow Stampacchia's method. For k ≥ , we de ne the function Clearly, G k ( ) = and G k (σ) is Lipschitz continuous. For u ∈ X s , we get G k (u) ∈ X s . Using G k (u) as a test function, we get Since (u(x) − u(y))(G k (u(x)) − G k (u(y))) ≥ (G k (u(x)) − G k (u(y))) , from (2.27) we get where A k := {x ∈ Ω : |u(x)| ≥ k}. By applying Sobolev and Hardy inequalities, we have where S is the best constant of the embedding X s → L The assumption q > N implies * ( − * − q ) > and so, by Lemma 20, there exists k > such that ψ(k ) = |A k | = . Consequently, we get |u(x)| ≤ k a.e. in Ω.
(2) For any T > , we consider the function where α > will be xed later. Note that Φ( ) = , Φ ∈ C , is convex and its rst derivative is Lipschitz continuous.Thus, if u ∈ X s , then Φ(u) ∈ X s as well.
provided that αT > . So, xing α small enough and taking T large, we obtain Φ(u) * ≤ C, and the conclusion follows.
(3) For any T > , consider the function where β = m * * > . It is straightforward to observe that Φ enjoys the same properties as in item (2). Then, following the same steps of item (2), we have where /q + /q = and C > is independent of T. Thus, (2.28) and letting T → +∞ and recalling that * β = q ( β − ) we arrive at as desired.
Finally, we prove the boundedness of the solution u obtained in Corollary 6.

Lemma 21.
The solution u of (1.19) stated in Corollary 6 satis es u ∞ < C for some constant C > .

Proof of Theorems 10 and 11
Here we study the asymptotic behavior of ground states of problem (1.3) Let uϵ be a minimizer of Sϵ, as well as a ground state of (1.3), as shown in Section 1.
It is immediate to estimate the ϵ-norm and the L p -norm of such minimizers. Proof. By direct calculations, we get Next, we prove that the function {Sϵ} is bounded.
To be more precise, we show that {Sϵ} converges to di erent constants according to the range of p. We consider three di erent cases.
. The case < p < * By Sobolev embedding theorems, for < p < * s there exists a non-negative minimizer w of S. Analogously to the proof carried out in Section 1, the function u = S − −p w is a non-negative ground state of We point out that(3.1)) also admits a mountain pass type solution, see e.g. [24].
It is easy to check that u is radially symmetric. Moreover, applying the Lagrange multiplier rule, u satis es (−∆) s u − ∆u = µ|u| p− u in R N (3.23) for some µ > .
Proof. This is a direct consequence of Lemma 31, as