SEMILINEAR NONLOCAL ELLIPTIC EQUATIONS WITH CRITICAL AND SUPERCRITICAL EXPONENTS

. We study the problem N , where s ∈ (0 , 1) is a ﬁxed parameter, ( − ∆) s is the fractional Laplacian in R N , q > p ≥ N +2 s N − 2 s and N > 2 s . For every s ∈ (0 , 1), we establish regularity results of solutions of above equation (whenever solution exists) and we show that every solution is a classical solution. Next, we derive certain decay estimate of solutions and the gradient of solutions at inﬁnity for all s ∈ (0 , 1). Using those decay estimates, we prove Pohozaev type identity in R N and we show that the above problem does not have any solution when p = N +2 s N − 2 s . We also discuss radial symmetry and decreasing property of the solution and prove that when p > N +2 s N − 2 s , the above problem admits a solution . Moreover, if we consider the above equation in a bounded domain with Dirichlet boundary condition, we prove that it admits a solution for every p ≥ N +2 s N − 2 s and every solution is a

where s ∈ (0, 1) is a fixed parameter, (−∆) s is the fractional Laplacian in R N , q > p ≥ N +2s N −2s and N > 2s. For every s ∈ (0, 1), we establish regularity results of solutions of above equation (whenever solution exists) and we show that every solution is a classical solution. Next, we derive certain decay estimate of solutions and the gradient of solutions at infinity for all s ∈ (0, 1). Using those decay estimates, we prove Pohozaev type identity in R N and we show that the above problem does not have any solution when p = N +2s N −2s . We also discuss radial symmetry and decreasing property of the solution and prove that when p > N +2s N −2s , the above problem admits a solution . Moreover, if we consider the above equation in a bounded domain with Dirichlet boundary condition, we prove that it admits a solution for every p ≥ N +2s N −2s and every solution is a classical solution.
We denote by H s (Ω) the usual fractional Sobolev space endowed with the socalled Gagliardo norm g H s (Ω) = g L 2 (Ω) + Ω×Ω |g(x) − g(y)| 2 |x − y| N +2s dxdy For further details on the fractional Sobolev spaces we refer to [22] and the references therein. Note that in problem (1.2), Dirichlet boundary data is given in R N \ Ω and not simply on ∂Ω. Therefore, for the Dirichlet boundary value problem in the bounded domain, we need to introduce a new functional space X 0 , which, in our opinion, is the suitable space to work with.
Observe that, norms in (1.4) and (1.6) are not same, since Ω×Ω is strictly contained in Q. Clearly, the integral in (1.6) can be extended to whole R 2N as v = 0 in R N \Ω.
It is well known that the embedding X 0 → L r (R N ) is compact, for any r ∈ [1, 2 * ) (see [31,Lemma 8]) and X 0 → L 2 * (R N ) is continuous (see [30,Lemma 9]) . We set and we defineḢ s (R N ) as the completion of C ∞ 0 (R N ) w.r.t. the norm ||u||Ḣ s (R N ) + |u| L 2 * (R N ) (see [11] and [23]). Definition 1.1. We say that u ∈Ḣ s (R N ) ∩ L q+1 (R N ) is a weak solution of Eq. (1.1), if u > 0 in R N and for every ϕ ∈Ḣ s (R N ), Similarly, when Ω is a bounded domain, we say u ∈ X 0 ∩ L q+1 (Ω) is a weak solution of Eq. (1.2) if u > 0 in Ω and for every ϕ ∈ X 0 , the above integral expression holds. In recent years, a great deal of attention has been devoted to fractional and nonlocal operators of elliptic type. One of the main reasons comes from the fact that this operator naturally arises in several physical phenomenon like flames propagation and chemical reaction of liquids, population dynamics, geophysical fluid dynamics, mathematical finance etc (see [1,9,33,34] and the references therein).
When s = 1, it follows by celebrated Pohozaev identity that (1.1) does not have any solution when p = 2 * − 1 and q > p. In this paper we prove this result for all s ∈ (0, 1) by establishing the Pohozaev identity in R N for the equation (1.1). We recall that (1.1) has an equivalent formulation by Caffarelli-Silvestre harmonic extension method in R N +1 + . For spectral fractional laplace equation in bounded domain, some Pohozaev type identities were proved in [5,6,7]. In [13], Fall and Weth have proved some nonexistence results associated with the problem (−∆) s u = f (x, u) in Ω and u = 0 in R N \ Ω by applying method of moving spheres.
Very recently Ros-Oton and Serra [27, Theorem 1.1] have proved Pohozaev identity by direct method for the bounded solution of Dirichlet boundary value problem. More precisely they have proved the following: Let u be a bounded solution of (−∆) s u = f (u) in Ω, where Ω is a bounded C 1,1 domain in R N , f is locally Lipschitz and δ(x) = dist(x, ∂Ω). Then u satisfies the following identity: where F (t) = t 0 f and ν is the unit outward normal to ∂Ω at x and Γ is the Gamma function. For nonexistence result with general integro-differential operator we cite [28].
To apply the technique of [27] in the case of Ω = R N , one needs to know decay estimate of u and ∇u at infinity. In [27], Ros-Oton and Serra have remarked that assuming certain decay condition of u and ∇u, one can show that (−∆) s u = u p in R N does not have any nontrivial solution for p > N +2s N −2s . In this article for (1.1) we first establish decay estimate of u and ∇u at infinity and then using that we prove Pohozaev identity for the solution of (1.1) for all s ∈ (0, 1) and consequently we deduce the nonexistence of nontrivial solution when p = 2 * − 1. In the appendix, using harmonic extension method in the spirit of Cabré and Cinti [6], we give an alternative proof of Pohozaev identity in R N for the equation of the form The interesting fact about this proof is that, here we do not require decay estimate of u and ∇u at infinity as we use suitable cut-off function and in limit we take that cut-off function approaches to 1.
On the contrary to the nonexistence result for p = 2 * −1, we show using constrained minimization method that Eq.(1.1) admits a positive solution when p > 2 * − 1. Moreover, we study the qualitative properties of solution. More precisely, using Moser iteration technique we prove that any solution, u, of (1.1) is in L ∞ (R N ) and we establish decay estimate of u and ∇u. Then using the Schauder estimate from [25] and the L ∞ bound that we establish, we show that u ∈ C ∞ (R N ) if both p and q are integer and C 2ks+2s (R N ), where k is the largest integer satisfying 2ks < p if p ∈ N and 2ks < q if p ∈ N but q ∈ N, where 2ks denotes the greatest integer less than equal to 2ks . We also prove that u is a classical solution. Thanks to decay estimate of solution that we establish, we further show that solution of (1.1) is radially symmetric.
When Ω is a bounded domain, we prove that (1.2) admits a solution for every p ≥ 2 * − 1. For similar type of equations involving critical and supercritical exponents in the case of local operator such as −∆, we cite [4], [18]- [20]. For similar kind of equations with nonlocal operator we cite [3,10].
We turn now to a brief description of the main theorems presented below.
Note that the scaled function U = λ 1 p−1 u satisfies the equation We organise the paper as follows. In section 2, we recall equivalent formulation of (1.1) by the Caffarelli-Silvestre [8] associated extension problem-a local PDE in R N +1 + and we also recall Schauder estimate for the nonlocal equation proved by Ros-Oton and Serra [25]. In Section 3, we establish u ∈ L ∞ (R N ), decay estimate of solution and the gradient of solution at infinity. Section 4 deals with the proof of nonexistence result in R N when p = 2 * − 1. In section 5, we show that any solution of (1.1) is radially symmetric and strictly decreasing about some point in R N . While in section 6, we prove existence of solution to (1.1) for p > 2 * − 1 and to (1.2) when Ω is bounded and p ≥ 2 * − 1.
Notations: Throughout this paper we use the notation C β (R N ), with β > 0 to refer the space C k,β (R N ), where k is the greatest integer such that k < β and β = β − k. According to this, [.] C β (R N ) denotes the following seminorm Throughout this paper, C denotes the generic constant, which may vary from line to line and n denotes the unit outward normal.

2.
Preliminaries. In this section we recall the other useful representation of fractional laplacian (−∆) s , which we will use to prove decay estimate of solution at infinity. Using the celebrated Caffarelli and Silvestre extension method, (see [8]), fractional laplacian (−∆) s can be seen as a trace class operator (see [8,15,2]) . Let u ∈Ḣ s (R N ) be a solution of (1.1). Define w := E s (u) be its s− harmonic extension to the upper half space R N +1 + , that is, there is a solution to the following problem: is a normalizing constant, chosen in such a way that the extension operator E s : . (see [11]). Conversely, for a function w ∈ X 2s (R N +1 + ), we denote its trace on R N × {y = 0} as: This trace operator satisfies: Consequently, 3) is called the trace inequality. We note that H 1 (R N +1 + , y 1−2s ), up to a normalizing factor, is isometric to X 2s (R N +1 + ) (see [15]). In [8], it is shown that E s (u) satisfies the following: With this above representation, (2.1) can be rewritten as: (2.5) Note that for any weak solution w ∈ X 2s (R N +1 + ) to (2.4), the function u := Tr(w) = w(., 0) ∈Ḣ s (R N ) is a weak solution to (1.1).
Next, we recall Schauder estimate for the nonlocal equation by Ros-Oton and Serra [25].
We conclude this section by recalling some weighted embedding results from Tan and Xiong [32]. For this, we introduce the following notations where B R is a ball in R N with radius R and centered at origin. Note that, B R ×{0} ⊂ Q R . We define, We note that, s ∈ (0, 1) implies the weight y 1−2s belongs to the Muckenhoupt class A 2 (see [21]) which consists of all non-negative functions w on R N +1 satisfying for some constant C, the estimate where the supremum is taken over all balls B in R N +1 .
Then there exists constant C and δ > 0 depending only on N and s such that for any 1 ≤ k ≤ n+1 n + δ, Proof. It is known from [32, Lemma 2.1] that the lemma holds for f ∈ C 1 c (Q R ) (also see [12]). For general f , the lemma can be easily proved applying density argument and Fatou's lemma.
Then there exists a positive constant δ depending only on N and s such that for any ε > 0. Proof for any ε > 0. Clearly the 1st integral on RHS converges to Q R y 1−2s |∇f | 2 dxdy. Thanks to Lemma 2.2, it follows that the embedding X 2s 0 (Q R ) → L 2 (Q R , y 1−2s ) is continuous. Therefore, we can also pass to the limit in the 2nd integral of the RHS. On the other hand, using the trace embedding result, we can also pass to the limit on LHS. Hence, the lemma follows.
3. L ∞ estimate and decay estimates. Proof of Theorem 1.3 Let u be an arbitrary weak solution of Eq.(1.1). We first prove that u ∈ L ∞ loc (R N ) by Moser iterative technique (see, for example [17,32]). From Section-2, we know that w(x, y), the s−harmonic extension of u, is a solution of (2.4).
Let B r denote the ball in R N of radius r and centered at origin. We define For t > 1, we choose the test function ϕ in (2.5) as follows: ). Using this test function ϕ, we obtain from (2.5) k 2s 3) Here we observe that on the set {w < 0}, we have ϕ = 0 and ∇ϕ = 0. Thus (3.2) remains same if we change the domain of integration to {w ≥ 0}. Therefore, in the support of the integrand ∇w = ∇w. As a result, substituting (3.3) into (3.2), it follows k 2s Notice that in the support of the integrand of second integral on the LHS ∇w = ∇w and in the third integral w L =w, ∇w L = ∇w. Hence the above expression reduces to k 2s where for the RHS, we have used the fact that w ≤w.
where C depend only on N, s, p, q. Hence, by iteration we have Hence, u ∈ L ∞ (B 1 2 (0)). Translating the equation, similarly it follows that u ∈ L ∞ loc (R N ).
To show the L ∞ bound at infinity, we define the Kelvin transform of u by the functionũ as follows:ũ
Case 2: Ω is a bounded domain.
Arguing along the same line with minor modifications, it can be shown that u ∈ L ∞ (Ω). Therefore the conclusion follows as u = 0 in R N \ Ω.
As a result, applying Theorem 2.1(a) , we obtain for all ε > 0. Here the constants C are independent of u, but may depend on radius 1 2 and centre 0. Since the equation is invariant under translation, translating the equation, we obtain Note that in (3.28) and (3.29) constants C are same as in (3.26) and (3.27) respectively. Thus, in (3.28) and (3.29) constants do not depend on y. This implies u ∈ C 2s (R N ) when s = 1 2 and in C 2s−ε (R N ), when s = 1 2 . Hence, f (u) ∈ C 2s (R N ) when s = 1 2 and in C 2s−ε (R N ), when s = 1 2 . Therefore, applying Theorem 2.1(b), we have Similarly, Arguing as before, we can show that u ∈ C 4s (R N ) when s = 1 2 and in C 4s−ε (R N ), when s = 1 2 . We can repeat this argument to improve the regularity C ∞ (R N ) if both p and q are integer and C 2ks+2s (R N ), where k is the largest integer satisfying 2ks < p if p ∈ N and 2ks < q if p ∈ N but q ∈ N, where 2ks denotes the greatest integer less than equal to 2ks .
Proof. Case 1: Let u be a weak solution of (1.1).
First, we show that (−∆) s u(x) can be defined as in (1.3). Using u ∈ L ∞ (R N ), we see that On the other hand, since by Theorem 1.4, u ∈ C 2s+α loc (R N ) for some α ∈ (0, 1), it follows that B

This in turn implies
Hence, u is a classical solution of (1.1).
where ρ ε is the standard molifier. Namely, we take Then u ε , f ε ∈ C ∞ . Proceeding along the same line as in the proof of [29,Proposition 5], we can show that, for ε > 0 small enough it holds (3.32) in the classical sense, where U is any arbitrary subset of Ω with U ⊂⊂ Ω. Moreover, it is easy to note that u ε → u and f ε → f (u) locally uniformly and Taking the limit ε → 0 on both the sides of (3.32) and using the regularity estimate of u ε from theorem 1.4, we obtain, Using the arguments used before, it is not difficult to check that LHS of above relation converges to (−∆) s u as ε → 0 and hence the result follows.
Proof of Theorem 1.5. First, we observe that from Theorem 1.4, it follows u is differentiable as p > 1. Let R 0 be as in Theorem From this expression we obtain (3.36) Now, using the definition of v and the fact that u ∈ L ∞ (R N ), we get where C is independent of R (since, R −2s < 1). On the other hand, using (3.34) we have for some constant C > 0, which does not depend on R. Plugging (3.37) and (3.38) into (3.36) we have where C depends only on N, s, p, q, R 0 . Furthermore, we observe that if z ∈ B r (x 0 ) ⊂ A 1 then |Rz| > R > R 0 and thus |u(Rz)| < C |Rz| N −2s . Consequently, from (3.33), it follows that In the last inequality we have used the fact that N − p(N − 2s) < 0 and N − q(N − 2s) < 0, as q, p ≥ 2 * − 1. Hence, where C is independent of R. Combining (3.39) and (3.40) along with (3.35) yields ||(−∆) s (ηv)|| L ∞ (Br(x0)) < C, where C depends only on N, s, p, q, R 0 . Consequently, using [26, Proposition 2.3] (see also [25]), we obtain where C depends only on N, s, p, q, R 0 . As a consequence, Thus, thanks to [26,Corollary 2.4] we have We continue to apply this bootstrap argument and after a finitely many steps we have ||v|| C β+ks (Br 0 (x0)) ≤ C, for some r 0 > 0 and β + ks > 1. This in turn implies ||∇v|| L ∞ (Br 0 (x0)) ≤ C. This further yields to where C depends only on N, s, p, q, R 0 . Therefore, using the definition of v, we obtain From the above expression, it is easy to deduce that |∇u(y)| ≤ C |y| N −2s+1 for R < |y| < 2R.
As R > R 0 was arbitrary we get for some R large.

Pohozaev identity and nonexistence result.
Proof of Theorem 1.6. We prove this theorem by establishing Pohozaev identity in the spirit of Ros-Oton and Serra [27]. For λ > 0, define u λ (x) = u(λx). Multiplying the equation (1.1) by u λ yields, where, w(x) := (−∆) s 2 u(x) and w λ (x) = w(λx). With the change of variable x = √ λy, we have Therefore, Observe that using the decay estimate at infinity of u and ∇u from Theorem 1.3 and Theorem 1.5 , we get R N (u p − u q )(x · ∇u)dx is well defined and that integral can be written as R N x · ∇ u p+1 p+1 − u q+1 q+1 dx. Again using the decay estimate of u from Theorem 1.3, we justify the following integration by parts Thus, using (4.3) we simplify the LHS of above expression as follows: LHS of (4.4) = On the other hand, multiplying (1.1) by u we have, Combining this expression along with (4.5) we obtain the Pohozaev identity Clearly, from the above identity, it follows that (1.1) does not admit any solution when p = 2 * − 1 and q > p. This completes the theorem. Proof. By Proposition 1, u is a classical solution of (1.1). Define f (u) = u p − u q . Then clearly f is locally Lipschitz.
Claim: There exists s 0 , γ, C > 0 such that To see the claim, for some θ 1 , θ 2 ∈ (0, 1). Thus, for 0 < u < v Therefore, the claim holds with C = p and γ = p − 1 and for any positive s 0 . Moreover, from Theorem 1.4, we have Since p ≥ N +2s N −2s , it is easy to check that where γ = p − 1, as found in the above claim. Hence, the theorem follows from [ Proof. It is enough to show that if u ∈Ḣ s (R N ) with u(x) = u(|x|) and u(r 1 ) ≤ u(r 2 ), when r 1 ≥ r 2 , then it holds u(R) ≤ C R N −2s 2 for any R > 0. To see this, we note that by Sobolev inequality we can write, As u ∈Ḣ s (R N ) implies LHS is bounded above, the above inequality yields Proof of Theorem 1.7. We are going to work on the manifold and F (.) on N reduces as Let u n be a minimizing sequence in N such that Thus, {u n } is a bounded sequence inḢ s (R N ) and L q+1 (R N ). Therefore, there exists u ∈Ḣ s (R N ) and L q+1 (R N ) such that u n u inḢ s (R N ) and L q+1 (R N ). Consequently u n → u pointwise almost everywhere.
Using symmetric rearrangement technique, without loss of generality, we can assume that u n is radially symmetric and decreasing (see [24]). We claim that u n → u in L p+1 (R N ). To see the claim, we note that u p+1 n → u p+1 pointwise almost everywhere. Since {u n } is uniformly bounded in L q+1 (R N ), using Vitali's convergence theorem, it is easy to check that K |u n | p+1 dx → K |u| p+1 dx for any compact set K in R N containing the origin. Furthermore, applying Lemma 6.1 it follows, R N \K |u n | p+1 dx is very small and hence we have strong convergence. Moreover, We note that u → ||u|| 2 is weakly lower semicontinuous. Using this fact along with Fatou's lemma, we have This proves F (u) = K. Moreover, using the symmetric rearrangement technique via. Polya-Szego inequality (see [24]), it is easy to check that u is nonnegative, radially symmetric and radially decreasing Applying the Lagrange multiplier rule, we obtain u satisfies − ∆u + u q = λu p , for some λ > 0. This in turn implies Finally, if q > (p − 1) N 2s − 1, then we know that u is a classical solution. Therefore, if there exists x 0 ∈ R N such that u(x 0 ) = 0, that that would imply (−∆) s u(x 0 ) < 0 (since, u is a nontrivial solution). On the other hand, (λu p − u q )(x 0 ) = 0 and that yields a contradiction. Hence u > 0 in R N .
Furthermore, we observe that by setting v( Hence the theorem follows.
Proof of Theorem 1.8. We are going to work on the manifold : Then F Ω reduces to Let u n be a minimizing sequence inÑ such that F Ω (u n ) → S Ω , then Then u n is bounded in X 0 ∩ L q+1 (Ω). Consequently, u n u on H s (Ω) and u n → u on L 2 (Ω). As a result, u n → u pointwise almost everywhere. By the interpolation inequality, we must have u n → u on L p+1 (Ω). Hence, Ω |u| p+1 dx = 1. Now we show that S Ω = F Ω (u). Using Fatou's Lemma and the fact that u → ||u|| 2 is weakly lower semicontinuous , ≥F Ω (u).
Appendix A. . In this section we give an alternative proof of Pohozaev identity in R N for the following type of equations: where u ∈Ḣ s (R N ) ∩ L ∞ (R N ) and f ∈ C 2 . Here we do not require the decay estimate of u or ∇u at infinity.
be a positive solution of (A.1) and F (u) ∈ L 1 (R N ). Then We prove this theorem using the harmonic extension method introduced in Section 2.
Let u be a nontrivial positive solution of (A.1). Suppose, w is the harmonic extension of u. Then w is a solution of    div(y 1−2s ∇w) = 0 in R N +1 + , ∂w ∂ν 2s = f (w(., 0)) on R N , (see (2.4)). For r > 0, we define B r to be the ball in R N +1 , that is, x · ∇ x w ψ R ∂w ∂ν 2s dx (A. 4) where k 2s is as defined in Section 2. In the above steps we have used the fact that ψ R = 0 on ∂B 2R . From (A.2), we know ∂w ∂ν 2s = f (w(x, 0)) on R N . Therefore, RHS of (A.4) simplifies as RHS of (A.4) = k −1 (A.5) Since w(x, 0) = u(x), from (A.2), we find F (w) = F (u) on R N . Moreover, |∇ψ R | ≤ 2 R . Hence, the 2nd integral on RHS of (A.5) can be written as which converges to 0 as R → ∞ (since, F (u) ∈ L 1 (R N )). As a result, Next, we like to simplify LHS of (A.4). Towards this aim, let us first simplify the term ∇w∇ ((x, y) · ∇w)ψ R .