A fractional Laplacian problem with critical nonlinearity

: In this paper, using the mountain pass theorem we obtain a positive solution to the fractional Laplacian problem , where Ω ⊂ R n is a bounded smooth domain, 0 < s < 1 , 2 ≤ q < 2 n / ( n − 2 s ) and k ∈ (0 , ∞ ) is an arbitrary number. The function g : Ω → R is a nonnegative continuous function satisfying some integrability


Introduction
In recent years, motivated by problems that appear in anomalous diffusions in plasmas, flames propagation and chemical reactions in liquids etc, fractional Laplacian problems have received extensive attention in the literature. Also, due to its rich multi-origins, fractional Laplacian operators turn out to have many different definitions. When studying fractional problems in the whole space R n , the fractional Laplacian (−∆) s is usually defined via the Fourier transformation (−∆) s f (x) = (|2πiξ| 2sf (ξ)) ∨ , see e.g. Frank and Lenzmann [12] and Di Nezza, Palatucci and Valdinoci [11]. Equivalently, this operator can be written as the difference quotient (−∆) s f (x) = c n,s P.V. for sufficiently regular f , where c n,s is a normalization constant, for a proof see e.g. [11]. An advantage of the difference quotient type definition is that it can be directly extended to define the so-called fractional p-Laplacian operator (−∆) s p f (x) = c n,s P.V.
R n f (x) − f (y) |x − y| n+sp dy, and the so called regional fractional Laplacian operator in case the problem under consideration is restricted to a bounded domain Ω ⊂ R n , see e.g. Chen [10] and the references therein. It is beyond the size of this paper to present all the other definitions for fractional Laplacian operators in the literature. We refer the interested readers to the references mentioned above.
In this paper, we are concerned with the fractional Laplacian operator (−∆) s on a bounded domain defined via the spectral of the Laplacian operator −∆. This type of fractional Laplacian operator was first defined by Cabré and Tan [7] for s = 1/2 and then extended by Brändle, Colorado, de Pablo and Sánchez [4] to the whole range s ∈ (0, 1). This type of definition can be viewed as a discrete version of Fourier transformation for functions on a bounded domain. More precisely, let s ∈ (0, 1) and Ω ⊂ R n be a bounded smooth domain. Let {λ k , ϕ k } k≥1 be the corresponding eigenvalues and eigenfunctions of −∆ on Ω with Dirichlet boundary value ϕ k | ∂Ω = 0 and normalization ϕ k L 2 (Ω) = 1. Then, for u = k≥1 c k ϕ k satisfying k≥1 c 2 k λ 2s k < ∞, we define Therefore, a natural function space of (−∆) s in the sense of functional analysis is given by Observe that a remarkable feature of the operator (−∆) s is that it is nonlocal, in the sense that for any point x ∈ Ω, the value of (−∆) s u at point x can be obtained only if one knows the distribution of u in the whole domain, so that it is possible derive all of the coefficients c k ≡ Ω uϕ k dx. For a systematical study on this type of fractional Laplacian operators, we refer to e.g. Cabré and Tan [7] for the case s = 1/2 and Brändle et al. [4] for the general case s ∈ (0, 1). Now our problem under consideration can be stated as below. Fix s ∈ (0, 1) and consider the following fractional Laplacian problem in Ω, where 2 * s = 2n/(n − 2s), 2 ≤ q < 2 * s and k ∈ (0, ∞) is an arbitrary positive number. A positive function u ∈ H s 0 (Ω) is called a (weak) solution of problem (1.3) if for every v ∈ H s 0 (Ω), there holds Precise assumptions on the coefficient function g will be given soon. Our aim is to prove that under appropriate conditions on the function g and the parameters q, n, there does exist at least one positive solution to the above problem. Problem (1.3) is not totally new. Problems of type (1.3) have been extensively studied in the literature with different leading operators. In the local case (i.e., s = 1) with k = 0, problem of type (1.3) dates back to the famous work Brézis and Nirenberg [6], where positive solutions to the critical growth problem was obtained: In this case, the function g is the constant function λ. Since then, there are numerous extensions and important variants which have been studied. In [1], Ambrosetti, Brezis and Cerami considered the problem combined with concave and convex nonlinearities where 0 < q < 1 < p and λ > 0. They proved among many other results that when λ is sufficiently small, this problem has a positive solution, see [1, Theorem 2.1] for more results. Quite recently, this problem was extended by Barrios, Colorado, Servadei,and Soria [3] to the nonlocal setting where 0 < s < 1, 0 < q < p = n+2s n−2s and λ > 0, and (−∆) s u is defined via the difference quotient (1.1). Note that in this problem the boundary condition is in fact assumed on the complement of Ω. They obtained similar results [3, Theorem 1.1, 1.2] as that of [1, Theorem 2.1]. In [15] Servadei and Valdinoci considered the even more general integrodifferential problem of Brézis and Nirenberg type where s ∈ (0, 1) and L K is a nonlocal operator with kernel function K such that (−∆) s plays a model case. Under various natural assumptions they proved several existence results to this problem, see [15,. In particular, as an application, they obtained a positive solution in the model case [15,Theorem 4]. Similar to [3], but with the leading operator (−∆) s defined in the spectral way (1.2), Barrios, Colorado, de Pablo and Sánchez [2] considered the problem  [4].
In the case g constant, Gazzola [13] studied this type of problem with p-Laplacian operator −∆ p as the leading term, and positive solutions were obtained among other results, see [13,Theorem 2]. Inspired by the work of Gazzola [13] and Cabré and Tan [7], Wang studied problem (1.3) with s = 1/2 and obtained a positive solution to her problem. We mention that Li and Xiang [14] extended Wang's work [17] to the setting of regional fractional Laplacion problems. Inspired by Wang's work [17], in this paper we study problem (1.3) with s belonging to the full range (0, 1). Throughout we assume that 2 ≤ q < 2 * s and g satisfies (G1) g ∈ C(Ω) is a nonnegative nontrivial function, and (G2) g ∈ L 2n/(2n−(n−2s)q) (Ω). In addition, if q = 2, then g L n/2s (Ω) < S , where S is defined as in (1.6) in the below.
Our main result reads as follows.
It is clear that the nonlocality of (−∆) s makes it difficult to deal with problem (1.3) directly. Our strategy is to use a localization method. In the global case Ω = R n , a localization principle for fractional Laplacian problem was first systematically developed by Caffarelli and Silvestre [9]. In our local case, a localization principle was also developed by Cabré and Tan [7] for the case s = 1/2 and by Brändle et al. [4] for the general case s ∈ (0, 1). To state the localization principle, we need to introduce some notations. Denote with coordinates (x, y) ∈ C for x ∈ Ω and y > 0. According to Brändle et al. [4], for every u ∈ H s 0 (Ω), there exists a unique 2s-harmonic extensionū which is equal to u on Ω × {y = 0} in the sense of trace withū = 0 on the lateral boundary ∂ L C, and satisfies the equation div(y 1−2s ∇ū) = 0 in C.
Following Brändle et al. [4], introduce the function space is a normalization constant such that the 2s-harmonic extensionū of u satisfies ū X 2s 0 (C) = u H s 0 (Ω) . Then, problem (1.3) is equivalent to That is, u ∈ H s 0 (Ω) is a solution to problem (1.3) if and only ifū ∈ X 2s 0 (C) is a solution to problem (1.4) which means that holds for all Φ ∈ X 2s 0 (C). This explains the localization method. So, to prove Theorem 1.1, we turn to the equivalent problem (1.4). Note that problem (1.4) is variational with the energy functional J : X 2s 0 (C) → R being given by forū ∈ X 2s 0 (C). To find a critical point of J, we will use the mountain pass theorem (see e.g. Struwe [16,Chapter 6]).
Before ending the introduction, we record the following inequality (see [4, Theorem 2.1]) for later use: there exists C = C(n, s) > 0 such that, for every w ∈ X 2s (R n+1 + ), the closure of C ∞ 0 (R n+1 + ) under the seminorm u 2 X 2s (R n+1 be the best constant for inequality (1.5). It is known [4, Theorem 2.1] that S is attained by the functions U ,x 0 (x, y) = (P 1−2s for all > 0 and x 0 ∈ R n , where P 1−2s y (x) = k 1−2s y −n (1 + |x|/y) −(n+2s)/2 is the so-called s-Poisson kernel and Our notations are standard. We use C to denote positive constants that are different from line to line. For simplicity, we will use · to denote the norm · X 2s 0 (C) throughout.

Proof of the main result
This section is devoted to the proof of Theorem 1.1. Our method is to use the well known mountain pass theorem, see e.g. Struwe [16,Theorem 6.1]. The first lemma points out that Lemma 2.1. J satisfies the mountain pass geometry.
The proof is standard, and we omit the details. We shall also need Lemma 2.2. Under the assumptions (G 1 ) and (G 2 ), there hold for every (u m ) ⊂ X 2s 0 (C) that converges weakly to u ∈ X 2s 0 (C). Lemma 2.2 is a consequence of Vitali's convergence theorem. We omit the details, see also Wang [17, Lemma 2.1] and Gazzola [13,Lemma 1].
We say that (u m ) ∈ X 2s 0 (C) is a (PS ) c sequence, if J(u m ) → c ∈ R and J (u m ) → 0 as m → ∞. Say that the functional J satisfies (PS ) c compactness condition if (u m ) ∈ X 2s 0 (C) is a (PS ) c sequence, then it contains a convergent subsequence in X 2s 0 (C). Next lemma says that this happens when c is below a critical value, as that was observed by Brezis and Nirenberg [6]. Proof. Let (u m ) ⊂ X 2s 0 (C) be a (PS ) c sequence. First we claim that (u m ) is a bounded sequence in X 2s 0 (C). This will follow from a simple combination of the (PS ) c assumption. Let β ≥ 2. Then Taking β = q when q > 2 yields Then the assumption g ≥ 0 implies that When q = 2, take 2 * s > β > 2. Then using the assumption (G 2 ) and Hölder's inequality gives Now the boundedness of (u m ) ⊂ X 2s 0 (C) follows from (2.1) and (2.2). Next we prove that (u m ) has a convergent subsequence in X 2s 0 (C). Since we have proved the boundedness of (u m ) in X 2s 0 (C), we may assume without loss of generality that, up to a subsequence, u m u in X 2s 0 (C), u m (·, 0) → u(·, 0) in L 2 (Ω), u m (·, 0) → u(·, 0) a.e. in Ω, for some u ∈ X 2s 0 (C). Put v m = u m − u. It suffices to prove that v m → 0. It is standard to find that J (u) = 0. To examine the sequence closer, note that Using Lemma 2.2 we find that By Lemma 2 of Brézis and Lieb [5], As a result, we obtain the first decomposition Similarly, we also have the second decomposition where we used used the assumption J (u m ) → 0 in the first equality and the fact J (u) = 0 in the last line.
To continue, note that v m ≤ C( u m + u ) is a bounded sequence. So we may assume v m 2 → b ≥ 0. Then the above second decomposition gives Recall the trace inequality (1.5). This implies S b As a consequence, we infer that either b = 0 or b ≥ S n 2s holds. We have to exclude the case b ≥ S n 2s . To this end, note that J (u) = 0 implies J(u) ≥ 0. Thus the first decomposition leads to If b ≥ S n 2s , then taking limit in this inequality gives c ≥ s n S n 2s , which contradicts with our assumption c < s n S n 2s . Hence b = 0. That is v m → 0 in X 2s 0 (C). The proof is finished. To proceed, we may assume that 0 ∈ Ω and g(0) > 0 with no loss of generality. By the assumption (G1) we can assume B r (0) ⊂ {x ∈ Ω : g(x) > g(0)} for some r > 0 sufficiently small. For simplicity, write u = u ,0 , U = U ,0 , where u ,x 0 , U ,x 0 are defined as in (1.7). Let 0 < ρ < r and take a cut-off function η ∈ C ∞ (C) such that 0 ≤ η ≤ 1 and η(x, y) ≡ 1 for |x| < ρ 2 and y ≥ 0, and η(x, y) ≡ 0 for |x| ≥ ρ. These auxiliary functions and parameters are used to construct a special path in X 2s 0 (C) starting from the origin such that the following lemma holds. Proof. Write v = ηU and γ(t) = tv for t ≥ 0. Suffices to show that max t>0 J(γ(t)) < s n S n 2s for sufficiently small. By a direct computation we have Since the second term on the right hand side is nonnegative, there holds Also note that v is a bounded function. So for t sufficiently small, we have (tv − k) + ≡ 0. Hence for t sufficiently small, for some C > 0 independent of , where we have used the estimate (2.3) for t . In the case q = 2, n = 4s, note that on the set { < |x| < 3/4 }, there holds for sufficiently small. Hence, |x| 4s = C 2s | ln | for some C > 0 independent of . Thus, there exists C > 0 such that C n− n−2s 2 q , if q ≥ 2 and n > 2s(1 + 2/q) C 2s ln if q = 2, n = 4s. In the case q ≥ 2 and n > 2s(1 + 2/q), it holds n−2s 2 q − 2s > 0. Therefore, in both cases we can deduce that J(γt ) < s n S n 2s , provided is sufficiently small. The proof is complete. By Lemma 2.4, we have c 0 < s n S n/2s . Since J satisfies the geometry of mountain pass, there exists a sequence (u m ) ⊂ X 2s 0 (C) satisfying J(u m ) → c 0 and J (u m ) → 0 as m → ∞. Therefore, Lemma 2.3 implies that problem (1.4) admits a nonnegative nontrivial solution. Finally, a maximum principle of Cabré and Sire [8,Remark 4.2] implies that the solution is positive in Ω. The proof is complete.