FRACTIONAL ELLIPTIC EQUATIONS WITH SIGN-CHANGING AND SINGULAR NONLINEARITY

Transcription

1 Electronic Journal of Differential Equations, Vol (2016), No. 145, pp IN: URL: or FRACTIONAL ELLIPTIC EQUATION WITH IGN-CHANGING AND INGULAR NONLINEARITY ARIKA GOYAL, KONIJETI REENADH Abstract. In this article, we study the fractional Laplacian equation with singular nonlinearity ( ) s u = a(x)u q + λb(x)u p in, u > 0 in, u = 0 in, where is a bounded domain in R n with smooth boundary, n > 2s, s (0, 1), λ > 0. Using variational methods, we show existence and multiplicity of positive solutions. 1. Introduction Let R n be a bounded domain with smooth boundary, n > 2s and s (0, 1). We consider the fractional elliptic problem with singular nonlinearity ( ) s u = a(x)u q + λb(x)u p in, u > 0 in, u = 0 in. We use the following assumptions on a and b: (1.1) (A1) a : R n R such that 0 < a L 2 2 s (). (A2) b : R n R is a sign-changing function such that b + 0 and b(x) L 2 s 2 s 1 p (). Here λ > 0 is a parameter, 0 < q < 1 < p < 2 s 1, with 2 s = 2n n 2s, nown as fractional critical obolev exponent and where ( ) s is the fractional Laplacian operator in with zero Dirichlet boundary values on. To define the fractional Laplacian operator ( ) s in, let {λ, φ } be the eigenvalues and the corresponding eigenfunctions of in with zero Dirichlet boundary values on ( ) s φ = λ φ in, s φ = 0 on. normalized by φ L2 () = 1. Then one can define the fractional Laplacian ( ) s for s (0, 1) by ( ) s u = λ s c φ, 2010 Mathematics ubject Classification. 35A15, 35J75, 35R11. Key words and phrases. Non-local operator; singular nonlinearity; Nehari manifold; sign-changing weight function. c 2016 Texas tate University. ubmitted January, 21, Published June 14, =1 1

2 2. GOYAL, K. REENADH EJDE-2016/145 which clearly maps H s 0() := { u = ( ) 1/2 c φ L 2 () : u H s 0 () = λ s c 2 } < =1 into L 2 (). Moreover H0() s is a Hilbert space endowed with an inner product c φ, d φ H0 s() = λ s c d φ, if c φ, d φ H0(). s =1 =1 =1 Definition 1.1. A function u H0() s such that u(x) > 0 in is a solution of (1.1) such that for every function v H0(), s it holds ( ) s/2 u( ) s/2 vdx = a(x)u q vdx + λ b(x)u p vdx. Associated with (1.1), we consider the energy functional for u H0(), s u > 0 in such that I λ (u) = ( ) s/2 u 2 dx 1 a(x) u 1 q dx λ b(x) u p+1 dx. 1 q p + 1 The fractional power of Laplacian is the infinitesimal generator of Lévy stable diffusion process and arise in anomalous diffusions in plasma, population dynamics, geophysical fluid dynamics, flames propagation, chemical reactions in liquids and American options in finance. For more details, we refer to 3, 14 and reference therein. Recently the study of existence, multiplicity of solutions for fractional elliptic equations attracted a lot of interest by many researchers. Among the wors dealing with fractional elliptic equations we cite 6, 9, 21, 22, 23, 24, 25, 26 and references therein, with no attempt to provide a complete list. Caffarelli and ilvestre 8 gave a new formulation of fractional Laplacian through Dirichlet-Neumann maps. This formulation transforms problems involving the fractional Laplacian into a local problem which allows one to use the variational methods. On the other hand, there are some wors where multiplicity results are shown using the structure of associated Nehari manifold. In 15, 16 authors studied subcritical problems and in 28 the authors obtained the existence of multiplicity for critical growth nonlinearity. In the case of the square root of Laplacian, the multiplicity results for sublinear and superlinear type of nonlinearity with signchanging weight functions are studied in 7, 27. In the local setting, s = 1, the paper by Crandall, Robinowitz and Tartar 12 is the starting point on semilinear problem with singular nonlinearity. There is a large body of literature on singular problems, see 1, 2, 11, 12, 17, 18, 19, 20 and reference therein. Recently, Chen and Chen in 10 studied the following problem with singular nonlinearity u λ x 2 = h(x)u q + µw (x)u p in \ {0}, u > 0 in \ {0}, u = 0 on, where 0 R n (n 3) is a bounded smooth domain with smooth boundary, 0 < λ < (n 2)2 4 and 0 < q < 1 < p < n+2 n 2. =1 =1 =1 Also h, W both are continuous functions in with h > 0 and W is sign-changing. By variational methods, they showed that there exists T λ such that for µ (0, T λ ) the above problem has two positive solutions.

3 EJDE-2016/145 FRACTIONAL LAPLACIAN WITH INGULAR NONLINEARITY 3 In case of the fractional Laplacian, Fang 13 proved the existence of a solution of the singular problem ( ) s w = w p, u > 0 in, u = 0 in R n \, with 0 < p < 1, using the method of sub and super solution. Recently, Barrios, Peral and et al 4 extend the result of 13. They studied the existence result for the singular problem ( ) s u = λ f(x) u γ + Mu p, u > 0 in, u = 0 in R n \, where is a bounded smooth domain of R n, n > 2s, 0 < s < 1, γ > 0, λ > 0, p > 1 and f L m (), m 1 is a nonnegative function. For M = 0, they proved the existence of solution for every γ > 0 and λ > 0. For M = 1 and f 1, they showed that there exist Λ such that it has a solution for every 0 < λ < Λ, and have no solution for λ > Λ. Here the authors first studied the uniform estimates of solutions {u n } of the regularized problems ( ) s u = λ f(x) (u up, u > 0 in, u = 0 in R n \. (1.2) n )γ Then they obtained the solutions by taing limit in the regularized problem (1.2). As far as we now, there is no wor related to fractional Laplacian for singular nonlinearity and sign-changing weight functions. o, in this paper, we study the multiplicity results for problem (1.1) for 0 < q < 1 < p < 2 s 1 and λ > 0. This wor is motivated by the wor of Chen and Chen in 10. Due to the singularity of problem, it is not easy to deal the problem (1.1) as the associated functional is not differentiable even in sense of Gâteaux and the strong maximum principle is not applicable to show the positivity of solutions. Moreover one can not directly extend all the results from Laplacian case to fractional Laplacian, due to the nonlocal behavior of the operator and the bounded support of the test function is not preserved. To overcome these difficulties, we first use the Cafferelli and ilvestre 9 approach to convert the problem (1.1) into the local problem. Then we use the variational technique to study the local problem as in 10. In this paper, the proofs of some Lemmas follow the similar lines as in 10 but for completeness, we give the details. The article is organized as follows: In section 2 we present some preliminaries on extension problem and necessary weighted trace inequalities required for variational settings. We also state our main results. In section 3, we study the decomposition of Nehari manifold and local charts using the fibering maps. In ection 4, we show the existence of a nontrivial solutions and show how these solutions arise out of nature of Nehari manifold. We will use the following notation throughout this paper: The same symbol denotes the norms in the three spaces L 2 2 s 1 q (), L 2 s 2 s 1 p (), and H0,L s (C ) defined by (2.1). Also := s (s, n) where (s, n) is the best constant of obolev embedding (see (2.2)). 2. Preliminaries and main results In this section we give some definitions and functional settings. At the end of this section, we state our main results. To state our main result, we introduce some s

4 4. GOYAL, K. REENADH EJDE-2016/145 notation and basic preliminaries results. Denote the upper half-space in R n+1 by R n+1 + := {z = (x, y) = (x 1, x 2,..., x n, y) R n+1 y > 0}, the half cylinder standing on a bounded smooth domain R n by C := (0, ) R n+1 + and its lateral boundary is denoted by L C = 0, ). Define the function space H0,L s (C ) as the completion of C0,L (C ) = {w C (C ) : w = 0 on L C } under the norm ) 1/2, w H s 0,L (C ) = ( s y 1 2s w(x, y) 2 dx dy (2.1) C where s := 21 2s Γ(1 s) Γ(s) is a normalization constant. Then it is a Hilbert space endowed with the inner product w, v H s 0,L (C ) = s y 1 2s w v dx dy. If is a smooth bounded domain then it is verified that (see 8, Proposition 2.1, 6, Proposition 2.1, 26, section 2) H s 0() := {u = tr w : w H s 0,L(C )} and there exists a constant C > 0 such that w(, 0) H s 0 () C w H s 0,L (C ) for all w H s 0,L(C ). Now we define the extension operator and fractional Laplacian for functions in H s 0(). Definition 2.1. Given a function u H s 0(), we define its s-harmonic extension w = E s (u) to the cylinder C as a solution of the problem div(y 1 2s w) = 0 in C, w = 0 on L C, w = u on {0}. Definition 2.2. For any regular function u H s 0(), the fractional Laplacian ( ) s acting on u is defined by ( ) s u(x) = s w lim y1 2s y 0 + y (x, y) for all (x, y) C. From 5 and 9, the map E s ( ) is an isometry between H s 0() and H s 0,L (C ). Furthermore, we have (1) ( ) s u H s () = u H s 0 () = E s (u) H s 0,L (C ), where H s () denotes the dual space of H s 0(); (2) For any w H s 0,L (C ), there exists a constant C independent of w such that tr w L r () C w H s 0,L (C ) 2n holds for every r 2, n 2s. Moreover, Hs 0,L (C ) is compactly embedded into L r 2n () for r 2, n 2s ).

5 EJDE-2016/145 FRACTIONAL LAPLACIAN WITH INGULAR NONLINEARITY 5 2n Lemma 2.3. For every 1 r n 2s and every w Hs 0,L (C ), it holds ( ) 2/r w(x, 0) r dx Cs y 1 2s w(x, y) 2 dx dy, C where the constant C depends on r, s, n and. Lemma 2.4. For every w H s (R n+1 + ), it holds ( ) n 2s 2n n (s, n) u(x) n 2s dx y 1 2s w(x, y) 2 dx dy, (2.2) R n where u = tr w. The constant (s, n) is nown as the best constant and taes the value (s, n) = 2πs Γ( 2 2s 2 )Γ( n+2s 2 )(Γ( n 2s 2 )) n. Γ(s)Γ( n 2s 2 )(Γ(n)) 2s n R n+1 + Now we can transform the nonlocal problem (1.1) into the local problem div(y 1 2s w) = 0 in C := (0, ), w = 0 on L C, w > 0 on {0}, w v 2s = a(x)w q + λb(x)w p on {0}, where w 1 2s w v := 2s s lim y 0 + y y (x, y), for all x. (2.3) Definition 2.5. A wea solution of (2.3) is a function w H s 0,L (C ), w > 0 in {0} such that for every v H s 0,L (C ), s y C 1 2s w v dx dy = a(x)(w q v)(x, 0)dx + λ b(x)(w p v)(x, 0)dx. If w satisfies (2.3), then u = tr w = w(x, 0) H s 0() is a wea solution to problem (1.1). Let c = 1 2s, then the associated functional J λ : H0,L s (C ) R to the problem (2.3) is J λ (w) = s y c w 2 dx dy 1 a(x) w 1 q dx 2 C 1 q λ b(x) w(x, 0) p+1 dx. p + 1 Now for w H0,L s (C ), we define the fiber map φ w : R + R as φ w (t) = J λ (tw) = t2 2 w 2 1 a(x) tw 1 q dx 1 q λtp+1 b(x) w(x, 0) p+1 dx. p + 1 Then φ w(t) = t w 2 t q a(x) w(x, 0) 1 q dx λt p b(x) w(x, 0) p+1 dx,

6 6. GOYAL, K. REENADH EJDE-2016/145 φ w(t) = w 2 + qt q 1 pλt p 1 a(x) w(x, 0) 1 q dx b(x) w(x, 0) p+1 dx. It is easy to see that the energy functional J λ is not bounded below on the space H s 0,L (C ). But we will show that it is bounded below on an appropriate subset of H s 0,L (C ) and a minimizer on subsets of this set gives rise to solutions of (2.3). In order to obtain the existence results, we define N λ : = {w H0,L(C s ) : J λ(w), w = 0} { = w H0,L(C s ) : w 2 = a(x) w(x, 0) 1 q dx } + λ b(x) w(x, 0) p+1 dx. Note that w N λ if w is a solution of (2.3). Also one can easily see that tw N λ if and only if φ w(t) = 0 and in particular, w N λ if and only if φ w(1) = 0. In order to obtain our result, we decompose N λ with N ± λ, N λ 0 as follows: N ± λ = {w N λ : φ w(1) 0} = { w N λ : (1 + q) w 2 λ(p + q) Nλ 0 = {w N λ : φ w(1) = 0} = { w N λ : (1 + q) w 2 = λ(p + q) b(x) w(x, 0) p+1 dx }, b(x) w(x, 0) p+1 dx }. Inspired by 9 and 10, we show that how variational methods can be used to established some existence and multiplicity results. Our results are as follows: Theorem 2.6. uppose that λ (0, Λ), where Λ := 1 + q ( p 1 ) p 1 1 ( p+q ) 1/(). p + q p + q b a p 1 Then problem (2.3) has at least two solutions w 0 N + λ, W 0 N λ with W 0 > w 0. Moreover, u 0 (x) = w 0 (, 0) H0() s and U 0 (x) = W 0 (, 0) H0() s are positive solutions of the problem (1.1). Next, we obtain the blow up behavior of the solution W ɛ N λ of problem (2.3) with p = 1 + ɛ as ɛ 0 +. Theorem 2.7. let W ɛ N λ λ (0, Λ), then where ( C ɛ = q ɛ be the solution of problem (2.3) with p = 1 + ɛ, where (Λ) 1/ɛ, W ɛ > C ɛ λ ) 1/() a /(q+1) 1 ) 1 q as ɛ 0 +. Namely, W ɛ blows up faster than exponentially with respect to ɛ.

7 EJDE-2016/145 FRACTIONAL LAPLACIAN WITH INGULAR NONLINEARITY 7 We remar that if w is a positive solution of the problem ( ) s u = a(x)u q + λb(x)u p u > 0 in, u = 0 on. in Then one can easily see that v = λ 1/(p 1) u is a positive solution of the problem ( ) s v = λ p 1 a(x)v q + b(x)v p v > 0 in, v = 0 in. in That is, the problem (2.4) has two positive solution for λ (0, Λ p 1 ). In this section, we show that N ± λ bounded below and coercive. 3. Fibering map analysis (2.4) is nonempty and N 0 λ = {0}. Moreover, J λ is Lemma 3.1. Let λ (0, Λ). Then for each w H0,L s (C ) with a(x) w(x, 0) 1 q dx > 0, we have the following: (i) b(x) w(x, 0) p+1 dx 0, then there exists a unique t 1 < t max such that t 1 w N + λ and J λ(t 1 w) = inf t>0 J λ (tw); (ii) b(x) w(x, 0) p+1 dx > 0, then there exists a unique t 1 and t 2 with 0 < t 1 < t max < t 2 such that t 1 w N + λ, t 2w N λ and J λ(t 1 w) = inf 0 t tmax J λ (tw), J λ (t 2 w) = sup t t1 J λ (tw). Proof. For t > 0, we define ψ w (t) = t 1 p w 2 t p q One can easily see that ψ w (t) as t 0 +. Now ψ w(t) = (1 p)t p w 2 + (p + q)t p q 1 ψ w(t) = p(1 p)t p 1 w 2 (p + q)(p + q + 1)t p q 2 Then ψ w(t) = 0 if and only if t = t max := a(x) w(x, 0) 1 q dx λ b(x) w(x, 0) p+1 dx. Also ψ w(t (p 1) w 2 max ) = p(p 1) (p + q) a(x) w(x, 0) 1 q dx (p + q)(p + q + 1) a(x) w(x, 0) 1 q dx = w 2 (p 1)(1 + q) a(x) w(x, 0) 1 q dx. a(x) w(x, 0) 1 q dx. (p 1) w 2 (p+q) R a(x) w(x,0) 1 q dx 1/(). p+1 q+1 w 2 (p 1) w 2 (p + q) a(x) w(x, 0) 1 q dx (p 1) w 2 (p + q) a(x) w(x, 0) 1 q dx p+q+2 q+1 p+1 q+1 < 0.

8 8. GOYAL, K. REENADH EJDE-2016/145 Thus ψ w achieves its maximum at t = t max. Now using the Hölder s inequality and obolev inequality (2.2), we obtain a(x) w(x, 0) 1 q dx a(x) 2 2 s +q 1 s 2 s 2 s ( w ) 1 q. a b(x) w(x, 0) p+1 dx b(x) ( w ) p+1. b Using (3.1) and (3.2), we obtain ψ w (t max ) 2 2 s p 1 s 2 s 1 p 2 s = 1 + q ( p 1 ) p 1 w 2(p+q) p + q p + q a(x) w(x, 0) 1 q dx p q ( p 1 ) p 1 p + q p + q E λ w p+1, where E λ := ( ( ) (1 q) a 1 + q ( p 1 ) p 1 p + q p + q 1 q w(x, 0) 2 2 s dx s p+1 w(x, 0) 2 2 s dx s ) p 1 λ b ) p+1 w p+1 ( ( ) (1 q) a (3.1) (3.2) λ b(x) w(x, 0) p+1 dx ) p 1 ( 1 ) p+1 λ b. Then we see that E λ = 0 if and only if λ = Λ, where Λ := 1 + q ( p 1 ) p 1 1 ( p+q ) 1/(). p + q p + q b a p 1 (3.3) Thus for λ (0, Λ), we have E λ > 0, and therefore it follows from (3.3) that ψ w (t max ) > 0. (i) If b(x) w(x, 0) p+1 dx 0, then ψ w (t) λ b(x) w(x, 0) p+1 dx < 0 as t. Consequently, ψ w (t) has exactly two points 0 < t 1 < t max < t 2 such that ψ w (t 1 ) = 0 = ψ w (t 2 ) and ψ w(t 1 ) > 0 > ψ w(t 2 ). From the definition of N ± λ, we get t 1w N + λ and t 2w N λ. Now φ w(t) = t p ψ w (t). Thus φ w(t) < 0 in (0, t 1 ), φ w(t) > 0 in (t 1, t 2 ) and φ w(t) < 0 in (t 2, ). Hence J λ (t 1 w) = inf 0 t tmax J λ (tw), J λ (t 2 w) = sup t t1 J λ (tw).

9 EJDE-2016/145 FRACTIONAL LAPLACIAN WITH INGULAR NONLINEARITY 9 (ii) If b(x) w(x, 0) p+1 dx < 0 and ψ w (t) λ b(x) w(x, 0) p+1 dx > 0 as t. Consequently, ψ w (t) has exactly one point 0 < t 1 < t max such that ψ w (t 1 ) = 0 and ψ w(t 1 ) > 0. Now φ w(t) = t p ψ w (t). Then φ w(t) < 0 in (0, t 1 ), φ w(t) > 0 in (t 1, ). Thus J λ (t 1 w) = inf t 0 J λ (tw). Hence, it follows that t 1 w N + λ. Corollary 3.2. uppose that λ (0, Λ), then N ± λ. Proof. From (A1) and (A2), we can choose w H0,L s (C ) \ {0} such that a(x) w(x, 0) 1 q dx > 0 and b(x) w(x, 0) p+1 dx > 0. Then by (ii) of Lemma 3.1, there exists a unique t 1 and t 2 such that t 1 w N + λ, t 2 w N λ. In conclusion, N ± λ. Lemma 3.3. For λ (0, Λ), we have N 0 λ = {0}. Proof. We prove this by contradiction. Assume that there exists 0 w Nλ 0. Then it follows from w Nλ 0 that (1 + q) w 2 = λ(p + q) b(x) w(x, 0) p+1 dx and consequently 0 = w 2 = p 1 p + q w 2 a(x) w(x, 0) 1 q dx λ b(x) w(x, 0) p+1 dx a(x) w(x, 0) 1 q dx. Therefore, as λ (0, Λ) and w 0, we use similar arguments as those in (3.3) to obain 0 < E λ w p q ( p 1 ) p 1 w 2(p+q) p + q p + q a(x) w(x, 0) 1 q dx p 1 λ b(x) w(x, 0) p+1 dx = 1 + q p + q ( p 1 p + q ) p 1 w 2(p+q) ( p 1 p+q w 2) p 1 a contradiction. Hence w 0. That is, N 0 λ = {0}. We note that Λ is also related to a gap structure in N λ : 1 + q p + q w 2 = 0,

10 10. GOYAL, K. REENADH EJDE-2016/145 Lemma 3.4. uppose that λ (0, Λ), then there exist a gap structure in N λ : W > A λ > A 0 > w for all w N + λ, W N λ, where 1 + q A λ = λ(p + q) b ( ) p+1 1/(p 1) Proof. If w N + λ N λ, then 0 < (1 + q) w 2 λ(p + q) and A 0 = = (1 + q) w 2 (p + q) w 2 p + q p 1 a ) 1 q 1/(q+1). b(x) w(x, 0) p+1 dx a(x) w(x, 0) 1 q dx = (1 p) w 2 + (p + q) a(x) w(x, 0) 1 q dx. Hence it follows from (3.1) that (p 1) w 2 < (p + q) which yields w < ( w ) 1 q a(x) w(x, 0) 1 q dx (p + q) a p + q p 1 a ) 1 q 1/(q+1) A0. If W N λ, then it follows from (3.2) that ( W ) p+1 (1 + q) W 2 < λ(p + q) b(x) W (x, 0) p+1 dx λ(p + q) b which yields 1 + q W > λ(p + q) b ( ) p+1 1/(p 1) Aλ. Now we show that A λ = A 0 if and only if λ = Λ. λ = Λ = 1 + q ( p 1 ) p 1 1 ( p+q ) 1/() p + q p + q b a p 1 if and only if A λ = λ 1/(p 1) + q ) 1/(p 1) ) 1/(p 1) ( ) p+1 p 1 p + q b ( p + q ) 1/() a ( ) 2(p+q) 1/(q+1) ()(p 1) = + p+1 p 1 p 1 (p + q) a 1/(q+1) = (p 1)( A0. ) 1 q Thus for all λ (0, Λ), we can conclude that This completes the proof. W > A λ > A 0 > w for all w N + λ, W N λ. Lemma 3.5. uppose that λ (0, Λ), then N λ is a closed set in Hs 0,L (C )-topology.

11 EJDE-2016/145 FRACTIONAL LAPLACIAN WITH INGULAR NONLINEARITY 11 Proof. Let {W } be a sequence in N λ and W 0 2 = lim W 2 = lim = with W W 0 in H s 0,L (C ). Then we have a(x) W (x, 0) 1 q dx + λ a(x) W 0 (x, 0) 1 q dx + λ (1 + q) W 0 2 λ(p + q) = lim (1 + q) W 2 λ(p + q) That is, W 0 N λ b(x) W 0 (x, 0) p+1 dx b(x) W (x, 0) p+1 dx b(x) W 0 (x, 0) p+1 dx b(x) W (x, 0) p+1 dx 0. N λ 0. ince {W } N λ, from Lemma 3.4 we have W 0 = lim W A 0 > 0, which imply, W 0 0. It follows from Lemma 3.1, that W 0 Nλ 0 for any λ (0, Λ). Thus W 0 N λ. Hence, N λ is a closed set in Hs 0,L (C )-topology for any λ (0, Λ). Lemma 3.6. Let w N ± λ. Then for any φ C 0,L (C ), there exists a number ɛ > 0 and a continuous function f : B ɛ (0) := {v H s 0,L (C ) : v < ɛ} R + such that f(v) > 0, f(0) = 1, f(v)(w + vφ) N ± λ for all v B ɛ (0). Proof. We give the proof only for the case w N + λ, the case N λ may be preceded exactly. For any φ C0,L (C ), we define F : H0,L s (C ) R + R as follows: F (v, r) = r w + vφ 2 a(x) (w + vφ)(x, 0) 1 q λr p+q b(x) (w + vφ)(x, 0) p+1. ince w N + λ ( N λ), we have F (0, 1) = w 2 a(x) w(x, 0) 1 q dx λ b(x) w(x, 0) p+1 dx = 0, and F r (0, 1) = (1 + q) w 2 λ(p + q) b(x) w(x, 0) p+1 dx > 0. Applying the implicit function theorem at (0, 1), we have that there exists ɛ > 0 such that for v < ɛ, v H0,L s (C ), the equation F (v, r) = 0 has a unique continuous solution r = f(v) > 0. It follows from F (0, 1) = 0 that f(0) = 1 and from F (v, f(v)) = 0 for v < ɛ, v H0,L s (C ) that 0 = f (v) w + vφ 2 a(x) (w + vφ)(x, 0) 1 q λf p+q (v) b(x) (w + vφ)(x, 0) p+1

12 12. GOYAL, K. REENADH EJDE-2016/145 that is, = ( f(v)(w + vφ) 2 a(x) f(v)(w + vφ)(x, 0) 1 q λ b(x) f(v)(w + vφ)(x, 0) p+1) /f 1 q (v); f(v)(w + vφ) N λ ince F r (0, 1) > 0 and F (v, f(v)) r = (1 + q)f q (v) w + vφ 2 λ(p + q)f p+q 1 (v) for all v H s 0,L(C ), v < ɛ. b(x) (w + vφ)(x, 0) p+1 dx = (1 + q) f(v)(w + vφ) 2 λ(p + q) b(x) f(v)(w + vφ)(x, 0) p+1 dx f 2 q, (v) we can tae ɛ > 0 possibly smaller (ɛ < ɛ) such that for any v H0,L s (C ), v < ɛ, (1 + q) f(v)(w + vφ) 2 λ(p + q) b(x) f(v)(w + vφ)(x, 0) p+1 dx > 0; that is, This completes the proof. f(v)(w + vφ) N + λ for all v B ɛ (0). Lemma 3.7. J λ is bounded below and coercive on N λ. Proof. For w N λ, from (3.1), we obtain J λ (w) = 2 1 ) w 2 p q 1 ) a(x) w(x, 0) 1 q dx p ) w 2 p q 1 ) ( w ) (3.4) 1 q. a p + 1 Now consider the function ρ : R + R as ρ(t) = αt 2 βt 1 q, where α, β are both positive constants. One can easily show that ρ is convex(ρ (t) > 0 for all t > 0) with ρ(t) 0 as t 0 and ρ(t) as t. ρ achieves its minimum at t min = β(1 q) 2α 1/() and ρ(t min ) = α β(1 q) 2α 2 Applying ρ(t) with α = 2 1 p+1 w N λ, we obtain from (3.4) that β β(1 q) 2α ), β = 1 q 1 p+1 1 q = 1 + q 2 β 2 lim J λ(w) lim ρ(t) =. w t Thus J λ is coercive on N λ. Moreover, it follows from (3.4) that i.e., J λ (w) 1 + q ( 2 β 2 1 q ) 1 q 2α q 2α ) 1 q. ) a ) 1 q and t = w, J λ (w) ρ(t) ρ(t min ) (a constant), (3.5) 1 + q = (1 q)(p + 1) ( (p + q) a ) 2 ( 2( 1 ) 1 q p 1 ) 1 q.

13 EJDE-2016/145 FRACTIONAL LAPLACIAN WITH INGULAR NONLINEARITY 13 Thus J λ is bounded below on N λ. 4. Existence of solutions in N ± λ Now from Lemma 3.5, N + λ N λ 0 and N λ are two closed sets in Hs 0,L (C ) provided λ (0, Λ). Consequently, the Eeland variational principle can be applied to the problem of finding the infimum of J λ on both N + λ N λ 0 and N λ. First, consider {w } N + λ N λ 0 with the following properties: J λ (w ) < inf w N + λ N 0 λ J λ (w) + 1 (4.1) J λ (w) J λ (w ) 1 w w, w N + λ N 0 λ. (4.2) From J λ ( w ) = J λ (w), we may assume that w 0 on C. Lemma 4.1. how that the sequence {w } is bounded in N λ. exists 0 w 0 H0,L s (C ) such that w w 0 wealy in H0,L s (C ). Proof. By equations (3.5) and (4.1), we have Moreover, there at 2 bt 1 q = ρ(t) J λ (w) < inf J λ (w) + 1 w N + λ N λ 0 C 5, for sufficiently large and a suitable positive constant. Hence putting t = w in the above equation, we obtain {w } is bounded. Let {w } is bounded in H0,L s (C ). Then, there exists a subsequence of {w }, still denoted by {w } and w 0 H0,L s (C ) such that w w 0 wealy in H0,L s (C ), w (, 0) w 0 (, 0) strongly in L p () for 1 p < 2 s and w (, 0) w 0 (, 0) a.e. in. For any w N + λ, we have from 0 < q < 1 < p that J λ (w) = 2 1 ) w q 1 q 1 ) λ p + 1 < 2 1 ) w q 1 q 1 p + 1 = p ) 1 + q 2 1 q w 2 < 0, ) 1 + q p + q w 2 b(x) w(x, 0) p+1 dx which means that inf N + J λ < 0. Now for λ (0, Λ), we now from Lemma 3.1, λ that Nλ 0 = {0}. Together, these imply that w N + λ for large and inf w N + λ N λ 0 J λ (w) inf w N + λ J λ (w) < 0. Therefore, by wea lower semi-continuity of norm, that is, w 0 0, w 0 0. J λ (w 0 ) lim inf J λ(w ) = inf J λ < 0, N + λ N λ 0 Lemma 4.2. uppose w N + λ such that w w 0 wealy in H0,L s (C ). Then for λ (0, Λ), (1 + q) a(x)w 1 q 0 (x, 0)dx λ(p 1) 0 (x, 0)dx > 0. (4.3)

14 14. GOYAL, K. REENADH EJDE-2016/145 Moreover, there exists a constant C 2 > 0 such that (1 + q) w 2 λ(p + q) (x, 0) dx C 2 > 0. (4.4) Proof. For {w } N + λ ( N λ), since (1 + q) a(x)w 1 q 0 (x, 0)dx λ(p 1) 0 (x, 0)dx = lim (1 + q) a(x)w 1 q (x, 0) dx λ(p 1) (x, 0) dx = lim (1 + q) w 2 λ(p + q) (x, 0) dx 0, we can argue by a contradiction and assume that (1 + q) a(x)w 1 q 0 (x, 0)dx λ(p 1) 0 (x, 0)dx = 0. (4.5) ince w N λ, from the wea lower semi continuity of norm and (4.5) we have 0 = lim w 2 a(x)w 1 q (x, 0) dx λ (x, 0) dx w 0 2 a(x)w 1 q 0 (x, 0)dx λ { w0 2 λ p+q = b(x)wp+1 0 (x, 0)dx w 0 2 p+q p 1 a(x)w1 q 0 (x, 0)dx. 0 (x, 0)dx Thus for any λ (0, Λ) and w 0 0, by similar arguments as those in (3.3) we have that 0 < E λ w 0 p q ( p 1 ) p 1 w 0 2(p+q) p + q p + q a(x)w1 q 0 (x, 0)dx p 1 = 1 + q p + q ( p 1 ) p 1 p + q w 0 2(p+q) ( p 1 p+q w 0 2) p q p + q w 0 2 = 0, λ 0 (x, 0)dx which is clearly impossible. Now by (4.3), we have that (1 + q) a(x)w 1 q (x, 0) dx λ(p 1) (x, 0) dx C 2 for sufficiently large and a suitable positive constant C 2. This, together with the fact that w N λ we obtain equation (4.4). Fix φ C0,L (C ) with φ 0. Then we apply Lemma 3.6 with w = w N + λ ( large enough such that (1 q)c1 < C 2 ), we obtain a sequence of functions f : Bɛ (0) R such that f (0) = 1 and f (w)(w + wφ) N + λ for all w B ɛ (0). It follows from w N λ and f (w)(w + wφ) N λ that w 2 a(x)w 1 q (x, 0) dx λ (x, 0) dx = 0 (4.6)

15 EJDE-2016/145 FRACTIONAL LAPLACIAN WITH INGULAR NONLINEARITY 15 and f 2 (w) w + wφ 2 f 1 q (w) a(x)(w + wφ) 1 q (x, 0)dx (4.7) λf p+1 (w) b(x)(w + wφ) p+1 (x, 0)dx = 0. Choose 0 < ρ < ɛ and w = ρv with v < 1. Then we find f (w) such that f (0) = 1 and f (w)(w + wφ) N + λ for all w B ρ(0). Lemma 4.3. For λ (0, Λ) we have f (0), v is finite for every 0 v Hs 0,L (C ) with v 1. Proof. By (4.6) and (4.7) we have 0 = f 2 (w) 1 w + wφ 2 + w + wφ 2 w 2 f 1 q (w) 1 a(x)(w + wφ) 1 q (x, 0)dx a(x)((w + wφ) 1 q w 1 q )(x, 0)dx λf p+1 (w) 1 b(x)(w + wφ) p+1 (x, 0)dx λ b(x)((w + wφ) p+1 w p+1 )(x, 0)dx f 2 (ρv) 1 w + ρvφ 2 + w + ρvφ 2 w 2 f 1 q (ρv) 1 a(x)(w + ρvφ) 1 q (x, 0) λf p+1 (ρv) 1 b(x)(w + ρvφ) p+1 (x, 0) λ b(x)((w + ρvφ) p+1 w p+1 )(x, 0) Dividing by ρ > 0 and passing to the limit ρ 0, we derive that 0 2 f (0), v w s C y c w (vφ) dx dy (1 q) f (0), v λ(p + 1) f (0), v λ(p + 1) = f (0), v 2 w 2 (1 q) λ(p + 1) a(x)w 1 q (x, 0)dx (x, 0)dx b(x)(w p vφ)(x, 0)dx (x, 0)dx + 2 s y C c w (vφ) dx dy λ(p + 1) a(x)w 1 q (x, 0)dx b(x)(w p vφ)(x, 0)dx

16 16. GOYAL, K. REENADH EJDE-2016/145 = f (0), v (1 + q) w 2 λ(p + q) (x, 0)dx + 2 s y C c w (vφ) dx dy λ(p + 1) b(x)(w p vφ)(x, 0)dx. (4.8) From this inequality and (4.4) we now that f (0), v. Now we show that f (0), v +. Arguing by contradiction, we assume that f (0), v = +. Now we note that f (ρv) 1 w + f (ρv) ρvφ f (ρv) 1w + ρvf (ρv)φ = f (ρv)(w + ρvφ) w (4.9) and f (ρv) > f (0) = 1 for sufficiently large. From the definition of derivative f (0), v, applying equation (4.2) with w = f (ρv)(w + ρvφ) N + λ, we clearly have f (ρv) 1 w + f (ρv) ρvφ 1 f (ρv)(w + ρvφ) w J λ (w ) J λ (f (ρv)(w + ρvφ)) = 2 1 ) w 2 + λ 1 q 1 q 1 ) (x, 0)dx p q 1 ) f 2 (ρv) w + ρvφ 2 2 λ(p + q) (1 q)(p + 1) f p+1 (ρv) b(x)(w + ρvφ) p+1 (x, 0)dx = 1 q 1 ) ( w + ρvφ 2 w 2 ) q 1 ) f 2 (ρv) 1 w + ρvφ 2 2 λ 1 q 1 ) f p+1 (ρv) b(x)((w + ρvφ) p+1 w p+1 )(x, 0)dx p + 1 λ 1 q 1 ) f p+1 (ρv) 1 (x, 0)dx. p + 1 Dividing by ρ > 0 and passing to the limit as ρ 0, we can obtain f (0), v w + vφ ( 1 + q ) s y c w (vφ) dx dy + 1 q C ( p + q ) λ f 1 q (0), v ( p + q ) λ b(x)(w p vφ)(x, 0)dx 1 q = f (0), v (1 + q) w 2 λ(p + q) 1 q + q ) f 1 q (0), v w 2 (x, 0)dx (x, 0) dx

17 EJDE-2016/145 FRACTIONAL LAPLACIAN WITH INGULAR NONLINEARITY 17 That is, ( 1 + q ( p + q ) + ) s y c w (vφ) dx dy λ 1 q C 1 q vφ f (0), v 1 q (1 q) w ( p + q ) λ 1 q (1 + q) w 2 λ(p + q) + b(x)(w p vφ)(x, 0)dx. (x, 0) dx ( 1 + q ) s y c w (vφ) dx dy 1 q C b(x)(w p vφ)(x, 0)dx, (4.10) which is impossible because f (0), v = + and () w 2 λ(p+q) (x, 0) dx (1 q) w C 2 (1 q)c 1 > 0. In conclusion, f (0), v < +. Furthermore, (4.4) with w C 1 and two inequalities (4.8) and (4.10) also imply that f (0), v C 3 for sufficiently large and a suitable constant C 3. Lemma 4.4. For each 0 φ C0,L (C ) and for every 0 v H0,L s (C ) with v 1, we have a(x)w q 0 vφ L1 () and s y C c w 0 (vφ) dx dy a(x)(w q 0 vφ)(x, 0)dx (4.11) λ b(x)(w p 0 vφ)(x, 0)dx 0. Proof. Applying (4.9) and (4.2) again, we obtain f (ρv) 1 w + f (ρv) ρvφ J λ (w ) J λ (f (ρv)(w + ρvφ)) = f 2(ρv) 1 w 2 f 2(ρv) ( w + ρvφ 2 w 2 ) f 1 q (ρv) 1 a(x)(w + ρvφ) 1 q (x, 0) 1 q q + λ f p+1 (ρv) 1 p λ p + 1 a(x)((w + ρvφ) 1 q w 1 q )(x, 0) b(x)(w + ρvφ) p+1 (x, 0) b(x)((w + ρvφ) p+1 w p+1 )(x, 0). Dividing by ρ > 0 and passing to the limit ρ 0 +, we obtain f (0), v w + vφ

18 18. GOYAL, K. REENADH EJDE-2016/145 f (0), v w 2 s C y c w (vφ) dx dy + f (0), v 1 + lim inf ρ q + λ f (0), v = f (0), v w 2 a(x)w 1 q (x, 0)dx a(x)((w + ρvφ) 1 q w 1 q )(x, 0) dx ρ (x, 0)dx + λ b(x)(w p vφ)(x, 0)dx. s C y c w (vφ) dx dy + λ a(x)w 1 q (x, 0) dx λ b(x)(w p φ)(x, 0)dx 1 a(x)(w + ρvφ) 1 q w 1 q )(x, 0) + lim inf dx ρ q ρ = s y C c w (vφ) dx dy + λ b(x)(w p vφ)(x, 0)dx 1 + lim inf ρ q a(x)((w + ρvφ) 1 q w 1 q )(x, 0) dx, ρ Using above inequality, we have a(x)((w + ρvφ) 1 q w 1 q )(x, 0) lim inf dx ρ 0 + ρ (x, 0) dx is finite. Now, since a(x)((w +vφ) 1 q w 1 q )(x, 0) 0, then by Fatou s Lemma, we have a(x)(w q vφ)(x, 0)dx 1 lim inf ρ q a(x)((w + ρvφ) 1 q w 1 q )(x, 0) dx ρ f (0), v w + vφ + s y C c w (vφ) dx dy λ b(x)(w p vφ)(x, 0)dx C 1C 3 + vφ + s y C c w (vφ) dx dy λ b(x)(w p vφ)(x, 0)dx. Again using Fatou s Lemma and this inequality, we have a(x)(w q 0 vφ)(x, 0)dx lim inf a(x)(w q vφ)(x, 0) dx lim inf a(x)(w q vφ)(x, 0)dx

19 EJDE-2016/145 FRACTIONAL LAPLACIAN WITH INGULAR NONLINEARITY 19 s y C c w 0 (vφ) dx dy λ b(x)(w p 0vφ)(x, 0)dx <, which completes the proof. Corollary 4.5. For every 0 φ H0,L s (C ), we have a(x)w q 0 φ L1 (), w 0 > 0 in C and s y C c w 0 φ dx dy a(x)(w q 0 φ)(x, 0)dx (4.12) λ b(x)(w p 0 φ)(x, 0)dx 0. Proof. Choosing v H0,L s (C ) such that v 0, v l in the neighborhood of support of φ and v 1, for some l > 0 is a constant. Then by the Lemma 4.4, we note that a(x)(w q 0 φ)(x, 0)dx <, for every 0 φ C 0,L (C ), which guarantees that w 0 > 0 a.e. in {0}. Also by the strong maximum principle 9, we obtain w 0 > 0 in C. Putting this choice of v in (4.11) we have s y C c w 0 φ dx dy λ b(x)(w p 0φ)(x, 0)dx a(x)(w q 0 φ)(x, 0)dx 0. for every 0 φ C 0,L (C ). Hence by density argument, (4.12) holds for every 0 φ H s 0,L (C ), which completes the proof. Lemma 4.6. We have w 0 N + λ. Proof. Using (4.12) with φ = w 0, we obtain w 0 2 a(x)w 1 q 0 (x, 0)dx + λ 0 (x, 0)dx. On the other hand, by the wea lower semi-continuity of the norm, we have Thus w 0 2 lim inf = w 0 2 = w 2 lim sup w 2 a(x)w 1 q 0 (x, 0)dx + λ 0 (x, 0)dx. a(x)w 1 q 0 (x, 0)dx + λ 0 (x, 0)dx. (4.13) Consequently, w w 0 in H0,L s (C ) and w 0 N λ. Moreover, from (4.3) it follows that (1 + q) w 0 2 λ(p + q) 0 (x, 0)dx =(1 + q) a(x)w 1 q 0 (x, 0)dx λ(p 1) 0 (x, 0)dx > 0; that is, w 0 N + λ. Lemma 4.7. The function w 0 is a positive wea solution of problem (2.3).

20 20. GOYAL, K. REENADH EJDE-2016/145 Proof. uppose that φ H0,L s (C ) and ɛ > 0, we define Ψ = (w 0 + ɛφ) +. C = Γ 1 Γ 2 with Let {0} = 1 2 with Γ 1 := { (x, y) C : (w 0 + ɛφ)(x, y) > 0}, Γ 2 := {(x, y) C : (w 0 + ɛφ)(x, y) 0 }. 1 := {(x, 0) {0} : (w 0 + ɛφ)(x, 0) > 0}, 2 := {(x, 0) {0} : (w 0 + ɛφ)(x, 0) 0}. Then 1 Γ 1, 2 Γ 2, Ψ Γ1 = w 0 + ɛφ, Ψ Γ2 = 0, Ψ 1 (x, 0) = (w 0 + ɛφ)(x, 0), and Ψ 2 (x, 0) = 0. Putting Ψ into (4.12) and using (4.13), we see that 0 s y C c w 0 Ψ dx dy a(x)(w q 0 Ψ)(x, 0)dx λ b(x)(w p 0Ψ)(x, 0)dx = s y Γ c w 0 (w 0 + ɛφ) dx dy a(x)(w q 0 (w 0 + ɛφ))(x, 0)dx 1 1 λ b(x)(w p 0 (w 0 + ɛφ))(x, 0)dx 1 = s y C c w 0 (w 0 + ɛφ) dx dy λ b(x) (w p 0 (w 0 + ɛφ)) (x, 0)dx a(x) ( w q 0 (w 0 + ɛφ) ) (x, 0)dx s Γ 2 y c w 0 (w 0 + ɛφ) dx dy a(x) ( w q 0 (w 0 + ɛφ) ) (x, 0)dx λ b(x) (w p 0 (w 0 + ɛφ)) (x, 0)dx 2 2 = s y C c w 0 2 dx dy + ɛ s y c w 0 φ dx dy C λ b(x)(w p 0 φ)(x, 0)dx a(x)w 1 q 0 (x, 0)dx λ s y Γ c w 0 2 a(x)w 1 q 2 2 ɛ s y c w 0 φ dx dy Γ 2 λ b(x)(w p 0φ)(x, 0)dx 2 ɛ s y c w 0 φ dx dy C λ b(x)(w p 0 φ)(x, 0)dx ɛ s Γ 2 y c w 0 φ dx dy + a(x)(w q 0 φ)(x, 0)dx 0 (x, 0)dx λ a(x)(w q 0 2 φ)(x, 0)dx a(x)(w q 0 φ)(x, 0)dx 2 a(x)w q 0 (w 0 + ɛφ)dx 2 0 (x, 0)dx Let 0 (x, 0)dx

21 EJDE-2016/145 FRACTIONAL LAPLACIAN WITH INGULAR NONLINEARITY 21 + λ b(x) ɛφ p+1 (x, 0)dx + ɛλ b(x)(w p 0φ)(x, 0)dx 2 2 ɛ s y c w 0 φ dx dy a(x)(w q 0 φ)(x, 0)dx C λ b(x)(w p 0 φ)(x, 0)dx ( ɛ s y Γ c w 0 φ dx dy + ɛλɛ p b 2 s φ ɛλ b(x)(w p 0φ)(x, 0)dx. 2 2 L s p 1 ( 2) ) p+1 2 s dx s ince the measure of Γ 2 and 2 tend to zero as ɛ 0, it follows that Γ 2 y c w 0 φ dx dy 0 as ɛ 0, and similarly for λɛ p b 2 s 2 L s p 1 ( 2) ( ) p+1 φ 2 2 s dx s 2 and λ 2 b(x)(w p 0φ)(x, 0)dx. Dividing by ɛ and letting ɛ 0, we obtain s y C c w 0 φ dx dy a(x)(w q 0 φ)(x, 0)dx λ b(x)(w p 0φ)(x, 0)dx 0 and since this holds equally well for φ, it follows that w 0 is indeed a positive wea solution of problem (2.3). Lemma 4.8. There exists a minimizing sequence {W } in N λ such that W W 0 strongly in N λ. Moreover W 0 is a positive wea solution of (2.3). Proof. Using the Eeland variational principle again, we may find a minimizing sequence {W } N λ for the minimizing problem inf N J λ such that for W λ H0,L s (C ), we have W W 0 wealy in H0,L s (C ) and pointwise a.e. in {0}. We can now repeat the argument used in Lemma 4.2 to derive that when λ (0, Λ) () a(x) W 0 (x, 0) 1 q dx λ(p 1) b(x) W 0 (x, 0) p+1 dx < 0 (4.14) which yields (1 + q) a(x) W (x, 0) 1 q dx λ(p 1) b(x) W (x, 0) p+1 dx C 4 for sufficiently large and a suitable positive constant C 4. At this point we may proceed exactly as in Lemmas 4.3, 4.4, 4.6, 4.7 and Corollary 4.5, and conclude that 0 < W 0 N λ is the required positive wea solution of problem (2.3). Moreover from (4.14) it follows that (1 + q) W 0 2 λ(p + q) b(x)w p+1 0 (x, 0)dx

22 22. GOYAL, K. REENADH EJDE-2016/145 = (1 + q) λ(p + q) a(x)w 1 q 0 (x, 0)dx + λ b(x)w p+1 0 (x, 0)dx b(x)w p+1 0 (x, 0)dx = (1 + q) a(x)w 1 q 0 (x, 0)dx λ(p 1) that is W 0 N λ. b(x)w p+1 0 (x, 0)dx < 0, Proof of the Theorem 2.6. From Lemmas 4.7, 4.8 and 3.4, we can conclude that problem (2.3) has two positive wea solutions w 0 N + λ, W 0 N λ with W 0 > w 0 for any λ (0, Λ). Hence, u 0 ( ) = w 0 (, 0) H0() s and U 0 ( ) = W 0 (, 0) H0() s are positive solutions of the problem (1.1). Proof of the Corollary 2.7. For any W N λ, it follows from Lemma 3.4 that ( W > A λ = Λ 1 p q ) 1/(p 1) ) 1/(p 1) ( ) p+1 ( p 1 Λ ) 1/(p 1). p + q b λ Thus by the definition of Λ, and using ( W > Hence, let W ɛ N λ λ (0, Λ), we have q p 1 2(p+q) ()(p 1) p+1 p 1 = 1 q ) 1/() a /(q+1) 1 ) 1 q ( Λ λ, we obtain, ) 1/(p 1). be the solution of problem (2.3) with p = 1 + ɛ, where W > C ɛ ( Λ λ ) 1/ɛ, where C ɛ = + ) 1/() a ( ) 1 q 1/(q+1) 1 ɛ as ɛ 0. This completes the proof of Corollary 2.7. Acnowledgements. The first author is supported by the National Board for Higher Mathematics, Govt. of India, grant number: 2/40(2)/2015/R& D-II/5488. The second author is also supported by the National Board for Higher Mathematics, Govt. of India, 2/48(12)/2012/NBHM (R.P.)/R& D II/ References 1 Adimurthi, J. Giacomoni; Multiplicity of positive solutions for a singular and critical elliptic problem in R 2, Comm. in Contemporary Mathematics, 8 (5) (2006), Ahmed Mohammed; Positive solutions of the p-laplace equation with singular nonlinearity, J. Math. Anal. Appl., 352 (2009), D. Applebaum; Lèvy process-from probability to finance and quantum groups, Notices Amer. Math. oc., 51 (2004), B. Barrios, I. D. Bonis, M. Medina, I. Peral; emilinear problems for the fractional Laplacian with a singular nonlinearity, Open Math., 13 (2015), B. Brändle, E. Colorado, A. de Pablo, U. ànchez; A concave-convex elliptic problem involving the fractional Laplacian, Proc. Roy. oc. Edinburgh. ect. A, 143 (2013), A. Capella, J. Dávila, L. Dupaigne, Y. ire; Regularity of radial extremal solutions for some non-local semilinear equations, Comm. Partial Differential Equations, 36 (2011), X. Cabré, J. Tan; Positive solutions for nonlinear problems involving the square root of the Laplacian, Adv. Math., 224 (2010), L. Caffarelli, L. ilvestre; An extension problem related to the fractional Laplacian, Comm. Partial Differential Equations, 32 (2007),

23 EJDE-2016/145 FRACTIONAL LAPLACIAN WITH INGULAR NONLINEARITY 23 9 L. ilvestre; Regularity of the obstacle problem for a fractional power of the Laplace operator, Comm. Pure Appl. Math., 60 (2007) Y. Chen, J. Chen; Existence of multiple positive wea solutions and estimates for extremal values to a class of elliptic problems with Hardy term and singular nonlinearity, J. Math. Anal. Appl., 429 (2015), M. M. Coclite, G. Palmieri; On a singular nonlinear Dirichlet problem, Comm. Partial Differential Equations, 14 (10) (1989), M. G. Crandall, P. H. Rabinowitz, L. Tartar; On a Dirichlet problem with a singular nonlinearity, Comm. Partial Differential Equations, 2 (1977), Y. Fang; Existence, uniqueness of positive solution to a fractional laplacians with singular nonlinearity, arxiv: v1. (2014). 14 A. Garroni,. Müller; Γ-limit of a phase-field model of dislocations, IAM J. Math. Anal., 36 (2005) Goyal, K. reenadh; Existence of multiple solutions of p-fractional Laplace operator with sign-changing weight function, Adv. Nonlinear Anal., 4 (1) (2015) Goyal, K. reenadh; The Nehari manifold for non-local elliptic operator with concaveconvex nonlinearities and sign-changing weight functions, Proc. Indian Acad. ci. Math. ci., PMC-D N. Hirano, C. accon, N. hioji; Existence of multiple positive solutions for singular elliptic problems with concave and convex nonlinearities, Adv. Differential Equations, 9 (2) (2004) N. Hirano, C. accon, N. hioji; Brezis-Nirenberg type theorems and multiplicity of positive solutions for a singular elliptic problem, J. Differential Equations, 245 (2008) A. C. Lazer, P. J. McKenna; On a singular nonlinear elliptic boundary-value problem, Proc. of Amer. Math. oc., 111 (3) (1991) A. C. Lazer, P. J. McKenna; On singular boundary value problems for the Monge-Ampère operator, J. Math. Anal. Appl., 197 (1996) E. Di Nezza, G. Palatucci, E. Valdinoci; Hitchhier s guide to the fractional obolev spaces, Bull. ci. Math., 136 (2012) R. ervadei, E. Valdinoci; Mountain pass solutions for non-local elliptic operators, J. Math. Anal. Appl., 389 (2012) R. ervadei, E. Valdinoci; Variational methods for non-local operators of elliptic type, Discrete Contin. Dyn. yst., 33 (5) (2013) R. ervadei, E. Valdinoci; A Brezis-Nirenberg result for non-local critical equations in low dimension, Commun. Pure Appl. Anal., 12 (6) (2013) R. ervadei, E. Valdinoci; The Brezis-Nirenberg result for the fractional Laplacian, Trans. Amer. Math. oc., 367 (2015) J. Tan; Positive solutions for nonlocal elliptic problems, Discrete Contin. Dyn. yst., 33 (2013) Xiaohui Yu; The Nehari manifold for elliptic equation involving the square root of the Laplacian, J. Differential Equations, 252 (2012) J. Zhang, X. Liu, H. Jiao; Multiplicity of positive solutions for a fractional Laplacian equations involving critical nonlinearity, arxiv: v1 math.ap. aria Goyal Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-16, India address: saria1.iitd@gmail.com Konijeti reenadh Department of Mathematics, Indian Institute of Technology Delhi, Hauz Khas, New Delhi-16, India address: sreenadh@gmail.com