Semilinear problems for the fractional laplacian with a singular nonlinearity

Abstract The aim of this paper is to study the solvability of the problem where Ω is a bounded smooth domain of RN, N > 2s, M ε {0, 1}, 0 < s < 1, γ > 0, λ > 0, p > 1 and f is a nonnegative function. We distinguish two cases: – For M = 0, we prove the existence of a solution for every γ > 0 and λ > 0. A1 – For M = 1, we consider f ≡ 1 and we find a threshold ʌ such that there exists a solution for every 0 < λ < ʌ ƒ, and there does not for λ > ʌ ƒ


Motivation and organization of the paper
In this paper we study the existence, regularity and multiplicity of solutions of the following nonlocal problem .P/ D

<
: where is a bounded smooth domain of R N , N > 2s, M 2 f0; 1g, 0 < s < 1, > 0, > 0, p > 1, and f is a nonnegative function that belongs to some L m . /, m 1. In this context, . / s is the fractional Laplace operator defined, up to a normalization factor, by the Riesz potential as . / s u.x/ WD P:V: u.x C y/ C u.x y/ 2u.x/ y N C2s dy; x 2 R N when s 2 .0; 1/ is a fixed parameter (see for example [16,24,33]). Such elliptic problems with a singular nonlinearity have a large history in the local case, that is, with a principal part of Laplacian type. The seminal paper by Crandall, Rabinowitz and Tartar [13] is the starting point of a large literature, see for instance [3,4,7,8,12,14,19,21,23,25,26,34,35]. The philosophy to deal with this framework is similar to the one used for problems with a concave or a concave-convex nonlinearity. See for instance the papers [1,9,10,17,18] and their corresponding references.
In this work we deal with two problems: .P / D . / s u D u C u p in ; u > 0 in ; u D 0 in R N n : Problem .P / is strongly inspired by the following semilinear elliptic problem with a singular nonlinearity (see [7]) where is a bounded open subset of R N , N 2, > 0 is a real number and f is a nonnegative function that belongs to some Lebesgue space. The singular problem .L 1 / appears by considering problems with a convection term via a change of variable, that is, this problem is related to singular problems with a term depending on jrvj 2 . See for instance the references [19,35].
In [7], the authors study existence and regularity results of the solutions to problem .L 1 /, depending on (splitting in the cases D 1, > 1 and < 1) and on the summability of f . Our aim will be to prove, using similar techniques as in the local case, this kind of results for the nonlocal framework. In particular, to study problem .P /, we will work by approximation, that is, analyzing the problems obtained truncating the singular term 1 u and the datum f , so that the first one becomes non singular at the origin and the second one belongs to L 1 . / (see [7,8] in the local setting). It is worthy to point out that the solvability results obtained for problem .P / when s D 1, have been recently extended by D. Giachetti, P. Martinez Aparicio and F. Murat (personal communication), to nonlinearities f .x; u/ where f is a function verifying that is, without the monotonicity assumption. This kind of results can be also extended to the non-local framework. We omit the details. Regarding problem .D ; ;p /, the motivation arises from the following semilinear problem, whose nonlinearity combines a singular term and a convex one where , p and are positive numbers (see, among other papers, [3,4,8,12,14,21,35] for an extensive analysis of this kind of problems). The multiplicity behavior in this case is essentially the same as in concave-convex type problems. Firstly, the approach of [9] based on the use of sub/supersolutions method still works in this context (see also [5,6] where this monotonicity techniques are applied in the nonlocal framework). Then, as in [3], we are also able to show, with some technical variations, the existence of a second solution for the problem .D ; ;p / when we add a convex term if > 0 is small enough. More precisely, we prove the existence of a first solution for > 0, p > 1, up to some threshold on , ƒ, considering the approximating problems corresponding to .D ; ;p /, by applying the sub/supersolutions and Sattinger methods.
When we consider the subcritical case, that is, 0 < < 1 and 1 < p < 2 s 1 D N C2s N 2s , we can show that for 0 < small enough the problem .D ; ;p / has indeed two positive energy solutions, obtained by means of variational techniques: the first one by minimization and the second one by the Mountain Pass Lemma (see [2,20]). As far as we know, the global multiplicity result, that is, the proof of the existence of a second positive solution in the whole range until the threshold ƒ, remains open in the nonlocal setting.
The paper is organized as follows: in Section 2, an introduction of the necessary functional framework is presented, as well as the problems that will be treated, and the different type of solutions we will work with. This distinction is related to the assumptions on the parameter of the singularity and on the summability of f . Section 3 is devoted to prove the existence and regularity of solution of .P /. With this purpose, firstly we need to introduce and analyze the associated approximating problems.
Finally, in Section 4 we deal with problem .D ; ;p /. The goal of this part of the work is to prove the existence of at least two positive solutions for some range of values of > 0.

Introduction and functional setting
Consider the space endowed with the natural norm In this framework, we can calculate h. / s u; 'i for every u 2 L s and ' in the Schwartz class. Moreover, we introduce the Sobolev space and the space X s 0 . /, defined as endowed with the norm where Q D R 2N n .C C /. This space allows us to deal with problems proposed in a bounded domain , as we need. The pair .X s 0 . /; k k X s 0 . / / yields a Hilbert space (see [29,Lemma 7] for more details). Moreover, it can be seen that . / s W X s 0 . / ! X s . / is a continuous operator.
In what follows we will use the relation between the norm in the space X s 0 . / and the L 2 norm of the fractional Laplacian, see [16,Proposition 3.6], Here C.N; s/ is a normalization constant, that appears in the definition of . / s , see [16,31], chosen to guarantee that 2 . / s u. / D j j 2s b u. /; 2 R N : We will also need to make use of the classical Sobolev Theorem: It is easy to check that for u and ' smooth enough, with vanishing condition outside , we have the following duality product, what means that . / s is selfadjoint in X s 0 . /. Before stating the results contained in this work, we need to precise the sense of solutions that we will handle here. In particular, if we consider the general problem we will distinguish two types of solutions, attending to the regularity of F and u. Indeed, Definition 2.2. We say that u 2 X s 0 . / is a positive energy supersolution (respectively subsolution) of problem .D/ if F 2 L for every nonnegative ' 2 X s 0 . / with compact support contained in . If u is a supersolution and a subsolution of .D/, we say that it is a positive energy solution.
Analogously, when we have less regularity on u, we will make use of a weaker notion of solution. Define first the set Notice that every 2 T belongs in particular to L 1 . / (see [27]).
Definition 2.3. We say that u 2 L 1 . / is a positive weak supersolution (respectively subsolution for every nonnegative 2 T . We say that u is a positive weak solution of problem .D/ if it is at the same time a supersolution and a subsolution of such a problem. Notice that, due to the singular term in (P), only by asking u 2 X s 0 . / we cannot expect F .x; u/ to belong to L .2 s / 0 . / in the first case, or F .x; u/ 2 L 1 . / if u 2 L 1 . / in the second one. Thus, the right hand side is not well defined if we test in the natural spaces X s 0 . / and T with functions supported in respectively. To deal with this difficulty we restrict our test sets to the functions with compact support.
Remark 2.4. If F 2 L .2 s / 0 . / in .D/, we can extend Definition 2.2, saying that (3) holds for every ' 2 X s 0 . /. This will be the case of the approximating problems, that we will use to build solutions of .P/. Notice that here the first term in the right hand side is no longer singular (it is bounded indeed), so we do not need to restrict our test set to functions compactly supported in .

Solvability of the elliptic problem with the singular nonlinearity
Let be a bounded domain in R N . We consider the following problem where N > 2s, 0 < s < 1, > 0, > 0 and f is a nonnegative function whose summability conditions will be specified later. By a rescaling argument is sufficient to consider D 1.

Approximating problems
In order to study the solvability of problem .P /, we will analyze the associated approximating problems. Indeed, suppose f 2 L 1 . /, f 0, and for every n 2 N, let us define f n WD min.f .x/; n/; and consider the problem First, we prove the existence of solution to this problem. .
Notice that the existence and uniqueness of solution to this problem is given by the Lax-Milgram Theorem, since the right hand side belongs to the dual space X s . /. Testing now in (5) with w, we get and thus, by the Sobolev embedding given in Theorem 2.1, it follows that with C D C.N; s; / independent of v, so that the ball of radius C n C1 is invariant under S in X s 0 . /. In order to apply the Schauder's Fixed Point Theorem over S to guarantee the existence of a solution of .P n; /, apart from the invariance, we need to check the continuity and compactness of S as an operator from X s 0 . / to X s 0 . /. First, we prove the continuity. In order to do this, we want to check that, if we denote w k WD S.v k / and w WD S.v/, then lim In fact, let fv k g k2N be a sequence in X s 0 . / converging to v 2 X s 0 . /. Thus we get where .2 s / 0 D 2N N C2s < 2 s . Thus, by Hölder's inequality and Theorem 2.1 again, we obtain and therefore, by the Dominated Convergence Theorem and (9), we conclude that kw k wk X s 0 . / ! 0 as k ! 1; and hence, S is continuous from X s 0 . / to X s 0 . /. To see that S is compact, we take a sequence fv k g k2N such that kv k k X s 0 . / Ä C . Therefore, by Rellich-Kondrachov Theorem (see [16,Theorem 7.1]) we conclude that, up to a subsequence, Furthermore, since S is continuous, with C a positive constant independent of k, and hence, Because of the continuity of S , necessarily Q w D S.v/. Thus, proceeding as in (10) one can reach and by (12) we conclude lim and therefore S is compact from X s 0 . / to X s 0 . /. Given these conditions on S, Schauder's Fixed Point Theorem provides the existence of u n 2 X s 0 . / such that u n D S.u n /, i.e. u n solves 8 < : By the weak maximum principle ( [31] and [6, Lemma 2.1]), u n 0, and hence u n solves .P n; /. Finally, since the right hand side of .P n; / belongs to L 1 . /, by [27] we also get that u n 2 L 1 . /.
Moreover, we can prove the following result, Lemma 3.2. fu n g n2N is an increasing sequence, u n > 0 in , and for every set Q there exists a positive constant c Q , independent of n, such that u n .x/ c Q > 0; for every x 2 and every n 2 N.
Proof. Consider the problem satisfied by u n and u nC1 and substracting them we get: Now we choose, in the problems satisfied by u n u nC1 , .u n u nC1 / C as test function. Since f nC1 0, and , by [27], we have, On the other hand, from Lemma 3.1 we know that u 1 belongs to L 1 . /, that is, ku 1 k L 1 . / Ä C , and thus, Once again, since f 1 .C C 1/ is not identically zero, a further use of the Strong Maximum Principle ( [31]) gives us that u 1 > 0 in and hence, since fu n g is an increasing sequence, u n verifies (14), for every n 1.
Corollary 3.3. The solution u n to problem .P n; / is unique.
Proof. Let us consider v n ¤ u n a solution of .P n; /. Taking .u n v n / C (.v n u n / C resp.) as a test function in .P n; /, we conclude v n Ä u n (u n Ä v n resp.), and the uniqueness follows. See [10] for more details and general results.
Now, the goal is passing to the limit in the sequence fu n g n2N to achieve a solution of .P /. With this purpose, we must distinguish three cases, attending to the value of the power . that is, ku n k X s 0 . / Ä C , with C independent of n. In the case < 1 taking again u n as a test function in .P n; /, by Theorem 2.1 and Hölder inequality, we get

Case Ä 1
Since 2 2 s > 1 m 0 , we get an uniform estimate of u n in the space X s 0 . /, as desired. Proof. First of all, notice that Therefore, for both cases D 1 and < 1, since, by Lemma 3.4, u n is bounded in the Hilbert space X s 0 . /, u n weakly converges to u in X s 0 . /. Therefore we are able to pass to the limit in the left hand side of .P n; /, that is, Therefore Sinceˆ0.u n /ˆ.u n / Ä C1 2 u n , from (18)  where C > 0 is independent of n. Letting T ! C1 we conclude. Proof. Consider the sequence fu n g n2N of solutions to the problem .P n; /. By Lemma 3.2 we know that this sequence is increasing, and thus we can define u WD lim n!1 u n . Hence, due to the weak lower semicontinuity of the norm, by Lemma 3.6 we obtain with C a positive constant independent of n, i.e., u Therefore, by the Dominated Convergence Theorem we can pass to the limit in the right hand side of the weak formulation of .P n; /, concluding the existence of a weak solution of problem .P /. Proof. We begin proving the case D 1. Following the idea given in [27,Theorem 3.10] we considerˆˇ.r/, the convex function defined in (16) as a test function in .P n; /, forˇD 2. Doing the same as in Lemma 3.6, sincê 0 2 .u n /ˆ2.u n / Ä 2u 3 n , we get that

Regularity of solutions of problem .P /
The integral in the left hand side of (19) can be estimated, by Theorem 2.1, in the following way Since u n 2 L 1 . / then lim .measfu n T g/ D 0: Thus, from (19) and (20), we find that  The summability of the solution obtained in the previous proposition could be improved requesting more regularity to the function f . In order to prove this result, we will adapt to the nonlocal framework the ideas given in [4,Lemma 1], to obtain the following: Lemma 3.10. Let > 0 and > 0. If w 2 X s 0 . /, w > 0 in , satisfies then there exists a constant C > 0, independent of w, such that Proof. Let us first consider the case D 1. For k 1, define the function G k . / WD maxf k; minfk; gg: Thus, testing with G k .w/ in (21), by [27,Proposition 2.7 ii)], we get that where A k WD fx 2 W w.x/ > kg: Since, from the previous inequality and Theorem 2.1, we have that using Stampacchia's method ([32, Lemma 4.1]), we conclude that there exists a positive constant C D C.N; s/ such that kwk L 1 . / Ä C: The general case, > 0, follows easily using that the fractional Laplacian is a linear operator. In fact, if we consider the modified function for every ' 2 X s 0 . /, we get that That is, e w is an energy solution of the Dirichlet problem .P / with D 1. Therefore, for the calculations done before for the case D 1, there exists C > 0 such that ke wk L 1 . / Ä C: That is, kwk L 1 . / Ä C 1 C1 : Using the previous Lemma we get the next result. Proof. Following the same ideas done in the proof of Lemma 3.10, applying Hölder inequality and the Stampacchia's method for the solutions u n of the truncated problems .P n; /. That is, one proves that there exists C D C.N; s; kf k L m . / ; j j/ > 0 such that so, recalling that u WD lim n!1 u n we conclude.

Some remarks on solvability and multiplicity of solutions for the elliptic problem when adding a convex term
In this section, we consider the previous Dirichlet problem, but adding a convex term of the form u p , with p > 1.

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: . / s u D u C u p in ; u > 0 in ; u D 0 in R N n :

First solution
We will prove the following Theorem 4.1. Assume > 0 and p > 1. Then, there exists 0 < ƒ < 1 such that, for every 0 < < ƒ, there exists a positive solution u to the problem .D ; ;p / in the following sense: Proof.

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: First, we prove that there exists a solution of this problem, by applying the Sattinger method.
Step 2: We construct a subsolution. Consider the solution u n 2 X s 0 . / to the problem 8 < : . / s u n D u n C C 1 n in ; that is constructed applying Schauder's fixed point theorem, in the same way as the solution of problem (5). By the maximum principle, u n 0, and thus that is, u n is a subsolution of .D n; ; ;p /.
Step 3: We find a supersolution. Let t > , that will be chosen later, and let 0 Ä u n be the energy solution to the problem By Lemma 3.10, we know that there exists C 0 > 0 such that ku n k L 1 . / Ä C 0 t 1 C1 . In order to have that u n is a supersolution of .D n; ; ;p /, we need to prove that we can choose t large enough so that t u n C 1 n u n C 1 n C u n p : But this is equivalent to Notice first that, since C p C 1 > 1, for small enough one can find t > 0 satisfying : (24) holds. Thus, we have proved the existence of ƒ 0 > 0 such that for 0 < < ƒ 0 , u n is a supersolution of problem .D n; ; ;p /.
Step 4: We prove that u n Ä u n . We procced as in the proof of Lemma 3.2, that is, we consider the problem satisfied by u n u n and we use .u n u n / C as test function. Therefore, since t > and 1 r C 1 n is decreasing for r > 0, we get that Thus, using that Z R N .u n u n / C . / s .u n u n / it follows that u n Ä u n .
Step 5: Sattinger method. Consider the function g.r/ D r C 1 n C r p C n C1 r; r 2 OE0; C1/: It can be easily checked that g is an increasing function. Thus, consider u n;1 2 X s 0 . / the solution of the problem 8 < : . / s u n;1 C n C1 u n;1 D g.u n / in ; u n;1 > 0 in ; u n;1 D 0 in R N n : Since the added term in the left hand side is linear, it can be seen ([6, Lemma 2.1]) that the new operator satisfies a comparison principle. Hence, using that . / s u n C n C1 u n Ä g.u n / in ; we conclude that u n Ä u n;1 . Likewise, it can be proved that u n;1 Ä u n . Consider now for every k 2 N the iterated problems 8 < : . / s u n;kC1 C n C1 u n;kC1 D g.u n;k / in ; u n;kC1 > 0 in ; u n;kC1 D 0 in R N n : Then, since g is increasing, the solutions of these problems satisfy u n Ä u n;1 Ä Ä u n;k Ä u n;kC1 Ä u n : Thus, we can define the pointwise limit u n WD lim k!1 u n;k . Moreover, using u n;kC1 as a test function in (28), by Lemma 3.10, we get that where C n is a constant dependent on n but independent of k. Hence, up to a subsequence, we can conclude that u n;k * u n in X s 0 . /. Thus, u n is an energy solution of problem .D n; ; ;p /. Furthermore, by Lemma 3.2, since the sequence of subsolutions fu n g n2N is increasing with respect to n, there exists a constant c Q > 0, independent of n, such that u n u n u 1 c Q > 0; for every x 2 Q and every n: Remark 4.2. Note that, by construction, the solution u n of problem .D n; ; ;p / is a minimal solution, that is, if f u n is another solution of .D n; ; ;p / then u n Ä f u n .
The idea now is passing to the limit in the sequence fu n g in order to get a solution to problem .D ; ;p /. Consider first the case Ä 1. Using u n 2 X s 0 . / as a test function in the problem .D n; ; ;p /, one gets Z where, by Lemma 3.10, Q C is a constant independent of n. Therefore, u n is uniformly bounded in X s 0 . / and, up to a subsequence, u n * u in X s 0 . /. Moreover, for every ' 2 X s 0 . / with supp.'/ Q , by (29), we have that Thus, we can pass to the limit in the energy formulation of the approximated problems .D n; ; ;p / to conclude that u is an energy solution of .D ; ;p /.
Consider now > 1. We proceed as in the proof of Lemma 3.6, considering the Lipschitz convex function .r/ for r > 0, defined in (16). Therefore, by Lemma 3.10, we have that where C is a constant independent of n. Letting T ! C1 in the definition ofˆ C1 2 , we conclude that u C1 2 n is uniformly bounded in X s 0 . /, and thus, by the Rellich-Kondrachov Theorem (see [16,Theorem 7.1]), there exists we could construct the supersolution of Step 3. The purpose now is to prove that indeed we can find a solution for every 0 < < ƒ, where ƒ was defined in (30). Take 0 < < ƒ. Given the definition of ƒ, we can find N as close as we want to ƒ so that problem .D N ; ;p / has a solution u N . In particular, taking < N < ƒ, it is easy to check that u N is a supersolution of .D n; ; ;p /. Proceeding as in Step 4, one can prove that u n Ä u N , where u n is the subsolution of .D n; ; ;p / constructed at Step 2. Therefore, repeating Step 5 and Step 6 using the new supersolution u N , the result follows.

About existence of second solution
For the truncated problems .D n; ; ;p / it is easy to check that in fact there exists at least a second solution for every 2 .0; ƒ n /, where ƒ n is the critical parameter for the existence of minimal solution for every n 2 N and p < 2 s 1. Notice that fƒ n g # ƒ as n ! 1.
More precisely, by means of variational techniques, one can find two different energy solutions, the first one by minimization, and the second one by the Mountain Pass Lemma (see [2,20]). In fact it would be natural to consider the energy functional J n W X s 0 . / ! R N defined as J n .u/ D C.N; s/ 4 kuk 2 To find the critical points of J n , it can be used an adaptation of the ideas developed in [1] (see [5] for the details in the non local case). Indeed, let us define the class where ı s .x/ Ddist.x; @ /. Notice that, by the Hopf Lemma (see [22,Lemma 1.2]), C s . / is a natural space where one can separate the minimal solutions (see [15]). More precisely, let u n;1 and u n;2 be the minimal solutions of .D n; 1 ; ;p / and .D n; 2 ; ;p /, with 0 < 1 < 2 . Thus, w WD u n;2 u n;1 0. The idea now is to apply a Hopf lemma on w but, since we cannot assure that . / s w 0 in , we will apply again the idea of adding a linear term in the problem (see Step 5 in the proof Theorem 4.1). That is, considering the increasing function g given in (26), it follows that 8 < : . / s w C n C1 1 w g.u n;2 / g.u n;1 / 0 in ; w 0 in ; w D 0 in R N n ; Therefore, since we are under the hypotheses of [22, Lemma 1.2], we conclude that u n;2 .x/ u n;1 .x/ C ı s .x/; x 2 : Moreover, by Theorem 4.1, we know that the solutions to the truncated problems are bounded, and thus the right hand side of .D n; ; ;p / is bounded as well. Hence, by [28, Proposition 1.1] we obtain u n;2 u n;1 2 C s . / Â C s . /, and therefore the minimal solutions are separated in C s . /.
It is worth to remark that this strategy cannot be followed to get the second solution of .D ; ;p / for every 0 < < ƒ because, due to the singular term, we cannot apply any bootstrapping result to reach a Hölder regularity up to boundary.
Using the ideas in [11], one can prove that there exists a local minimum of the problem .D n; ; ;p / in the C s -topology, and then that every minimum of J n in C s . / is indeed a minimum in X s 0 . /, separating the minimal solutions of this problem for different . Finally, following the ideas of [1] (see also [5]), it can be made a translation of the truncated functional and to use the Mountain Pass Lemma, checking that the translated functional satisfies the suitable geometry and compactness properties to prove the existence of a non-trivial critical point.
Despite the fact that we cannot apply this strategy for the non truncated problem, for p < 2 s 1 and small enough a second solution can be still obtained using the Mountain Pass Theorem (see for example Section 3 of [6], where these ideas are developed for a nonlocal concave-convex problem). The analysis of multiplicity of solutions for the whole range 0 < < ƒ, as far as we know, remains as an open problem.