Quasilinear nonlocal elliptic problems with variable singular exponent

In this article, we provide existence results to the following nonlocal equation \begin{document}$ \begin{align*} \begin{cases} (-\Delta)_p^{s} u = g(x,u),\;u>0\; \text{in}\; \Omega,\\ u = 0 \; \text{in}\; \mathbb{R}^N\setminus \Omega, \end{cases} \end{align*}\quad\quad(P_ \lambda)$ \end{document} where \begin{document}$ (-\Delta)_{p}^{s} $\end{document} is the fractional \begin{document}$ p $\end{document} -Laplacian operator. Here \begin{document}$ \Omega \subset \mathbb R^N $\end{document} is a smooth bounded domain, \begin{document}$ s\in(0,1) $\end{document} , \begin{document}$ p>1 $\end{document} and \begin{document}$ N>sp $\end{document} . We establish existence of at least one weak solution for \begin{document}$ (P_ \lambda) $\end{document} when \begin{document}$ g(x,u) = f(x)u^{-q(x)} $\end{document} and existence of at least two weak solutions when \begin{document}$ g(x,u) = \lambda u^{-q(x)}+ u^{r} $\end{document} for a suitable range of \begin{document}$ \lambda>0 $\end{document} . Here \begin{document}$ r\in(p-1,p_{s}^{*}-1) $\end{document} where \begin{document}$ p_s^{*} $\end{document} is the critical Sobolev exponent and \begin{document}$ 0 .

Here Ω ⊂ R N is a smooth bounded domain, s ∈ (0, 1), p > 1 and N > sp. We consider the following two type of nonlinearities: Case (I) g(x, u) = f (x)u −q(x) , and Case (II) g(x, u) = λf (x)u −q(x) + u r . Throughout the paper, we assume λ > 0, q ∈ C 1 (Ω) and f ∈ L m (Ω) (m > 1 is appropriately chosen in each case (I) and (II)) both being positive and p − 1 < r < p * s − 1. The problem (1.1) is singular in nature due to the fact that the nonlinearity g(x, t) in both Case (I) and Case (II) blows up as t → 0 + . The novelty in our work relies on the fact that the exponent q is variable and we provide sufficient conditions on q to assure the existence of at least one solution in Case (I) and at least two solutions in Case (II). Additionally, we also establish a regularity result in Case (I). Before proceeding to state our main results, let us briefly mention the state of the art concerning singular elliptic problems both in the local and nonlocal case.
Under the hypothesis that q(x) ≡ q, singular problems with constant exponent has been investigated widely both in the local and nonlocal case. Starting point of study of singular problems is the pioneering work of Crandall et al. [13]. Let us briefly discuss the local case first. Consider the following p-Laplace equation u q + µu r in Ω; u > 0 in Ω; u = 0 on ∂Ω. (1.2) For p = 2, µ = 0 and f (x) ≡ 1, existence of a classical solution to the problem (1.2) is proved in [13] for any q > 0. Later for certain restricted range of q, the existence of weak solution was proved in [25]. This restriction on q was removed in [8] to obtain existence of at least one weak solution. Indeed the authors in [8] proved the existence of solution in H 1 0 (Ω) for 0 < q ≤ 1 and in H 1 loc (Ω) for q > 1. The case of p ∈ (1, ∞) was settled in [11], where existence of weak solution in W 1,p 0 (Ω) was proved for 0 < q ≤ 1 and in W 1,p loc (Ω) for q > 1. On the other hand, for p = 2, f (x) ≡ µ = 1 and 1 < r ≤ 2 * = 2N N −2 , multiplicity of weak solutions was established using Nehari manifold and sub super solution techniques in [21,22] for 0 < q < 1. Whereas the case of any q > 0 was settled in [4,23]. In the nonlinear case that is for p ∈ (1, ∞), authors in [20] answered the question of existence, multiplicity and regularity of weak solutions in the case of 0 < q < 1 which was further extended to the case of q ≥ 1 in [6] deducing the multiplicity of weak solutions. Recently for the weighted p-Laplace operator with Muckenhoupt class of weights existence and multiplicity is deduced in [16,17].
In the nonlocal case, after immense activeness in the study of elliptic problems involving the fractional Laplace operator by researchers, it was a natural question to study the nonlocal singular problems. In this context, authors in [7] studied the following problem Here authors studied the existence of distributional solutions for small λ using the uniform estimates of {u n } which are solutions of the regularized problems with singular term u −γ replaced by (u + 1 n ) −γ . This was extended for the p-fractional Laplace operator by Canino et al. in [10]. In the critical case for 0 < q < 1, the question of existence and multiplicity of weak solutions to nonlocal singular problems has been answered in [19,27,28] whereas q ≥ 1 case has been dealt in [18]. We also refer a recent article [5] related to nonlocal singular problem with exponential nonlinearity. Moreover, we refer readers to [1,2] concerning the existence and multiplicity results for the fractional p-Laplacian problems.
Naturally, now a question arises that what is the result when q is a function depending on x. In this direction, for the following model problem existence of at least one weak solution was proved in [12]. Motivated by this and the current interest in the study of nonlocal problems involving fractional Laplacian, we study the singular problem (1.1) where the singular exponent is a function of x. The case (I) is treated using the solutions to an approximated nonsingular formulation of (1.1) whereas case (II) has been dealt using the critical point theory and Mountain pass Lemma. We have also established that every weak solution is bounded which contributes to the regularity part. This article is divided into 5 sections-Section 2 contains the preliminaries and main results. We prove the existence result for (1.1) in case (I) in Section 3. The regularity of such weak solutions has been proved in Section 4. Finally, the existence and multiplicity of weak solution to (1.1) in case (II) have been established in Section 5.
Moreover, we define the space W s,p 0 (Ω) := {u ∈ W s,p (R N ) : u = 0 a.e. in R N \ Ω} endowed with the norm [·] s,p which we denote by · W s,p 0 (Ω) . Both the spaces W s,p (Ω) and W s,p 0 (Ω) are reflexive Banach spaces. We have the following useful embedding result.
Moreover, the above embedding is compact except for r = p * s . For more detailed discussion on such spaces, we refer the interested reader to the article [15].
Definition 2.2. We say that u ∈ W s,p 0 (Ω) is a weak solution of (1.1) if u > 0 in Ω and for all φ ∈ C ∞ c (Ω), one has Moreover we say a function u ∈ W s,p 0 (Ω) is a subsolution (or supersolution) of (1.1) if for every 0 ≤ φ ∈ C ∞ c (Ω) respectively. Therefore every weak solution is both a sub and supersolution. We shall use the following notations throughout the article.
Proof. Following the proof of [22, Lemma A.1], we get for any v ∈ X + , there exists a sequence {v n } ∈ W s,p 0 (Ω) such that each v n has a compact support in Ω, 0 ≤ v 1 ≤ v 2 ≤ . . . and {v n } converges strongly to v in W s,p 0 (Ω). Now arguing similarly as in [22,Lemma 9] we get the result.
For a given δ > 0, we denote by Our main results in this paper reads as follows: . Then the problem (1.1) admits a weak solution u ∈ X in Case (I).

Existence results in case (I).
To prove our main results we make use of the following approximated problem: where f n (x) = min{f (x), n} for n ∈ N and q ∈ C 1 (Ω) is positive.
Proof. Since Therefore we can define the operator S : X → X by S(u) = w where w satisfies (3.2). Since q ∈ C 1 (Ω) is positive, proceeding as in the proof [10, Proposition 2.3], it follows that the operator S is both continuous and compact. Choosing w as a test function in (3.2) and using the embedding result, Lemma 2.1 we get for some constant C = C(p, s, N, Ω) (independent of u). Therefore, by Schauder fixed point theorem we get the existence of a solution u n of the problem (3.1). Since the R.H.S of (3.2) belong to L ∞ (Ω), by [10, Lemma 2.2], we get u n ∈ L ∞ (Ω). Next, we present the proof of our first main theorem.
Proof of Theorem 2.4: By Lemma 3.1, choosing u n as a test function in (3.1), we get Using Hölder's inequality and Lemma 2.1 we obtain u n p X ≤ f L 1 (Ω) + C f L m (Ω) u n X , which implies that the sequence {u n } is uniformly bounded in X. Thus up to a subsequence, by Lemma 2.1, we get u n u weakly in X, u n → u strongly in L r (Ω) for 1 ≤ r < p * s and u n → u pointwise a.e. in Ω as n → ∞. Since u n is a weak solutions of (3.1), we have Now the convergence of L.H.S of (3.3) follows proceeding similarly as in the proof of [10, Theorem 3.2]. Therefore we can pass to the limit as n → ∞ to conclude that Hence u ∈ X is a weak solution of (1.1). Finally, we prove our second main result below.
Proof of Theorem 2.5: Denote by Φ(t) := t q * +p−1 p , for t ≥ 0. Since u n ∈ L ∞ (Ω) for each n and Φ is Lipschitz, we have Φ(u n ) ∈ W s,p 0 (Ω). Also since u n solves (3.1) and the R.H.S of (3.1) belong to L ∞ (Ω), we can use [10, Proposition 3.3] to get By Lemma 3.2, since u n is monotone increasing, we can define the pointwise limit u of u n as n → ∞. Therefore by Fatou's lemma Hence u q * +p−1 p ∈ W s,p 0 (Ω) and since q * > 1 we have u ∈ L p (Ω). Moreover, by Lemma 3.2 for every K Ω, there exists a constant l(K) > 0 such that u(x) ≥ l(K) > 0 for a.e. x ∈ Ω. Now the fact that u ∈ W s,p loc (Ω) and is a weak solution of (1.1) follows from the lines of proof of [10, Theorem 3.6] while realising that the role of γ there is played by q * here.

4.
Regularity results in case (I). Following is a local regularity result which follows directly as a consequence of Lemma 3.2.
Theorem 4.1. Every weak solution of (1.1) in Case (I) as obtained through Theorem 2.4 and Theorem 2.5 belongs to C α loc (Ω) for some α ∈ (0, 1). Proof. Let u denotes a weak solution to (1.1) obtained in Theorem 2.4 and Theorem 2.5 then using Lemma 3.2 and u being a pointwise limit of {u n } we obtain that there exists a l K > 0 such that This implies u −q(x) (x) ≤ C K for a.e. K ⊂⊂ Ω for some positive constant C(K) depending on K. Therefore using [24,Corollary 4.2] we conclude that u ∈ L ∞ loc (Ω). Furthermore, from [24, Corollary 5.5] it follows that u ∈ C α loc (Ω) for some α ∈ (0, 1).
While restricting N > sp(p + 1), we can get that solutions of (1.1) belong to L ∞ (Ω). Before proving this result, we recall the following Lemmas.

P. GARAIN AND T. MUKHERJEE
As t → 0, z (t) → |t| and we have |z (t)| ≤ 1. So using Fatou's Lemma, we let → 0 in the above inequality to get for every 0 < ψ ∈ C ∞ c (Ω). The inequality (4.1) still holds for 0 ≤ ψ ∈ X (similar proof as of [28, Lemma 6.1]). Now, define u K = min{(u − 1) + , K} ∈ X, for K > 0. For β > 0 and ρ > 0, we take ψ = (u K + ρ) β − ρ β as test function in (4.1) and get Then, by using Lemma 4.3 with the function g(u) = (u K + ρ) β we get where U (x, y) =| |u(x)| − |u(y) | |. Now, using the support of u K we have where r = p * s and r = p * s p * s −1 . By using Sobolev inequality given in [26, Theorem 1], we get where T p,s is a nonnegative constant and the last inequality follows from triangle inequality and (u K + ρ) β+p−1 ≥ ρ p−1 (u K + ρ) β . Using all these estimates, we now have where C = C(p) > 0 is a constant. By convexity of the map t → t p , we can show that 1 β Using this we can also check that Hence we have for C = C(p) > 0 is constant. We now choose and let β ≥ 1 be such that In addition, if we let τ = βr and ν = r pr > 1 since we have assumed N > sp(p + 1), then the above inequality reduces to Now, we iterate (4.3) using τ 0 = r and let τ m+1 = ντ m = ν m+1 r which gives Taking limit as m → 0 in (4.4), we finally get Since u K ≤ (u − 1) + , using the triangle inequality in the above inequality we get, for some constant C = C(p) > 0. If we now let K → ∞, we get Hence in particular, we say that u ∈ L ∞ (Ω).

Multiplicity result in case (II).
This section is devoted to prove our second main result that is Theorem 2.6 using the method of approximation. We follow [3] here. Let us denote the energy functional I λ : X → R ∪ {±∞} corresponding to the problem (1.1) for Case (II) by Now for > 0, we consider the following approximated problem in Ω, for which the corresponding energy functional is given by It is easy to verify that I λ, ∈ C 1 (X, R), I λ, (0) = 0 and I λ, (v) ≤ I 0, (v) for all 0 ≤ v ∈ X. From [30], we have the existence of the first nonnegative eigenfunction e 1 ∈ X ∩ L ∞ (Ω) corresponding to the first eigenvalue λ 1 satisfying the equation W.l.o.g. we may assume that e 1 L ∞ (Ω) = 1. Our next Lemma states that I λ, satisfies the Mountain Pass Geometry.
Proof. We fix l = |Ω| 1/( p * s r+1 ) . Then using Hölder's inequality and Lemma 2.1 we get that for some positive constant C independent of v. We now observe that which implies that it is possible to choose k ∈ (0, 1) sufficiently small and to set which is a positive constant and since ρ, R depends on k, r, p, |Ω|, C so does Λ. We know that which gives if λ ∈ (0, Λ). Lastly, it is easy to see that I 0, (te 1 ) → −∞ as t → +∞, which implies that we can choose T > R such that I 0, (T e 1 ) < −1. Hence which completes the proof.
As a consequence of Lemma 5.1, we have Our next Lemma ensures that I λ, satisfies the Palais Smale (P S) c condition.
Proposition 6. I λ, satisfies the (P S) c condition, for any c ∈ R, that is if {u k } ⊂ X is a sequence satisfying I λ, (u k ) → c and I λ, (u k ) → 0 (6.1) as k → ∞, then {u k } contains a strongly convergent subsequence in X.
Proof. Let {u k } ⊂ X satisfies (6.1) then we claim that {u k } must be bounded in X. To see this using (5.2), we obtain for some positive constant C (independent of k), where we have used the embedding result, Lemma 2.1 and the fact 0 < q(x) < 1 in Ω. Due to the same reasoning, we obtain for some positive constant C independent of k. Thus inserting (6.3) into (6.2), for some positive constant C 1 (independent of k), we get Also from (6.1) it follows that for k large enough Combining (6.4) and (6.5), our claim follows since p > 1. By reflexivity of X, there exists u 0 ∈ X such that up to a subsequence, u k u 0 weakly in X as k → ∞. Claim: u k → u 0 strongly in X as k → ∞. For convenience, we denote by Then by (6.1), we have that Indeed, u k u 0 weakly in X implies that . Now it is easy to see that which proves our statement. Also |(u + k + ) −q(x) u 0 | ≤ −q(x) u 0 and Ω | −q(x) u 0 |dx ≤ (1 + − q L ∞ (Ω) ) Ω |u 0 | dx < +∞. Thus Lebesgue Dominated convergence theorem gives that Since u k → u 0 a.e. in Ω and for any measurable subset E of Ω we have so from Vitali convergence theorem, it follows that Similarly, we have for some positive constants C 3 , C 4 , α and β. Therefore Vitali convergence theorem gives and lim k→∞ Ω Using (6.7), (6.8), (6.9), (6.10) and (6.11) into (6.6), we obtain From Hölder's inequality, we get that which proves our claim.

P. GARAIN AND T. MUKHERJEE
Moreover, since 0 ≤ v (v + ) q(x) ≤ v 1−q(x) , using (6.3) together with Vitali convergence theorem, we get λ lim Therefore for every φ ∈ X, we have Using Lemma 2.3 we can choose φ = v 0 as a test function in (1.1) to deduce that Hence we obtain lim which gives the strong convergence of v to v 0 in X. Now by the Vitali convergence theorem, we get which together with the strong convergence of v implies lim →0 I λ, (v ) = I λ (v 0 ). Hence from (6.12) we get ζ 0 = ν 0 .