Fractional Sobolev inequalities associated with singular problems

TL;DR

This paper investigates Sobolev inequalities related to singular problems involving the fractional p-Laplacian operator within bounded domains, contributing to the understanding of fractional Sobolev spaces and singular differential operators.

Contribution

It introduces new Sobolev inequalities tailored for singular problems with the fractional p-Laplacian, expanding the theoretical framework for these operators.

Findings

Established fractional Sobolev inequalities for singular problems

Extended classical inequalities to fractional p-Laplacian context

Provided mathematical tools for analyzing singular fractional PDEs

Abstract

In this paper we consider Sobolev inequalities associated with singular problems for the fractional $p$-Laplacian operator in a bounded domain of $\mathbb{R}^{N}$, $N\geq 2$.

Equations

$$u\mapsto {\left({\left[u\right]}_{s,p}^p+{\Vert u\Vert }_p^p\right)}^{\frac{1}{p}},$$

$${\left[u\right]}_{s,p}:={\left(\int {\int }_{{\mathbb{R}}^{2N}}\frac{{\left|u(x)-u(y)\right|}^p}{{\left|x-y\right|}^{N+sp}}\mathrm{d}x\mathrm{d}y\right)}^{\frac{1}{p}}$$

$${\Vert u\Vert }_p^p\le C_{N,s,p,\Omega }{\left[u\right]}_{s,p}^p,\forall u\in C_c^{\infty }(\Omega ),$$

$$W_0^{s,p}(\Omega )=\left\{u\in L^p({\mathbb{R}}^N):{\left[u\right]}_{s,p}<\infty \mathrm{and}u=0\mathrm{in}{\mathbb{R}}^N\setminus \Omega \right\}$$

$$1\leq r<p_{s}^{\star}:=\left\{\begin{array}[c]{lcl}\frac{Np}{N-sp}&\mathrm{if}&N>sp\\ \infty&\mathrm{if}&N\leq sp,\end{array}\right.$$

$$\left\{\begin{array}[c]{lcl}(-\Delta_{p})^{s}\,u=\dfrac{\omega}{u^{\alpha}}&\mathrm{in}&\Omega\\ u>0&\mathrm{in}&\Omega\\ u=0&\mathrm{on}&\mathbb{R}^{N}\setminus\Omega,\end{array}\right.$$

$$(-{\Delta }_p{)}^{s}u(x)=\underset{ϵ\to 0^{+}}{\lim }{\int }_{{\mathbb{R}}^N\setminus B_{ϵ}(x)}\frac{|u(x)-u(y){|}^{p-2}(u(x)-u(y))}{|x-y{|}^{N+sp}}\mathrm{d}y.$$

$$C{\left({\int }_{\Omega }{\left|v\right|}^{1-\alpha }\omega \mathrm{d}x\right)}^{\frac{p}{1-\alpha }}\le {\left[v\right]}_{s,p}^p,\forall v\in W_0^{s,p}(\Omega ).$$

$${\lambda }_{\alpha }={\left[u_{\alpha }\right]}_{s,p}^{p(\frac{1-\alpha -p}{1-\alpha })},$$

$${\lambda }_{\alpha }{\left({\int }_{\Omega }{\left|v\right|}^{1-\alpha }\omega \mathrm{d}x\right)}^{\frac{p}{1-\alpha }}={\left[v\right]}_{s,p}^p$$

$$C\exp \left(\frac{p}{{\Vert \omega \Vert }_1}{\int }_{\Omega }(\log \left|v\right|)\omega \mathrm{d}x\right)\le {\left[v\right]}_{s,p}^p,\forall v\in W_0^{s,p}(\Omega ).$$

$$\mu :=\underset{\alpha \to 1^{-}}{\lim }{\lambda }_a{\Vert \omega \Vert }_1^{\frac{p}{1-\alpha }},$$

$$\mu \exp \left(\frac{p}{{\Vert \omega \Vert }_1}{\int }_{\Omega }(\log \left|v\right|)\omega \mathrm{d}x\right)={\left[v\right]}_{s,p}^p$$

$$\left\{\begin{array}[c]{lcl}(-\Delta_{p})^{s}\,u=\dfrac{\mu}{\left\|\omega\right\|_{1}}\dfrac{\omega}{u}&\mathrm{in}&\Omega\\ u>0&\mathrm{in}&\Omega\\ u=0&\mathrm{on}&\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$$

$$\left\{\begin{array}[c]{lcl}(-\Delta_{p})^{s}\,u_{n}=\dfrac{\omega_{n}}{(u_{n}+\frac{1}{n})^{\alpha}}&\mathrm{in}&\Omega\\ u_{n}>0&\mathrm{in}&\Omega\\ u_{n}=0&\mathrm{on}&\mathbb{R}^{N}\setminus\Omega.\end{array}\right.$$

$$⟨(-{\Delta }_p{)}^{s}u,v⟩:=\int {\int }_{{\mathbb{R}}^{2N}}\frac{{\left|u(x)-u(y)\right|}^{p-2}(u(x)-u(y))(v(x)-v(y))}{{\left|x-y\right|}^{N+sp}}\mathrm{d}x\mathrm{d}y,$$

$$\tilde{v}(x,y)=v(x)-v(y),$$

$$⟨(-{\Delta }_p{)}^{s}u,v⟩=\int {\int }_{{\mathbb{R}}^{2N}}\frac{{\left|\tilde{u}(x,y)\right|}^{p-2}\tilde{u}(x,y)\tilde{v}(x,y)}{{\left|x-y\right|}^{N+sp}}\mathrm{d}x\mathrm{d}y$$

$$⟨(-{\Delta }_p{)}^s{u}_2-(-{\Delta }_p{)}^s{u}_1,u_2-u_1⟩=\int {\int }_{{\mathbb{R}}^{2N}}\frac{{\left|\widetilde{u_2}\right|}^{p-2}\widetilde{u_2}-{\left|\widetilde{u_1}\right|}^{p-2}\widetilde{u_1}}{{\left|x-y\right|}^{N+sp}}(\widetilde{u_2}-\widetilde{u_1})\mathrm{d}x\mathrm{d}y.$$

$$v_{+}:=\max \left\{v,0\right\}\mathrm{and}{r}^{\prime }:=\frac{r}{r-1}.$$

$$\left|\left|u(x)\right|-\left|u(y)\right|\right|<\left|u(x)-u(y)\right|\mathrm{whenever}u(x)u(y)<0.$$

$${\Vert u\Vert }_{\theta }^p\le S_{\theta }{\left[u\right]}_{s,p}^p,\forall u\in W_0^{s,p}(\Omega ).$$

$$u\ge C_{{\Omega }^{\prime }}a.e.\mathrm{in}{\Omega }^{\prime }$$

$$⟨(-{\Delta }_p{)}^{s}u,\phi ⟩={\int }_{\Omega }\frac{\omega \phi }{u^{\alpha }}\mathrm{d}x.$$

$$⟨(-{\Delta }_p{)}^{s}u,\phi ⟩={\int }_{\Omega }\frac{\omega \phi }{u^{\alpha }}\mathrm{d}x,\forall \phi \in W_0^{s,p}(\Omega ).$$

$$\left|{\int }_{\Omega }\frac{\omega v}{u^{\alpha }}\mathrm{d}x\right|\le {\left[u\right]}_{s,p}^{p-1}{\left[v\right]}_{s,p},\forall v\in W_0^{s,p}(\Omega ).$$

$$\left|{\int }_{\Omega }\frac{\omega v}{u^{\alpha }}\mathrm{d}x\right|$$

$$\le \underset{n\to \infty }{\mathrm{liminf}}{\int }_{\Omega }\frac{\omega {\xi }_n}{u^{\alpha }}\mathrm{d}x$$

$$=\underset{n\to \infty }{\mathrm{liminf}}⟨(-{\Delta }_p{)}^{s}u,{\xi }_n⟩$$

$$\le {\left[u\right]}_{s,p}^{p-1}\underset{n\to \infty }{\lim }{\left[{\xi }_n\right]}_{s,p}={\left[u\right]}_{s,p}^{p-1}{\left[\left|v\right|\right]}_{s,p}\le {\left[u\right]}_{s,p}^{p-1}{\left[v\right]}_{s,p}.$$

$$\underset{n\to \infty }{\lim }\left|{\int }_{\Omega }\frac{\omega \left({\phi }_n-\phi \right)}{u^{\alpha }}\mathrm{d}x\right|\le \underset{n\to \infty }{\lim }{\left[u\right]}_{s,p}^{p-1}{\left[{\phi }_n-\phi \right]}_{s,p}=0,$$

$$\underset{n\to \infty }{\lim }{\int }_{\Omega }\frac{\omega {\phi }_n}{u^{\alpha }}\mathrm{d}x={\int }_{\Omega }\frac{\omega \phi }{u^{\alpha }}\mathrm{d}x.$$

Full text

2010 Mathematics Subject Classification. Primary 35A23; 35R11; Secondary 35A15.

Keywords: Best constants, fractional $p$-Laplacian, singular problem, Sobolev inequalities.

1 Introduction

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N}$ ($N\geq 2$) and, for $0<s<1<p<\infty,$ let $W_{0}^{s,p}(\Omega)$ denote the fractional Sobolev space defined as the completion of $C_{c}^{\infty}(\Omega)$ with respect to the norm

[TABLE]

where

[TABLE]

is the Gagliardo semi-norm and $\left\|\cdot\right\|_{r}$ denotes the standard norm of $L^{r}(\Omega),$ $1\leq r\leq\infty$ (a notation that will be used in the whole paper).

Thanks to the fractional Poincaré inequality (see [6, Lemma 2.4]),

[TABLE]

$\left[\cdot\right]_{s,p}$ is a norm in $W_{0}^{s,p}(\Omega)$ equivalent to (1). Thus,

[TABLE]

equipped with the norm $\left[\cdot\right]_{s,p}$ is a Banach space. Moreover, $W_{0}^{s,p}(\Omega)$ is uniformly convex and compactly embedded into $L^{r}(\Omega),$ for all

[TABLE]

continuously embedded into $L^{p_{s}^{\star}}(\Omega)$ when $N>sp,$ and compactly embedded into the Hölder space $C^{s-\frac{N}{p}}(\Omega)$ when $N<sp$ (see [6, Lemma 2.9]). We refer the reader to [13] for a self-contained exposition on the fractional Sobolev spaces.

In this paper we will consider the Sobolev inequalities associated with the fractional, singular problem

[TABLE]

where $0<\alpha\leq 1,$ $\omega$ is a nonnegative (weight) function in $L^{r}(\Omega)\setminus\left\{0\right\},$ for some $r\geq 1,$ and $(-\Delta_{p})^{s}$ denotes the fractional $p$-Laplacian, formally defined by

[TABLE]

In the case $0<\alpha<1$ the Sobolev inequality associated with (4) takes the form

[TABLE]

We will prove that the best (i.e. the larger) constant $C$ in (5) is

[TABLE]

where $u_{\alpha}$ denotes the only weak solution of (4). We also will show that

[TABLE]

if, and only if, $v$ is a scalar multiple of $u_{\alpha}.$

By means of a limit procedure (when $\alpha\rightarrow 1^{-}$) we will deduce the following Sobolev inequality

[TABLE]

Moreover, we will prove that the best constant $C$ in this inequality is

[TABLE]

provided that it is finite, and that

[TABLE]

if, and only if, $v$ is a scalar multiple of the only weak solution of the singular problem

[TABLE]

Our approach here is based on that developed in [14], where we have considered the local, singular equation $-\operatorname{div}\left(\left|\nabla u\right|^{p-2}\nabla u\right)=u^{-1}.$ Here, besides the technical difficulties related to the nonlocal operator, we also have to deal with a non-constant weight $\omega\in L^{r}(\Omega).$

The literature on singular problems for equations of the form $\mathcal{L}u=\omega u^{-\alpha}$ has primarily focused on local operators as the Laplacian, $\mathcal{L}u=-\operatorname{div}\nabla u$ (see [2, 4, 7, 11, 18, 19, 23]), or the $p$-Laplacian, $\mathcal{L}u=-\operatorname{div}\left(\left|\nabla u\right|^{p-2}\nabla u\right),$ $p>1$ (see [1, 10, 14, 15, 16, 21]).

As regarding to nonlocal (fractional) operators, the literature on singular problems is quite recent and more restricted to $\mathcal{L}u=(-\Delta_{p})^{s}\,u$ (see [3, 8]). Furthermore, according to our knowledge, Sobolev-type inequalities associated with fractional singular problems have not been investigated up to now.

In general, the energy functional associated with a singular problem of the form $\mathcal{L}u=\omega u^{-\alpha}$ is not differentiable. This fact makes very difficult the direct application of variational methods for proving existence of solutions for this kind of problem. In order to overcome this issue (in the cases where $\mathcal{L}$ is a local operator), authors have employed the sub-super solutions method (see [7, 19, 21]) or a method of approximation by nonsigular problems introduced in [4] by Boccardo and Orsina (see [2, 10]). Recently, in [8], the latter method was applied to (4) in order to obtain the existence of a weak solution, in the case $0<\alpha\leq 1,$ and also the existence of a solution in $W_{loc}^{1,s}(\Omega),$ in the case $\alpha>1.$ We remark that singular problems for equations of the form $\mathcal{L}u=\omega u^{-\alpha}$ might not have weak solutions (in the standard sense) when $\alpha>1$ and $\omega$ is a general positive weight (see [19]). This fact is related to the singularity of the problem when the support of $\omega$ intercepts the boundary $\partial\Omega.$ In fact, if $\alpha>1$ and the support of $\omega$ is contained in a proper subdomain of $\Omega,$ the singular problem (4) has a unique weak solution (see Remark 2.5.3).

In order to make this paper self-contained we will present, in Section 2, results of existence, uniqueness and boundedness (in $L^{\infty}$) for the singular problem (4). The existence will be proved by applying the approximation method by Boccardo and Orsina, which consists in finding a solution as the limit of the sequence $\left\{u_{n}\right\}_{n\in\mathbb{N}}\subset W_{0}^{s,p}(\Omega)$ satisfying

[TABLE]

Many of the results presented in Section 2 are contained in [3] (for $p=2$) and [8] (for $p>1$), but we will contribute with some additional information. For example, we will prove that $\left[u_{n}\right]_{s,p}\leq\left[u_{n+1}\right]_{s,p}$ for all $n\in\mathbb{N}.$ This property makes simpler the proof that $u_{n}$ converges strongly to a solution of (4) when $\left\{u_{n}\right\}_{n\in\mathbb{N}}$ is bounded in $W_{0}^{s,p}(\Omega).$ It also holds true for the local version of the problem.

Our main results, related to the Sobolev inequalities (5) and (6), will be proved in the Sections 3 and 4, respectively.

2 The fractional singular problem

In this section we will provide a framework for the fractional singular problem (4). First, we will present results of uniqueness and boundedness for the singular problem (4). In the sequence we will study a family of nonsingular problems whose solutions approach the solution of (4) when it exists. At last, we will present a result of existence for (4) in the case $0<\alpha\leq 1.$

2.1 Preliminaries

Let us first fix the notation that will be used in the whole paper.

The duality pairing corresponding to the fractional $p$-Laplacian is defined as

[TABLE]

where $u,v\in W_{0}^{s,p}(\Omega).$ For the sake of clarity we will use the following notation

[TABLE]

which allows us to write

[TABLE]

and

[TABLE]

We will adopt the standard notations $v_{+}$ and $r^{\prime}$ for, respectively, the positive part of a function $v$ and the Hölder conjugate of a number $r>1.$ Thus,

[TABLE]

Remark 2.1.1

If a function $u\in W_{0}^{s,p}(\Omega)$ changes sign in $\Omega$ then $\left[\left|u\right|\right]_{s,p}^{p}<\left[u\right]_{s,p}^{p}.$ This stems from the following fact

[TABLE]

The symbol $S_{\theta}$ will denote, for each $\theta\in[1,p_{s}^{\star}),$ a positive constant satisfying

[TABLE]

The existence of such a constant comes from the continuity of the embedding $W_{0}^{s,p}(\Omega)\hookrightarrow L^{\theta}(\Omega).$ Accordingly, the symbol $S_{p_{s}^{\star}}$ will be used to denote the constant relative to the combined case $r=p_{s}^{\star}$ and $N>sp,$ since the embedding $W_{0}^{s,p}(\Omega)\hookrightarrow L^{p_{s}^{\star}}(\Omega)$ is also continuous in this case.

Definition 2.1.2

We say that $u\in W_{0}^{s,p}(\Omega)$ is a weak solution of the singular, fractional Dirichlet problem (4), with $\alpha>0,$ if the following conditions are satisfied:

1. (i)

for each subdomain $\Omega^{\prime}$ compactly contained in $\Omega$ there exists a positive constant $C_{\Omega^{\prime}}$ such that

[TABLE] 2. (ii)

for each $\varphi\in C_{c}^{\infty}(\Omega),$ one has

[TABLE]

Condition (i) arises from the singular nature of (4) and guarantees that the right-hand term of (11) is well defined. The following proposition shows that the distributional formulation (ii) leads to the traditional notion of weak solution, according to which the set of testing functions is taken to be $W_{0}^{s,p}(\Omega).$

Proposition 2.1.3

Let $u\in W_{0}^{s,p}(\Omega)$ be a weak solution as defined above. Then

[TABLE]

Proof. First we show, by using Fatou’s Lemma and Hölder inequality, that

[TABLE]

Let $v$ be an arbitrary function in $W_{0}^{s,p}(\Omega)$ and take $\left\{\xi_{n}\right\}_{n\in\mathbb{N}}\subset C_{c}^{\infty}(\Omega)$ such that $0\leq\xi_{n}\rightarrow\left|v\right|$ in $W_{0}^{s,p}(\Omega)$ and also pointwise almost everywhere. Thus,

[TABLE]

Now, we fix $\varphi\in W_{0}^{s,p}(\Omega)$ and $\left\{\varphi_{n}\right\}_{n\in\mathbb{N}}\subset C_{c}^{\infty}(\Omega)$ such that $\varphi_{n}\rightarrow\varphi$ in $W_{0}^{s,p}(\Omega).$ Then, by taking $v=\varphi_{n}-\varphi$ in (12) we obtain

[TABLE]

that is,

[TABLE]

Combining this fact with the strong convergence $\varphi_{n}\rightarrow\varphi$ we can make $n\rightarrow\infty$ in the inequality

[TABLE]

(recall that $u$ is a distributional solution), in order to obtain

[TABLE]

2.2 Uniqueness

The following lemma is well-known.

Lemma 2.2.1

Let $p>1$ and $X,Y\in\mathbb{R}^{N}\setminus\{0\},$ $N\geq 1.$ There exist positive constants $c_{p}$ and $C_{p},$ depending only on $p,$ such that

[TABLE]

and

[TABLE]

Lemma 2.2.2

Let $v_{1},v_{2}\in W_{0}^{s,p}(\Omega)\setminus\{0\}.$ There exists a positive constant $C,$ depending at most on $\Omega,N,s$ and $p,$ such that

[TABLE]

Proof. When $p\geq 2$ estimates (14) and (10) yield

[TABLE]

Now, let us consider the case $1<p<2.$ It follows from (14) that

[TABLE]

Hölder inequality yields

[TABLE]

where

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

At this point we can already prove that weak solutions are unique.

Theorem 2.2.3

The singular fractional Dirichlet problem (4), with $\alpha>0,$ has at most one weak solution.

Proof. Let us suppose that $u,v\in W_{0}^{s,p}(\Omega)$ are weak solutions of (4). Then, according to Proposition 2.1.3, we have

[TABLE]

since the integrand of the right-hand term is not positive in $\Omega.$ Thus, according to Lemma 2.2.2, we must have $\left[u-v\right]_{s,p}=0,$ showing that $u=v$ almost everywhere.

2.3 $L^{\infty}$ bounds

The following lemma can be found in [22, Lemma 2.1]. For the sake of completeness, we sketch its proof.

Lemma 2.3.1

Let $g$ be a nonnegative and nonincreasing function defined for all $t\geq k_{0}$ and such that

[TABLE]

where $C,\theta$ and $b$ are constants, $C,\theta>0$ and $b>1$. Then,

[TABLE]

where $d^{\theta}=C[g(k_{0})]^{b-1}2^{\theta b/(b-1)}.$

Proof. Let $\left\{k_{n}\right\}_{n\in\mathbb{N}}$ be the increasing sequence defined by $k_{n}:=k_{0}+d-\dfrac{d}{2^{n}}<k_{0}+d.$ Using (15) one can show, by induction, that

[TABLE]

Hence, since $0\leq g(k_{0}+d)\leq g(k_{n})$ we obtain (16), after making $n\rightarrow\infty.$

Theorem 2.3.2

Let $\alpha>0$ and $\omega\in L^{r}(\Omega),$ with $pr^{\prime}<p_{s}^{\star}.$ If $u\in W_{0}^{s,p}(\Omega)$ is positive in $\Omega$ and satisfies

[TABLE]

then $u\in L^{\infty}(\Omega).$ Moreover, for each $pr^{\prime}<\theta<p_{s}^{\star},$ one has

[TABLE]

where

[TABLE]

Proof. Let

[TABLE]

be the $k$-super-level set of $u,$ for each $k\geq 0.$ Since $(u-k)_{+}\in W_{0}^{s,p}(\Omega)$ we obtain

[TABLE]

where the first inequality can be easily checked.

Let $\theta$ be such that $pr^{\prime}<\theta<p_{s}^{\star}.$ Then, the continuity of the Sobolev embedding $W_{0}^{s,p}(\Omega)\hookrightarrow L^{\theta}(\Omega)$ and the Hölder inequality imply that

[TABLE]

so that

[TABLE]

Let $0<k_{0}\leq k<h.$ Then, $A_{h}\subset A_{k}$ and

[TABLE]

After combining this with (19) we get (recall that $k^{\alpha}\geq(k_{0})^{\alpha}$)

[TABLE]

which can be rewritten as

[TABLE]

where

[TABLE]

and

[TABLE]

It follows from Lemma 2.3.1, with $d^{\theta}=C[\left|A_{1}\right|]^{b-1}2^{\theta b/(b-1)},$ that

[TABLE]

This fact shows that $u\in L^{\infty}(\Omega)$ and

[TABLE]

where

[TABLE]

After choosing the optimal value of $k_{0}$ we obtain

[TABLE]

Remark 2.3.3

When $sp<N$ the proof of Theorem 2.3.2 applies if $r>\dfrac{N}{sp}$ and $\theta=p_{s}^{\star}.$ In this case, the estimate (17) becomes

[TABLE]

When $sp\geq N$ the condition $pr^{\prime}<p_{s}^{\star}=\infty$ naturally holds true if $r>1,$ in which case the estimate (17) is valid for any fixed $\theta>pr^{\prime}.$

2.4 A family of approximating problems

The following lemma is inspired by the proof of Lemma 9 of [20].

Lemma 2.4.1

Let $v_{1},v_{2}\in W_{0}^{s,p}(\Omega)$ and denote $v=v_{1}-v_{2}.$ Then,

[TABLE]

where

[TABLE]

Proof. Making use of the identity

[TABLE]

we obtain

[TABLE]

where $Q$ is given by (20).

Hence, we can write (recall that $v=v_{1}-v_{2}$)

[TABLE]

Since

[TABLE]

the proof is complete. (The latter inequality is very simple to check.)

In the sequel we will show that, for each $n\in\mathbb{N},$ there exists a unique function $u_{n}\in W_{0}^{s,p}(\Omega)\cap L^{\infty}(\Omega)$ such that

[TABLE]

where

[TABLE]

Proposition 2.4.2

Let $\alpha>0$ and $\omega\in L^{1}(\Omega)\diagdown\left\{0\right\},\;\omega\geq 0.$ For each $n\in\mathbb{N}$ there exists a unique function $u_{n}\in W_{0}^{s,p}(\Omega)$ satisfying (21) in the weak sense, that is,

[TABLE]

Moreover, $u_{n}$ is strictly positive in $\Omega,$ belongs to $C^{\beta_{s}}(\overline{\Omega}),$ for some $\beta_{s}\in(0,s]$ and

[TABLE]

Proof. We will obtain $u_{n}$ as a fixed point of the operator $T:L^{p}(\Omega)\longrightarrow W_{0}^{s,p}(\Omega)\hookrightarrow L^{p}(\Omega)$ that associates to each $w\in L^{p}(\Omega)$ the only weak solution $v=T(w)\in W_{0}^{s,p}(\Omega)$ of the nonsingular Dirichlet problem

[TABLE]

The function $v$ is obtained through a direct minimization method applied to the functional

[TABLE]

which is strictly convex and of class $C^{1}.$ Thus, $v$ is both the only minimizer and the only critical point of this functional. Hence,

[TABLE]

and

[TABLE]

It follows that

[TABLE]

where $S_{1}$ is a positive constant that is uniform with respect to $v$ (we have used the continuity of the embedding $W_{0}^{s,p}(\Omega)\hookrightarrow L^{1}(\Omega)$).

It follows from (26) that

[TABLE]

and thus, by taking into account the compactness of the embedding $W_{0}^{s,p}(\Omega)\hookrightarrow L^{p}(\Omega),$ we conclude that the operator $T:L^{p}(\Omega)\longrightarrow L^{p}(\Omega)$ is compact.

We are going to show, by contradiction, that $T$ is also continuous. Thus, we assume that there exist $\epsilon>0$ and $w_{k}\rightarrow w$ in $L^{p}(\Omega)$ such that

[TABLE]

where $v_{k}:=T(w_{k})$ and $v:=T(w).$ We can also assume, without loss of generality, that $\left|w_{k}\right|\rightarrow\left|w\right|$ almost everywhere in $\Omega$ (this comes from the convergence in $L^{p}(\Omega)$).

It follows from (25), with $\varphi=v_{k}-v,$ that

[TABLE]

where

[TABLE]

Therefore,

[TABLE]

Since $\left|h_{k}\right|\leq n^{\alpha}$ and $\lim\limits_{k\rightarrow\infty}\left|h_{k}\right|\rightarrow 0$ almost everywhere in $\Omega$, Dominated Convergence Theorem guarantees that

[TABLE]

At this point we consider separately the cases $1<p<2$ and $p\geq 2.$

Case $1<p<2.$ In this case, it follows from Lemma 2.2.2 and (27) that

[TABLE]

that is,

[TABLE]

where the positive constant $C_{n}$ does not depend on $k.$

After combining this inequality with (29) and (30) we obtain

[TABLE]

which contradicts (28).

Case $p\geq 2.$ In this case Lemma 2.2.2 and (29) yield

[TABLE]

Hence, after using (30) we arrive at

[TABLE]

which also contradicts (28).

We have proved that $T:L^{p}(\Omega)\longrightarrow L^{p}(\Omega)$ is compact and continuous. Moreover, (27) implies that $T$ leaves invariant the ball $\left\{w\in L^{p}(\Omega):\left\|w\right\|_{p}\leq\left(S_{1}n^{\alpha+1}\right)^{\frac{1}{p-1}}\right\}.$ Therefore, by applying Schauder’s Fixed Point Theorem we conclude that $T$ has a fixed point $u_{n}$ in this ball. Of course,

[TABLE]

in the weak sense.

Since the right-hand term of the above equation is nonnegative and belongs to $L^{\infty}(\Omega)$, we can apply the comparison principle for the fractional $p$-Laplacian (see [20, Lemma 9]) and the main result of [17] to conclude, respectively, that $u_{n}$ is nonnegative and belongs to $C^{\beta_{s}}(\overline{\Omega})$ for some $\beta_{s}\in(0,s]$ ($\beta_{s}$ does not depend neither on $\alpha$ nor on $n$).

It follows from [5, Theorem A.1] that $u_{n}>0$ almost everywhere in $\Omega.$ Let us show, by employing a nonlocal Harnack inequality proved in [12], that $u_{n}(x)>0$ for all $x\in\Omega.$ Suppose, by the way of contradiction, that $u_{n}(x_{0})=0$ for some $x_{0}\in\Omega.$ According Lemma 4.1 of [12], there exist positive constants $\epsilon$ and $c$ (with $0<\epsilon<1\leq c$) such that

[TABLE]

where $B(x_{0})$ denotes a ball centered at $x_{0}$ and contained in $\Omega.$ Since, $\inf_{B(x_{0})}u_{n}=u_{n}(x_{0})=0$ the above inequality implies that $u$ is identically null in $B(x_{0}),$ contradicting thus the fact that $u>0$ almost everywhere.

In order to prove the uniqueness of $u_{n}$ we assume that $v_{i},$ $i\in\left\{1,2\right\},$ satisfies

[TABLE]

Then,

[TABLE]

since the integrand of the right-hand term is not positive in $\Omega.$

On the other hand, by applying Lemma 2.2.2, we conclude that $\left[v_{1}-v_{2}\right]_{s,p}=0,$ showing that $v_{1}=v_{2}$ almost everywhere in $\Omega.$

We finish this proof by observing that (23) follows directly from (24), with $w=u_{n}$ and $v=T(u_{n})=u_{n}:$

[TABLE]

Proposition 2.4.3

The sequences $\left\{u_{n}\right\}_{n\in\mathbb{N}}\subset W_{0}^{s,p}(\Omega)$ and $\left\{\left[u_{n}\right]_{s,p}\right\}_{n\in\mathbb{N}}\subset(0,\infty)$ are nondecreasing, that is

[TABLE]

Proof. Let $\varphi:=u_{n}-u_{n+1}.$ It follows from (22) that

[TABLE]

Since

[TABLE]

we have $\omega_{n}\varphi_{+}\leq\omega_{n+1}\varphi_{+}$ and, hence,

[TABLE]

since the integrand above is not positive.

On the other hand, it follows from Lemma 2.4.1 that

[TABLE]

where

[TABLE]

Note that $Q(x,y)=0$ implies that $\widetilde{u_{n+1}}(x,y)=\widetilde{u_{n}}(x,y)=0,$ a pair of equalities that lead to $\varphi(x)=\varphi(y).$

After comparing (32) with (31) we can conclude that

[TABLE]

at almost every point $(x,y)\in\mathbb{R}^{2N},$ implying that $\varphi_{+}(x)=\varphi_{+}(y)$ at almost every point $(x,y).$ Since $\varphi$ is zero out of $\Omega,$ this fact implies that $\varphi_{+}=0$ almost everywhere. That is, $u_{n}-u_{n+1}\leq 0$ almost everywhere.

The second conclusion follows then from (23) with $\varphi=u_{n+1}:$

[TABLE]

In what follows $\psi\in W_{0}^{s,p}(\Omega)$ is such that

[TABLE]

Since $0\leq\omega_{1}=\min\left\{\omega,1\right\}\in L^{\infty}(\Omega)\diagdown\left\{0\right\}$ we can check that \psi\in C^{\beta_{s}}(\overline{\Omega})\for some $\beta_{s}\in(0,s]$ and that $\psi(x)>0$ $\forall\,x\in\Omega.$ (See arguments in the proof of Proposition 2.4.2, based on [5, 12, 17]).

Proposition 2.4.4

Let $u_{n}\in W_{0}^{s,p}(\Omega)$ be the weak solution of (22), with $\alpha>0,$ and $\omega\in L^{1}(\Omega)\diagdown\left\{0\right\},\;\omega\geq 0.$ We have

[TABLE]

where

[TABLE]

Proof. Let $\varphi$ be any nonnegative function in $W_{0}^{s,p}(\Omega).$ Then,

[TABLE]

It follows from the comparison principle for the fractional $p$-Laplacian that $m_{\alpha}\psi\leq u_{1}.$ This concludes the proof since $u_{1}\leq u_{n}$ for all $n\in\mathbb{N}.$

The following corollary is immediate since $\psi$ is strictly positive in $\Omega$ and continuous in $\overline{\Omega}.$

Corollary 2.4.5

Let $\Omega^{\prime}$ be an arbitrary subdomain compactly contained in $\Omega.$ There exists a positive constant $C_{\Omega^{\prime}},$ that does not depend on $n,$ such that

[TABLE]

Taking into account the monotonicity of the sequence $\left\{u_{n}\right\}_{n\in\mathbb{N}},$ let us define, for each $\alpha>0,$ the function $u_{\alpha}:\overline{\Omega}\rightarrow[0,\infty]$ by

[TABLE]

We anticipate that $u_{\alpha}(x)<\infty$ for almost every $x\in\Omega$ (see Remark 2.5.2).

Proposition 2.4.6

Let $\alpha>0$ and $\omega\in L^{1}(\Omega)\diagdown\left\{0\right\},\;\omega\geq 0.$ If the sequence $\left\{u_{n}\right\}_{n\in\mathbb{N}}$ is bounded in $W_{0}^{s,p}(\Omega),$ then it converges in $W_{0}^{s,p}(\Omega)$ to $u_{\alpha}$ and this function is the weak solution of (4).

Proof. We note that the condition (i) of Definition 2.1.2 is fulfilled, according to Corollary 2.4.5. Thus, we need to check the condition (ii).

The boundedness of $\left\{u_{n}\right\}_{n\in\mathbb{N}}$ implies that there exists a subsequence $\left\{u_{n_{k}}\right\}_{k\in\mathbb{N}}$ converging to a function $u,$ weakly in $W_{0}^{s,p}(\Omega)$ and pointwise almost everywhere. This implies that $u=u_{\alpha}$ almost everywhere, so that $u_{\alpha}\in W_{0}^{s,p}(\Omega).$

Thus, by applying (23) with $\varphi=u_{\alpha}$ we obtain

[TABLE]

Combining this fact with the monotonicity of $\left\{\left[u_{n}\right]_{s,p}^{p}\right\}_{k\in\mathbb{N}}$ we get

[TABLE]

where the latter inequality stems from the weak convergence $u_{n_{k}}\rightharpoonup u_{\alpha}.$

We have concluded that

[TABLE]

and hence we obtain the strong convergence $u_{n}\rightarrow u_{\alpha}$.

This convergence and the Corollary 2.4.5 allow us to pass to the limit, when $n\rightarrow\infty,$ in

[TABLE]

in order to obtain

[TABLE]

This concludes the proof that $u_{\alpha}$ is a weak solution of (4).

The next result is a reciprocal of Proposition 2.4.6.

Proposition 2.4.7

Let $\alpha>0$ and $\omega\in L^{1}(\Omega)\diagdown\left\{0\right\},\;\omega\geq 0.$ Suppose that $u\in W_{0}^{s,p}(\Omega)$ is a weak solution of (4). Then, $\left\{u_{n}\right\}_{n\in\mathbb{N}}$ converges in $W_{0}^{s,p}(\Omega)$ to $u$ and $u=u_{\alpha}$.

Proof. Let $\varphi=(u_{n}-u)_{+}.$ On the one hand, according to Lemma 2.4.1, we have

[TABLE]

On the other hand,

[TABLE]

Thus, by repeating the arguments in the proof of Proposition 2.4.3 we can conclude that $u_{n}\leq u$ almost everywhere.

Hence, by using (23), we obtain the boundedness of the sequence $\left\{u_{n}\right\}_{n\in\mathbb{N}}$ in $W_{0}^{s,p}(\Omega):$

[TABLE]

Consequently, according to Proposition 2.4.6, $\left\{u_{n}\right\}_{n\in\mathbb{N}}$ converges in $W_{0}^{s,p}(\Omega)$ to $u_{\alpha}$ and this function is the only solution of (4). Therefore, $u=u_{\alpha}.$

2.5 Existence for the singular problem

In the sequel we will use the following notation

[TABLE]

Theorem 2.5.1

Let $0<\alpha\leq 1$ and $\omega\in L^{r}(\Omega),$ with $r\geq r_{\alpha}.$ The sequence $\left\{u_{n}\right\}_{n\in\mathbb{N}}$ is bounded in $W_{0}^{s,p}(\Omega).$ Consequently, it converges in $W_{0}^{s,p}(\Omega)$ to $u_{\alpha}$ and this function is the weak solution of (4).

Proof. We will assume in this proof, without loss of generality, that $r=r_{\alpha}$ (note that $L^{r}(\Omega)\hookrightarrow L^{r_{\alpha}}(\Omega)$ whenever $r\geq r_{\alpha}$).

According to Proposition 2.4.6, we need only to show that the sequence $\left\{u_{n}\right\}_{n\in\mathbb{N}}$ is bounded in $W_{0}^{s,p}(\Omega).$

We have

[TABLE]

where the equality follows from (22). Thus, $\left[u_{n}\right]_{s,p}^{p}\leq\left\|\omega\right\|_{1}=\left\|\omega\right\|_{r_{\alpha}},$ when $\alpha=1.$

In the case $0<\alpha<1,$ by applying Hölder inequality to (36), we obtain

[TABLE]

Hence, when $sp<N$ we have $(1-\alpha)(r_{\alpha})^{\prime}=p_{s}^{\star}$, so that

[TABLE]

It follows that $\left\{u_{n}\right\}_{n\in\mathbb{N}}$ is bounded in $W_{0}^{s,p}(\Omega)$ and

[TABLE]

At last, for $sp\geq N$ we have $(1-\alpha)(r_{\alpha})^{\prime}=1,$ so that, by (37),

[TABLE]

Therefore, $\left\{u_{n}\right\}_{n\in\mathbb{N}}$ is bounded in W_{0}^{s,p}(\Omega)\and

[TABLE]

Remark 2.5.2

Theorem 2.5.1 guarantees that if $0<\alpha\leq 1$ and $\omega\in L^{r}(\Omega),$ with $r\geq r_{\alpha},$ then $u_{\alpha}(x)<\infty$ for almost every $x\in\Omega.$ The same holds true if $\alpha>1$ and $\omega\in L^{1}(\Omega).$ Indeed, in [8, Lemma 3.4] the authors proved that, under these hypotheses, the sequence $\left\{u_{n}^{(\alpha-1+p)/p}\right\}_{n\in\mathbb{N}}$ is bounded in $W_{0}^{s,p}(\Omega).$ This fact and the monotonicity of $\left\{u_{n}\right\}_{n\in\mathbb{N}}$ imply that $u_{\alpha}^{(\alpha-1+p)/p}\in L^{1}(\Omega),$ so that $u_{\alpha}(x)<\infty$ for almost every $x\in\Omega.$

Remark 2.5.3

When $\alpha>1,$ we have

[TABLE]

Thus, if $\omega$ belongs to $L^{1}(\Omega)$ and vanishes in $\Omega\setminus\Omega^{\prime},$ for some proper subdomain $\Omega^{\prime}$ of $\Omega,$ then

[TABLE]

which shows that $u_{\alpha}$ is the only weak solution of (4).

3 Sobolev inequality associated with $0<\alpha<1$

In this section we consider $0<\alpha<1$ and $\omega\in L^{r}(\Omega),$ with $r\geq r_{\alpha},$ where $r_{\alpha}$ is defined by (35). Thus, according to Theorem 2.5.1, the existence of the unique weak solution $u_{\alpha}$ of the singular problem (4) is guaranteed.

In order to derive the Sobolev inequality (5) we will first show that $u_{\alpha}$ minimizes the energy functional $E_{\alpha}:W_{0}^{s,p}(\Omega)\longrightarrow\mathbb{R},$ associated with the singular problem (4), defined by

[TABLE]

Since $E_{\alpha}$ is not differentiable we will obtain its minimizer as the limit of the sequence $\left\{u_{n}\right\}_{n\in\mathbb{N}}$ by taking advantage that $u_{n}$ minimizes the energy functional $E_{n}:W_{0}^{s,p}(\Omega)\longrightarrow\mathbb{R}$ associated with (21), which is defined by

[TABLE]

where $G_{n}:\mathbb{R}\rightarrow\mathbb{R}$ is the increasing $C^{1}$ function

[TABLE]

(as usual, $t_{\pm}=\max\left\{\pm t,0\right\}$).

One can easily see that $E_{n}$ is of class $C^{1}$ and

[TABLE]

Thus, nonnegative critical points of $E_{n}$ are weak solutions of (21). Moreover, by making use of standard arguments one can also check that $E_{n}$ is coercive and bounded from below. All of these features of $E_{n}$ allow one to verify that $E_{n}$ attains its minimum value at a function $v_{n}\in W_{0}^{s,p}(\Omega).$ Since $E_{n}(v_{+})\leq E_{n}(v)$ for all $v\in W_{0}^{s,p}(\Omega)$ one has $v_{n}\geq 0.$ Of course, the minimizer $v_{n}$ is also a critical point of $E_{n},$ that is,

[TABLE]

Therefore, $v_{n}=u_{n}$ since $u_{n}$ is the only nonnegative function satisfying (22).

Proposition 3.0.1

The function $u_{\alpha}$ minimizes the energy functional $E_{\alpha}.$

Proof. Recall that $u_{n}\rightarrow u_{\alpha}$ strongly in $W_{0}^{s,p}(\Omega)$ and that $u_{n}\leq u_{\alpha}.$ Thus, $\left[u_{n}\right]_{s,p}^{p}\rightarrow\left[u_{\alpha}\right]_{s,p}^{p},$

[TABLE]

and

[TABLE]

These facts show that $E_{n}(u_{n})\rightarrow E_{\alpha}(u_{\alpha}).$

For each $v\in W_{0}^{s,p}(\Omega)$ we have

[TABLE]

and

[TABLE]

so that $E_{n}(v)\rightarrow E_{\alpha}(v).$

Therefore, observing that $E_{n}(u_{n})\leq E_{n}(v)$ we obtain

[TABLE]

In order to simply the notation in the sequence, let us define

[TABLE]

and

[TABLE]

Of course, $U_{\alpha}\in\mathcal{M}_{\alpha}.$

Theorem 3.0.2

One has

[TABLE]

Proof. Since $u_{\alpha}$ is a weak solution of (4) we have

[TABLE]

so that

[TABLE]

what is the first equality in (39).

In order to prove the second equality in (39) let us fix $v\in\mathcal{M}_{\alpha}.$ It follows from (40) that

[TABLE]

Now, for any $t>0$ we have

[TABLE]

that is

[TABLE]

By choosing

[TABLE]

we obtain

[TABLE]

so that

[TABLE]

This fact implies that

[TABLE]

and then the first equality in (39) shows that this infimum is reached at $U_{\alpha}.$

From now on we denote the minimum in (39) by $\lambda_{\alpha},$ that is,

[TABLE]

Corollary 3.0.3

The inequality

[TABLE]

holds if, and only if, $C\leq\lambda_{\alpha}.$

Proof. Since

[TABLE]

it follows from Theorem 3.0.2 that (42) holds for any $C\leq\lambda_{a}.$ We can see from (41) that if $C>\lambda_{\alpha}$ then (42) fails at some $v\in\mathcal{M}_{\alpha}.$

Proposition 3.0.4

The only minimizers of the functional $v\mapsto[v]_{s,p}^{p}$ on $\mathcal{M}_{\alpha}$ are $U_{\alpha}$ and $-U_{\alpha}.$ Therefore, if

[TABLE]

for some $v\in W_{0}^{s,p}(\Omega)\setminus\left\{0\right\},$ then $v=kU_{\alpha}$ for some constant $k.$

Proof. Let $\Phi\in\mathcal{M}_{\alpha}$ be such that $\lambda_{\alpha}=\left[\Phi\right]_{s,p}^{p}.$ We observe from Remark 2.1.1 that $\Phi$ does not change sign in $\Omega.$ Indeed, otherwise we would arrive at the following absurd, since $\left|\Phi\right|\in\mathcal{M}_{\alpha}:$

[TABLE]

Thus, without loss of generality, we assume that $\Phi\geq 0$ in $\Omega$ (otherwise, we proceed with $-\Phi$ instead of $\Phi$).

Since $\Phi,$ $U_{\alpha},$ $\omega\geq 0$ and $0<1-\alpha<1$ we have

[TABLE]

showing that

[TABLE]

Observing that $h^{-1}(\frac{1}{2}\Phi+\frac{1}{2}U_{\alpha})\in\mathcal{M}_{\alpha}$ and

[TABLE]

we can conclude that: $h=1,$ $\frac{1}{2}\Phi+\frac{1}{2}U_{\alpha}\in\mathcal{M}_{\alpha}$ and

[TABLE]

We recall that the functional $v\mapsto\left[v\right]_{s,p}$ is strictly convex over $W_{0}^{s,p}(\Omega).$ Thus, the second equality in (44) implies that $\Phi=U_{\alpha}.$

We have shown that $\lambda_{\alpha}=\left[\Phi\right]_{s,p}^{p}$ for some $\Phi\in\mathcal{M}_{\alpha}$ if, and only if, either $\Phi=U_{\alpha}$ or $\Phi=-U_{\alpha}.$ Thus, if (43) holds true for some $v\in W_{0}^{s,p}(\Omega)\setminus\left\{0\right\},$ then either $v=\theta^{-1}U_{\alpha}$ or $v=-\theta^{-1}U_{\alpha},$ where $\theta=\left({\displaystyle\int_{\Omega}}\left|v\right|^{1-\alpha}\omega\mathrm{d}x\right)^{-\frac{1}{1-\alpha}}$ (since $\Phi=\theta v\in\mathcal{M}_{\alpha}$ and $\lambda_{\alpha}=\left[\theta v\right]_{s,p}^{p}$).

4 Sobolev inequality associated with $\alpha=1$

According to (41)

[TABLE]

We would like to pass to the limit, as $\alpha\rightarrow 1^{-},$ in the above inequality. For this, we need the following two lemmas.

Lemma 4.0.1

Let $\omega\in L^{r}(\Omega),$ $r>1,$ and $v\in W_{0}^{s,p}(\Omega).$ The map

[TABLE]

is well-defined and nondecreasing.

Proof. For simplicity, let us denote $\overline{\omega}=\frac{\omega}{\left\|\omega\right\|_{1}},$ so that $\left\|\overline{\omega}\right\|_{1}=1.$ For each $q\in(0,\frac{p_{s}^{\star}}{r^{\prime}})$ we have, by Hölder’s inequality,

[TABLE]

Therefore, since the embedding $W_{0}^{s,p}(\Omega)\hookrightarrow L^{qr^{\prime}}(\Omega)$ is continuous, the map (46) is well-defined.

Now, in order to prove the monotonicity of this map, let $0<q_{1}<q_{2}<\frac{p_{s}^{\star}}{r^{\prime}}.$ By Hölder’s inequality we have

[TABLE]

implying that

[TABLE]

Lemma 4.0.2

Let $\omega\in L^{r}(\Omega),$ $r>1.$ The map

[TABLE]

is nondecreasing, for some $\alpha_{0}\in(0,1).$

Proof. Since $\lim_{\alpha\rightarrow 1^{-}}r_{\alpha}=1^{+},$ there exists $\alpha_{0}\in(0,1)$ such that $r\geq r_{\alpha}$ whenever $\alpha\in[\alpha_{0},1).$ Thus, according to Section 3, for each $\alpha\in[\alpha_{0},1)$ there exists $u_{\alpha}\in W_{0}^{s,p}(\Omega)$ such that

[TABLE]

Now, let $\alpha_{0}\leq\alpha_{1}<\alpha_{2}<1.$ We have

[TABLE]

where the second inequality comes from Lemma 4.0.1.

Remark 4.0.3

L’Hôpital’s rule and Lemma 4.0.1 show that

[TABLE]

As a consequence of Lemma 4.0.2, we can define

[TABLE]

and also conclude that

[TABLE]

However, we cannot guarantee, at least in principle, that $\mu<\infty.$ According to (45), one way of achieving this is to show the existence of a function $\varphi\in W_{0}^{s,p}(\Omega)$ satisfying

[TABLE]

or, equivalently,

[TABLE]

Apparently, the task of finding such a function $\varphi\in W_{0}^{s,p}(\Omega)$ is not simple when a general nonnegative function $\omega\in L^{r}(\Omega)$ is considered. Note, for example, that if $v\in W_{0}^{s,p}(\Omega)\cap L^{\infty}(\Omega)$ and vanishes over a part $\Omega^{\prime}$ of the support of $\omega$ in such a way that $0<{\displaystyle\int_{\Omega\setminus\Omega^{\prime}}}\omega\mathrm{d}x<\left\|\omega\right\|_{1},$ then

[TABLE]

This is what happens when $\omega\equiv 1,$ but in this case it is possible to built (see [14]) a suitable function $\varphi$ that vanishes only on $\partial\Omega$ and satisfies

[TABLE]

A simpler situation where (47) holds is when $\omega$ is compactly supported in $\Omega.$ In fact, if there exists a subdomain $\Omega^{\prime}\subset\Omega$ such that $\omega(x)=0$ for almost every $x\in\Omega\setminus\Omega^{\prime},$ then we can take a smooth function $\varphi\in W_{0}^{s,p}(\Omega)$ such that $\inf_{\Omega^{\prime}}\left|\varphi\right|=m>0$ in order to obtain

[TABLE]

Our feeling is that, in fact, $\mu<\infty$ whenever $\omega\in L^{r}(\Omega),$ with $r>1.$ But we were not able to prove this generically, even knowing that

[TABLE]

as (45) and Remark 4.0.3 show. Since this issue of generically determining the finiteness of $\mu$ goes beyond of our purposes in this paper, we will assume from now on that $\mu<\infty.$

Theorem 4.0.4

Let $\omega\in L^{r}(\Omega),$ $r>1,$ and suppose that $\mu<\infty.$ We have

[TABLE]

Proof. The proof follows immediately from (45), by making $\alpha\rightarrow 1^{-}.$

We are proceeding in the direction of proving that (49) becomes an equality for some $V\in W_{0}^{s,p}(\Omega).$ More precisely, we will show that $\mu$ is the minimum of the functional $v\mapsto\left[v\right]_{s,p}^{p}$ on the set

[TABLE]

Note that $\mathcal{M}\not=\emptyset$ if, and only if, there exists $\varphi\in W_{0}^{s,p}(\Omega)$ satisfying (48). Moreover, if $\mathcal{M}\not=\emptyset$ then $\mu<\infty.$ The reciprocal of this will follow immediately from our next theorem.

In the following results $V_{\alpha}$ denotes the function defined by

[TABLE]

where $U_{\alpha}$ is given by (38).

It is simple to check that

[TABLE]

and that

[TABLE]

Theorem 4.0.5

Let $\omega\in L^{r}(\Omega),$ $r>1,$ and suppose that $\mu<\infty.$ Then $V_{\alpha}$ converges in $W_{0}^{s,p}(\Omega)$ to a nonnegative function $V\in\mathcal{M},$ which minimizes the functional $v\mapsto\left[v\right]_{s,p}^{p}$ on $\mathcal{M}.$ Moreover, the only minimizers of this functional on $\mathcal{M}$ are $-V$ and $V.$ Consequently, the equality in (49) holds for some $v\in W_{0}^{s,p}(\Omega)$ if, and only if, $v=kV$ for some constant $k.$

Proof. Multiplying the equation in (52) by $V_{\alpha}$ and integrating over $\Omega$ we obtain

[TABLE]

Therefore,

[TABLE]

This fact implies that there exist $\alpha_{n}\rightarrow 1^{-}$ and a function $V\in W_{0}^{s,p}(\Omega)$ such that: $V_{\alpha_{n}}\rightharpoonup V$ (weakly) in $W_{0}^{s,p}(\Omega),$ $V_{\alpha_{n}}\rightarrow V$ in $L^{1}(\Omega)$ and $V_{\alpha_{n}}(x)\rightarrow V(x)$ for almost every $x\in\Omega.$ We remark that $V\geq 0$ in $\Omega$ since $V_{\alpha_{n}}>0$ in $\Omega.$

The weak convergence implies that

[TABLE]

Note from (49) that $\mu\leq\left[v\right]_{s,p}^{p}$ for every $v\in\mathcal{M}.$ Thus, by taking (53) into account, in order to conclude that $V$ minimizes the functional $v\mapsto\left[v\right]_{s,p}^{p}$ on $\mathcal{M}$ we need only to prove that $V\in\mathcal{M}.$

According to (45), we have

[TABLE]

so that

[TABLE]

Hence, in view of (53), we can conclude that

[TABLE]

Now, let us fix an arbitrary $t\in(\alpha_{0},1).$ Then, for all $n$ large enough (such that $a_{0}<t<\alpha_{n}$) we have

[TABLE]

according to (51) and Lemma 4.0.1. It is straightforward to check that the convergence $V_{\alpha_{n}}\rightarrow V$ in $L^{1}(\Omega)$ implies that

[TABLE]

Therefore,

[TABLE]

This fact and (55) show that

[TABLE]

Since

[TABLE]

we conclude that

[TABLE]

that is, $V\in\mathcal{M}.$ Thus, we have

[TABLE]

The (strong) convergence $V_{\alpha_{n}}\rightarrow V$ in $W_{0}^{s,p}(\Omega)$ then stems from the first equality in (56).

Now, let $\Phi$ be a function that attains the minimum $\mu$ on $\mathcal{M}.$ We emphasize that $\Phi$ does not change sign in $\Omega.$ Otherwise, since $\left|\Phi\right|$ also belongs to $\mathcal{M},$ we would arrive at the contradiction

[TABLE]

Thus, without loss of generality, we will assume that $\Phi\geq 0.$

Repeating the arguments developed in the proof of Proposition 3.0.4 we obtain

[TABLE]

so that

[TABLE]

Therefore,

[TABLE]

from what follows that

[TABLE]

The strict convexity of the Gagliardo semi-norm then implies that $V=\Phi.$

Since $V$ is the unique nonnegative function that attains the minimum $\mu$ on $\mathcal{M}$ we can conclude that the convergence $V_{\alpha_{n}}\rightarrow V$ in $W_{0}^{s,p}(\Omega)$ does not depend on the subsequence $\alpha_{n}$ going to $1^{-}.$

We would like to pass to the limit in (52), as $\alpha\rightarrow 1^{-},$ in order to conclude that the minimizer $V$ is the solution of the singular problem

[TABLE]

The convergence $V_{\alpha}\rightarrow V$ in $W_{0}^{s,p}(\Omega)$ shows that

[TABLE]

However, due the singular nature of the equation in (52), this convergence is not enough to directly obtain

[TABLE]

For this, we will assume that $r>\max\left\{1,\frac{N}{sp}\right\}$ in order to use the boundedness results of Subsection 2.3.

In the sequel $\psi\in W_{0}^{s,p}(\Omega)\cap C^{\beta_{s}}(\overline{\Omega})$ is the function satisfying (33).

Lemma 4.0.6

Let $\omega\in L^{r}(\Omega),$ with $r>\max\left\{1,\frac{N}{sp}\right\},$ and suppose that $\mu<\infty.$ There exist positive constants $m$ and $M$ such that

[TABLE]

Proof. Since $V_{\alpha}$ satisfies (52), we can apply Theorem 2.3.2 (with $\omega$ replaced by $\left\|\omega\right\|_{1}^{\frac{p}{1-\alpha}-1}\lambda_{\alpha}\omega$) to conclude that

[TABLE]

where $pr^{\prime}<\theta\leq p_{s}^{\star}$ (the equality only in the case $sp<N$) and

[TABLE]

Therefore,

[TABLE]

It follows that, by increasing $\alpha_{0}$ if necessary, there exists $M$ such that $\left\|V_{\alpha}\right\|_{\infty}\leq M$ for all $\alpha\in[\alpha_{0},1).$

Thus,

[TABLE]

where $\omega_{1}=\min\left\{\omega,1\right\}$ and

[TABLE]

Therefore, by the weak comparison principle we get the estimate

[TABLE]

valid in $\Omega$, for every $\alpha\in[\alpha_{0},1).$

Proposition 4.0.7

Let $\omega\in L^{r}(\Omega),$ with $r>\max\left\{1,\frac{N}{sp}\right\},$ and suppose that $\mu<\infty.$ The minimizer $V$ is the weak solution of the singular problem (57).

Proof. We recall that $\psi$ is positive in $\Omega$ and belongs to $C^{\beta_{s}}(\overline{\Omega})$ for some $0<\beta_{s}<1.$ Hence, according to the previous lemma, $V_{\alpha}$ is bounded from below by a positive constant (that is uniform with respect to $\alpha$) in each proper subdomain $\Omega^{\prime}\subset\Omega.$ This property guarantees that (59) holds. Since we have already obtained (58), the conclusion follows.

5 Acknowledgments

This work was supported by CNPq/Brazil (483970/2013-1 and 306590/2014-0) and Fapemig/Brazil (APQ-03372-16).