Abstract
We study the existence, nonexistence and qualitative properties of the solutions to the problem
where \(s\in (0,1)\), \(N>2s\), \(q>p\ge {(N+2s)}/{(N-2s)}\), \(\theta \in (0, \Lambda _{N,s})\) and \(\Lambda _{N,s}\) is the sharp constant in the fractional Hardy inequality. For qualitative properties of the solutions, we mean both the radial symmetry, that is obtained by using the moving plane method in a nonlocal setting on the whole \(\mathbb {R}^N\), and a suitable upper bound behavior of the solutions. To this last end, we use a representation result that allows us to transform the original problem into a new nonlocal problem in a weighted fractional space.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
In this paper, we consider the following singular problem
where \(s\in (0,1)\) is fixed, \((-\Delta )^s\) denotes the fractional Laplace operator which can be defined for the Schwartz class functions \(\mathcal {S}({\mathbb {R}}^N)\) as follows
\(N>2s\), \(q>p\ge 2^*_s-1=\frac{N+2s}{N-2s}\) and \(\theta \in (0, \Lambda _{N,s})\), where \(\Lambda _{N,s}\) is the sharp constant in the Hardy inequality
In the above inequality, \(\mathcal {F}(u)\) denotes the Fourier transform of u and
Moreover,
In recent years, considerable attention has been given to equations involving the fractional Laplacian operator. This motivation comes, apart from a pure mathematical point of view, also from the fact that these nonlocal structures have connection with many real-world phenomena. We refer the readers to the list (far from to be complete), [2, 3, 11,12,13,14,15, 29,30,31, 34, 37,38,39] where existence of solutions and/or regularity of solutions are studied for some nonlocal problems.
We start giving the following
Definition 1.1
Let \(s\in (0,1)\). We define the homogeneous fractional Sobolev space of order s as
namely the completion of \(C^\infty _0({\mathbb {R}}^N)\) under the norm
Using Plancherel’s identity, for \(s\in (0,1)\), \(N\ge 1\), we obtain an equivalent expression of the norm (1.3), namely
see [25]. Using the above norm-equivalence and a density argument, it follows that
see [19, Remark 1.3].
The notion of solutions to (\({\mathcal {P}}\)) that we consider in this paper is given in the following
Definition 1.2
We say \(u\in {\dot{H}}^s({\mathbb {R}}^N)\cap L^{q+1}({\mathbb {R}}^N)\) is a weak solution of (\({\mathcal {P}}\)) if for every \(\varphi \in {\dot{H}}^s({\mathbb {R}}^N)\cap L^{q+1}({\mathbb {R}}^N)\), we have
We note that in the above definition,
since \(2^*_s\le p+1<q+1\) implies \(u, \varphi \in L^{p+1}({\mathbb {R}}^N)\) by interpolation inequality.
Remark 1.1
Note that, \(0<\theta <\Lambda _{N,s}\). Therefore using (1.3), (1.4) and (1.5) we see that
is an equivalent norm to
Very recently, a great deal of attention is given to the mathematical study of the following class of semilinear elliptic problems
where f(x, u) is a superlinear function in u. The local case, i.e., when \(s = 1\) and \(f(x, u) = u^\frac{N+2}{N-2}\) has been thoroughly investigated by Terracini in [41]. Existence, uniqueness and qualitative properties of the solutions have been shown by the author. It is natural to study these type of problems in the nonlocal framework. The study of nonlocal Yamabe problem in the Euclidean space, i.e., for the case when \(\theta = 0\) and \(f(x,u)=u^{2^*_s-1}\) in (1.7) has been studied in [17]. The authors completely characterize the solutions using the moving plane technique. However for \(\theta \ne 0,\) the presence of Hardy potential requires a new understanding to deal with the problem.
In the last few years, there are several works related to Eq.(1.7); see for instance ([5, 6, 19, 22, 23] and references therein). Authors in [5] have studied the two-sided Green function estimate of fractional Hardy operator and the integral representation of solution. Dipierro et al. in [19] have considered \(f(x,u)=u^{2^*_s-1}\) and proved existence, qualitative properties and asymptotic behavior of solutions both at 0 and infinity. In [6], Bhakta et. al have studied (1.7) with \(f(x,u)=K(x){|u|^{2^*(t)-2}u}/{|x|^t}+f(x)\), where \(0\le t<2s\), \(2^*_s(t):={2(N-t)}/{N-2s}\), K is a positive continuous function on \(\mathbb {R}^{N}\), with \(K(0)=1=\lim _{|x|\rightarrow \infty }K(x)\) and f is a nonnegative nontrivial functional in the dual space of \({\dot{H}}^s({\mathbb {R}}^N)\). They have established existence of at least two positive solutions when \(K\ge 1\) and \(\Vert f\Vert _{({\dot{H}}^s)'}\) is small enough but \(f \not \equiv 0\). On the other hand, Fall and Felli in [23] have studied the more general relativistic Schrödinger type equations. We also quote the papers [7,8,9] for nonlocal equations (without Hardy potential) where the nonlinearity involved is of the form \(f(x,u)=u^p-u^q\). In the regular case, \(\theta =0\), (\({\mathcal {P}}\)) is reduced to the problem without Hardy term and it was studied in [8, 9].
In this paper, we are interested in studying the existence, nonexistence of solutions to problem (\({\mathcal {P}}\)) and we further investigate qualitative properties like radial symmetry and upper bound of solutions.
This kind of critical and supercritical exponent problem has been extensively studied when \(\theta =0\) and \(s = 1\) (local case) by Merle and Peletier in [32]. When \(\theta \ne 0\) and \(s=1\) (local case), a complete classification of the nature of singularities and of the asymptotic behavior of solutions near zero was studied in [10]. In the nonlocal case, \(s\in (0,1)\), the fractional framework introduces nontrivial difficulties that have interest in itself. Moreover due to nonlocal structure of the equation, none of the methods developed in the local case \(s= 1\) can be used for (\({\mathcal {P}}\)). There are many new novelties that need to be exploited to tackle the problem (\({\mathcal {P}}\)) which are described below.
Below we state the main results of this paper. The first one is an existence and nonexistence result, stated in the following
Theorem 1.1
Let \(0< \theta < \Lambda _{N, s}\). Then the following holds
- (i):
-
If \(q>p=2^{*}_s-1\), then (\({\mathcal {P}}\)) does not admit any solution.
- (ii):
-
If \(q>p>2^{*}_s-1\), then there exists a solution to (\({\mathcal {P}}\)).
In order to prove the first part of Theorem 1.1 we show a Pohozaev identity (see Proposition 2.2) using the representation of fractional laplacian \((-\Delta )^s\) as a harmonic extension (we give the details in Section 2). To prove the second part of our result, we deal with a constrained minimization problem (see formula 2.33) and then we use Lagrange multipliers technique to get a solution to (\({\mathcal {P}}\)). Moreover in Section 2, we show that, in some cases, weak solutions to (\({\mathcal {P}}\)) are indeed strong solutions in \({\mathbb {R}}^n \setminus \{0\}\), see Proposition 2.1.
In the next two results (see also Remark 2.1), we deduce qualitative properties of solutions to (\({\mathcal {P}}\)) when
In particular the first one is related to the radial symmetry of the solutions. We have the following
Theorem 1.2
Let \(p>2^{*}_s-1\) and
Assume that \(0< \theta < \Lambda _{N,s}\) and let u be a solution to (\({\mathcal {P}}\)). Then u is radial and radially decreasing with respect to the origin. Namely there exists some strictly decreasing function
such that
We prove the result exploiting the moving plane method, well understood in the case of local problems. In the nonlocal case we refer to [4, 17, 20, 24, 27, 28, 33, 40]. However the presence of the Hardy potential in equation (\({\mathcal {P}}\)), on the one hand, makes it difficult to use the technique developed in [17] where the authors exploited the equivalence of (\({\mathcal {P}}\)) to an appropriate integral equation and on the other hand, solutions to (\({\mathcal {P}}\)) lose (because the presence of Hardy potential) regularity at the origin. For this reason, in order to prove the radial symmetry of every solution to (\({\mathcal {P}}\)), we use an approach based on the moving plane method for weak solutions of the equation in all \(\mathbb {R}^N\) taking care of all difficulties introduced by the presence of Hardy potential.
In the second result, we establish an estimate from above of the asymptotic behavior of the solutions to (\({\mathcal {P}}\)). We have the following
Theorem 1.3
Let \(p>2^{*}_s-1\) and
Assume that \(0< \theta < \Lambda _{N,s}\) and let u be a solution to (\({\mathcal {P}}\)). Then there exists a positive constant \(C=C(p,q,s,N,\gamma _{\theta },\Vert u\Vert _{L^{q+1}({\mathbb {R}}^N)})\) such that
where \(\gamma _{\theta }\in (0, (N-2s)/2)\) is a parameter determined as the unique solution to (3.29).
This result is achieved by a delicate use of Moser iteration technique for an equivalent problem to (\({\mathcal {P}}\)) (see in particular problem (\({\mathcal {P}}\))) in a weighted fractional space. However due to the presence of super critical exponent and the unbounded domain, the standard Moser iteration technique seems to be insufficient to tackle the problem. There are many intriguing steps involved in the Moser iteration to prove the upper bound of solutions (see Proposition 3.1).
The rest of the paper is organized as follows. In Section 2, we provide a regularity result (i.e., Proposition 2.1) and we briefly discuss the different representation of fractional Laplacian using harmonic extension method. Then, we prove Theorem 1.1. In Section 3, we prove Theorem 1.2 and Theorem 1.3.
Notation. Throughout the present paper, we denote by \(C, C_1, C_2, C', {\tilde{C}}, \hat{C}, \cdots\) positive constants that may vary from line to line. If necessary, the dependence of these constants will be made precise. By \(u \gneqq 0\), we mean \(u\ge 0\) and there exists a set of positive measure where \(u>0\).
2 Existence, nonexistence and regularity results for solutions to (\({\mathcal {P}}\))
We start providing a useful result about the regularity of the solutions to the problem (\({\mathcal {P}}\)). We have the following
Proposition 2.1
Let u be a nonnegative solution of (\({\mathcal {P}}\)) with
Then \(u>0\) in \({\mathbb {R}}^N\setminus \{0\}\) and \(u\in C^\infty ({\mathbb {R}}^N\setminus \{0\})\).
Proof
Let \(x_0\in {\mathbb {R}}^N\setminus \{0\}\). Since \(|x|^{-2s}\) is bounded away from 0 and \(q+1>(p-1)\frac{N}{2s}\), using the Moser iteration method as in Theorem [8, Theorem 1.3], it can be shown that \(u\in L^\infty (B_r(x_0))\), for some \(r>0\). Since, \(x_0\) is arbitrary, it implies that \(u\in L^\infty _{loc}({\mathbb {R}}^N\setminus \{0\})\).
claim:
We prove this claim in two steps.
Step 1: In this step, we show that
Indeed, by Hölder and Hardy inequalities, we get
Step 2: In this step, we show that
For this first we note that
We also observe that since \(u\in {\dot{H}}^s({\mathbb {R}}^N)\), then
and Fubini’s theorem implies that
Fix some \(y \in {\mathbb {R}}^N\) for which the above expression holds and let \(R\in {\mathbb {R}}\) be such that \(R\ge 10|y|\). For \(|x|\ge R\), we deduce that
for some constant \(C>1\). Moreover
Therefore, using the inequality
the fact that by Sobolev inequality
and inequalities (2.3), (2.4), (2.5), it follows that
Substituting (2.6) in (2.2) completes Step 2.
Combining Step 1 and Step 2, we prove
Hence the claim follows.
Now using the Schauder estimate in the spirit of [36, Corollary 2.4 and 2.5], it follows that \(u\in C^{\beta +2s}\big (B_{\frac{r}{2}}(x_0)\big )\), for some \(\beta >0\). As a consequence, for any \(x\in {\mathbb {R}}^N\setminus \{0\}\) we can write (see problem (\({\mathcal {P}}\))) in a pointwise way that
and consequently we deduce that \(u>0\) in \({\mathbb {R}}^N\setminus \{0\}\). Indeed suppose, that is not true. That means there exists \(y_0\in {\mathbb {R}}^N\setminus \{0\}\) such that \(u(y_0)=0\), which in turn implies that u attains it’s minimum at \(y_0\). Since \((-\Delta )^su\) is defined in pointwise sense in \({\mathbb {R}}^N\setminus \{0\}\), from the integral representation (1.1) of \((-\Delta )^s\), we see that \((-\Delta )^su(y_0)<0\). On the other hand
and this leads to the contradiction.
Finally a bootstrap procedure using Schauder estimate in the spirit of [36, Corollary 2.4], yields \(u\in C^\infty ({\mathbb {R}}^N\setminus \{0\})\). \(\square\)
Remark 2.1
Proposition 2.1 indeed contains a strong maximum principle for (\({\mathcal {P}}\)). Actually all nonnegative solutions to (\({\mathcal {P}}\)) are positive providing \(q+1>(p-1){N}/{2s}\).
Moreover we have the following
Lemma 2.1
Let u be a solution of (\({\mathcal {P}}\)) with
Then \(u\rightarrow 0\) as \(|x|\rightarrow \infty\).
Proof
Thanks to the Sobolev inequality, we already have \(u\in L^{2^*_s}({\mathbb {R}}^N)\). Therefore, in order to prove the lemma, it is enough to show that \(u\in C^\sigma (\{|x|>1\})\) is uniformly continuous, for some \(\sigma >0\). To prove this, first we claim the following:
claim: There exists a constant \(C>0\) and an uniform radius \(r>0\) such that
where C and r do not depend on x.
Indeed the claim follows arguing along the same line as in the proof of [8, Theorem1.3]. A careful look on their proof would reveal that the constant C (\(L^\infty\) bound of the solution) depends only on \(p,q,s,N,\Vert u\Vert _{L^{q+1}({\mathbb {R}}^N)}\) and the radius of the support of the cut-off function that was chosen in the proof [8, Theorem1.3], more importantly it is independent of the choice of the point x. Therefore we can choose the radius of the support of the cut-off function there as 1/4, which in turn yields \(r={1}/{8}\). Hence the claim follows.
In particular, the above claim implies
where C is as above. Therefore from (\({\mathcal {P}}\)), we also have
Combining this with (2.1), it follows from the Schauder estimate [36, Corollary 2.5] that
for some \(\sigma \in (0,2s)\), where \(C_2\) does not depend on \(x_0\) and r is obtained as in the above claim. Since \(\sup _{|x|>1} u<C,\) then it follows that u is uniformly continuous for \(|x|>1\). Hence the lemma follows. \(\square\)
In the following, we recall now the other useful representation of fractional laplacian \((-\Delta )^s\) as harmonic extension [16], which we will use to establish the Pohozaev identity, Proposition 2.2. Using the harmonic extension method, the fractional laplacian \((-\Delta )^s\) can be seen as a trace class operator (see [16, 23, Appendix A]). Let \(u \in {\dot{H}}^s({\mathbb {R}}^N)\) be a solution of (\({\mathcal {P}}\)) and define \(w:=E_{s}(u)\in \dot{H}^1({\mathbb {R}}^{N+1}_+; y^{1-2s})\) its s-harmonic extension to the upper half space
that is the unique solution (see [23, Proposition 6.2]), to the following problem
where the space \(\dot{H}^1({\mathbb {R}}^{N+1}_+; y^{1-2s})\) is defined as the completion of \(C_0^\infty (\overline{{\mathbb {R}}_+^{N+1}})\) with respect to the norm
In addition
where \(\dot{H}^{-s} ({\mathbb {R}}^N)\) denotes the dual of \(\dot{H}^{s} ({\mathbb {R}}^N)\) and
Moreover, the extension operator \(E_s(u):{\dot{H}}^s({\mathbb {R}}^N)\rightarrow \dot{H}^1({\mathbb {R}}^{N+1}_+; y^{1-2s})\) is an isometry, that is, for any \(u\in {\dot{H}}^s({\mathbb {R}}^N)\), we have
see [26]. Conversely, for any \(w \in \dot{H}^1({\mathbb {R}}^{N+1}_+; y^{1-2s})\), we denote its trace on \({\mathbb {R}}^N\times \{y=0\}\) as \(Tr(w):=w(.,0)\). By [23, Proposition 6.2] this trace operator is well defined and belongs to \({\dot{H}}^{s}({\mathbb {R}}^N)\), namely there exists a (unique) linear trace operator
such that \(T(w)(x, y):=w(x,0)\) for any \(w\in C^{\infty }_c(\overline{{\mathbb {R}}^{N+1}_+})\). Moreover this trace operator satisfies
for all \(w\in \dot{H}^1({\mathbb {R}}^{N+1}_+; y^{1-2s})\). Consequently,
for all \(w\in \dot{H}^1({\mathbb {R}}^{N+1}_+; y^{1-2s})\). Inequality (2.12) is called the trace inequality. With the above representation, (2.8) can be rewritten as:
A function \(w\in \dot{H}^1({\mathbb {R}}^{N+1}_+; y^{1-2s})\) with \(Tr(w)\in L^{q+1}({\mathbb {R}}^N)\), is said to be a weak solution to (2.13) if for all \(\varphi \in \dot{H}^1({\mathbb {R}}^{N+1}_+; y^{1-2s})\) with \(Tr(\varphi )\in L^{q+1}({\mathbb {R}}^N)\), we have
Note that for any weak solution \(w \in \dot{H}^1({\mathbb {R}}^{N+1}_+; y^{1-2s})\) to (2.13), the function \(u:=\text{ Tr }(w)=w(.,0)\in {\dot{H}}^{s}({\mathbb {R}}^N)\) is a weak solution to (\({\mathcal {P}}\)).
We prove Theorem 1.1 using Pohozaev identity and constrained minimisation method. Before proving the theorem we need to prove a key proposition.
Proposition 2.2
Let \(q > 2_s^* - 1\) and u be a solution of
Then there holds the following identity
Proof
We prove this proposition using the harmonic extension method described in this section. Let us define
and
We point out that by Hardy inequality (1.5) and by (2.15) it follows \(F(u) \in L^{1}({\mathbb {R}}^N).\) Suppose, w is the harmonic extension of u. Then w is a solution of
where f is defined in (2.17). We denote a point in \({\mathbb {R}}^{N+1}_+\) as \(z = (x, y) \in {\mathbb {R}}^N\times \mathbb {R}^{+}.\) Let us denote the following sets, for \(\varepsilon> 0,\, R> 0 \ \text{ and }\ \rho > 0\):
The unit (outward) normal on \(\partial \mathcal {O}_{\varepsilon , R}\) is given by
On \(\mathcal {O}_{\varepsilon , R, \rho },\) there holds
Taking \((z, \nabla w)\) as test function and integrating by parts on \(\mathcal {O}_{\varepsilon , R, \rho }\), we obtain
Now consider
We use [23, Lemma 4.1 and Proposition 4] in our framework, i.e., we set \(m =0\) and \(g(u):= {\theta u}/{|x|^{2s}} + u^{2^* -1} -u^{q}\) in [23]. Clearly by Proposition 2.1, \(g\in C^{0,\gamma }(B_{R} \setminus B_{\rho })\) for \(\gamma \in [0, 2-2s).\) Therefore using (2.9), we can pass to the limit as \(\varepsilon \rightarrow 0,\) in the 1st integral of \(I_{\varepsilon , R, {\rho }}^{1}\) and obtain
Moreover given any \(r>0\) , we show that there exists a sequence \(\varepsilon _n\rightarrow 0\) such that
Indeed by contradiction, we would get
namely there exists \(\bar{\varepsilon }>0\) such that
for any \(\varepsilon \in (0,\bar{\varepsilon })\). This is a contradiction because integrating on \((0,\bar{\varepsilon })\), w would not belong to \(\dot{H}^1({\mathbb {R}}^{N+1}_+; y^{1-2s})\). Therefore combining (2.22) and (2.23), replacing \(\varepsilon\) with \(\varepsilon _n\) we conclude
and
Moreover we have
We claim that there exists a sequence \(R_{n'} \rightarrow +\infty ,\) such that
As we did above, if the claim is not true, then there exists \(\bar{R} > 0\) and \(C > 0\) such that
for any \(R \in (\bar{R},+\infty )\). This immediately implies that
which is a contradiction to the fact that \(w\in \dot{H}^1({\mathbb {R}}^{N+1}_+; y^{1-2s})\) and hence the claim holds. Hence
and therefore
In the same way, we define
Since \(w \in \dot{H}^1({\mathbb {R}}^{N+1}_+; y^{1-2s})\), arguing similarly as in the case of \(I^2_{R}\), we conclude there exists a sequence \(\rho _{{n''}} \rightarrow 0,\) such that
and then
Using integration by parts in (2.24) and recall (2.18), we obtain
Furthermore, letting \(\rho _{n''} \rightarrow 0\) in (2.28) along the same sequence as above (or eventually taking a subsequence), arguing as above using Hardy inequality (1.5) and the fact that \(F(u) \in L^{1}({\mathbb {R}}^N)\), we get
Similarly, along the same sequence (or eventually on a subsequence) \(\{ R_{n'} \}\) (chosen in \(I^2_{R}\)), we obtain
Combining equations (2.24), (2.28), (2.29) and (2.30), we obtain
Further, using (2.10), we also have
Finally combining (2.21), (2.25), (2.27), (2.31), (2.32), we get
namely
\(\square\)
We are ready to give the
Proof of Theorem 1.1
(i): Let u be a solution of (2.15), then using (2.16) we have
Rearranging the above terms, we get
Since \(q > 2^*_s - 1\) this immediately implies that \(u \equiv 0\). This proves the first part of the theorem.
(ii): We use a constrained minimization. Let us define the manifold
Define the functional F on \({\dot{H}}^{s}({\mathbb {R}}^N)\cap L^{q+1}({\mathbb {R}}^N)\) as
and define
Let \(u_{n}\) be a minimizing sequence in \(\mathcal {N}\) such that
We assume that \(u_{n} \ge 0\) a.e. in \({\mathbb {R}}^N\). This is not restrictive, since we can consider \(|u_{n}|\) as a minimizing sequence. Since \(\theta <\Lambda _{N,s}\), by Remark 1.1 we have that the quantity
is an equivalent norm in \({\dot{H}}^s({\mathbb {R}}^N)\). Hence it follows that \(\{u_n\}\) is a bounded sequence in \({\dot{H}}^s({\mathbb {R}}^N)\cap L^{q+1}({\mathbb {R}}^N)\). Therefore, there exists \(u\in {\dot{H}}^s({\mathbb {R}}^N)\cap L^{q+1}({\mathbb {R}}^N)\) such that \(u_n \rightharpoonup u\) in \({\dot{H}}^s({\mathbb {R}}^N)\) and \(u_n \rightharpoonup u\) in \(L^{q+1}({\mathbb {R}}^N)\). Consequently \(u_{n}\rightarrow u\) a. e. in \({\mathbb {R}}^N\), by the Sobolev embedding (see [18]).
By using a Polya-Szegö type inequality (see [35]), we have that
where by \(f^*\) we denote the decreasing rearrangement of a measurable function f. Therefore via a rearrangement technique and without loss of generality, we can assume that \(u_n\) is radially symmetric and decreasing. By [8, Lemma 6.1], for every \(R:=|x|>0\) we deduce
where C is a positive constant that does not depend on n. Moreover for every bounded measurable set \(S\subset {\mathbb {R}}^N\) we have
where C is a positive constant that does not depend on n, since \(\{u_n\}\) is a bounded sequence in \({\dot{H}}^s({\mathbb {R}}^N)\cap L^{q+1}({\mathbb {R}}^N)\). Using (2.35) together with Vitali’s theorem we see that \(u_{n} \rightarrow u \, \text {in}\,\, L^{p+1}(B_R(0))\) for any \(R>0\) and using equations (2.34) (recall that \(p>2^*_s -1\)) it follows that \(u_{n} \rightarrow u \, \text {in}\,\, L^{p+1}({\mathbb {R}}^N\setminus B_R(0))\). Therefore,
and as a consequence
namely \(u\in \mathcal {N}\). Further, using Fatou’s lemma and that fact that \(u_n \rightharpoonup u\) in \({\dot{H}}^s({\mathbb {R}}^N)\), it can be shown that
This proves \(F(u)=\mathcal {K}\). Applying the Lagrange multiplier rule, we obtain that u satisfies
for some \(\lambda >0\). Finally, setting
it holds
The solution v obviously is nonnegative in \({\mathbb {R}}^N\). We show that actually \(v>0\) in \({\mathbb {R}}^N\). Indeed the limit function u is radial and radial decreasing since by construction, the sequence \(\{u_n\}\) is radial and radial decreasing. From (2.36) the solution v does inherit the same properties. Therefore if by contradiction there exists some point \(x_0\in {\mathbb {R}}^N\) such that \(v(x_0)=0\), the support of v would be contained in a ball, say of radius \(R_0\). Let us take a nonnegative test function \(\varphi _{c}\in C^{\infty }({\mathbb {R}}^N)\) with compact support \({\mathcal {K}}\) contained in \({\mathbb {R}}^N\setminus B_{2R_0}\). Using equation (2.37) we deduce
that is
Because \(v\equiv 0\) on \({\mathcal {K}}\), the first term of (2.38)
The second term
since \(v,\varphi _c\ge 0\). This is a contradiction with (2.38). \(\square\)
3 Qualitative properties for solutions to (\({\mathcal {P}}\))
In this section we first prove Theorem 1.2. We show that all the solutions of (\({\mathcal {P}}\)) (see also Remark 2.1) are radial and radially decreasing with respect to the origin, as stated in Theorem 1.2. The proof will be carried out exploiting the moving plane method. Without loss of generality, we start considering the \(x_1\)-direction. We denote a point \(x\in \mathbb {R}^N\) as \(x=(x_1,x_2,\ldots ,x_N)\) and, for any \(\lambda \in \mathbb {R},\) we set
and
For any \(\lambda \in \mathbb {R}\), we also set
and we define
With these definitions, if u is a solution to (\({\mathcal {P}}\)), then \(u_\lambda\) weakly satisfies
Indeed for any \(\varphi \in {\dot{H}}^s({\mathbb {R}}^N) \cap L^{q+1}({\mathbb {R}}^N)\), we have
where the changes of variables \(t=x_\lambda\) and \(z=y_\lambda\) and (1.6) were used. Notice that, if \(\varphi \in {\dot{H}}^s({\mathbb {R}}^N)\cap L^{q+1}({\mathbb {R}}^N)\), then \(\varphi _\lambda \in {\dot{H}}^s({\mathbb {R}}^N)\cap L^{q+1}({\mathbb {R}}^N)\) and so \(\varphi _\lambda\) can be used as a test function in (1.6).
Proof of Theorem 1.2
Let \(\lambda <0\) and we define the following function
where \((u-u_\lambda )^+:=\max \{u-u_\lambda ,0\}\) and \((u-u_\lambda )^-:=-\min \{u-u_\lambda ,0\}\). Note that \(w_\lambda\) is anti-symmetric w.r.t. \(T_\lambda\). We set
Using (3.4), it is not difficult to see
Step 1 We claim that
We start noticing that, the function \(w_\lambda\) defined in (3.5) belongs to \({\dot{H}}^s(\mathbb {R}^N)\cap L^{q+1}({\mathbb {R}}^N)\). Therefore, we can use it as test function in the weak formulations of (\({\mathcal {P}}\)) and (\({\mathcal {P}}_\lambda\)). We have
Subtracting the two equations in (3.9), we obtain
since \(|x|\ge |x_\lambda |\) and \(w_\lambda \ge 0\) in \(\Sigma _\lambda\) and \(|x|\le |x_\lambda |\) and \(w_\lambda \le 0\) outside \(\Sigma _\lambda\). On the other hand, we have
where
Arguing as in [4, 19], we deduce that
and therefore from (3.11), it follows
To estimate the RHS of (3.10), we first note that
For the first term on the RHS of (3.10), we use Hardy inequality and we get
To estimate the second term on the RHS of (3.10), we observe that for any \(t>1\)
Therefore, recalling (3.6) and (3.7), we have
where in the last line we exploited a change of variable. By Hölder and Sobolev inequalities we deduce
Collecting the last inequality, (3.10), (3.12), (3.13) and (3.14), we finally obtain
Since \(p> 2^*_s-1\) and (1.8), we get
Therefore there exists \(R>0\) such that for \(\lambda <-R\) we have
This and (3.15) give that
This implies that \(w_\lambda\) is constant and the claim (3.8) follows since \(w_\lambda\) is zero on \(\{x_1=\lambda \}\).
Now we define the set
Notice that (3.8) implies that \(\Lambda \ne \emptyset\), and therefore we can consider
Step 2 We will show that
Let us assume by contradiction that \(\bar{\lambda }<0\). Now, in this case, we are going to show that we can move the plane a little further to the right reaching a contradiction with the definition (3.17). Indeed, by continuity, we have that \(u\le u_{\bar{\lambda }}\) in \(\Sigma _{\bar{\lambda }}\) (say outside the reflected point of the origin \(0_{\bar{\lambda }}\), since \(u_{\bar{\lambda }}\) may be not well defined there). We want to prove that
The case \(u\equiv u_{\bar{\lambda }}\) in \(\Sigma _{\bar{\lambda }}\), is not possible because of the presence of the Hardy potential. Indeed if this would be the case, \(|(-\Delta )^s u(0_{\bar{\lambda }})|<+\infty\), since in the point \(0_{\bar{\lambda }}\) the solution u is regular and consequently \(|(-\Delta )^s u_{\bar{\lambda }}(0_{\bar{\lambda }})|<+\infty\) as well. Moreover \((-\Delta )^s u(0_{\bar{\lambda }})=(-\Delta )^s u_{\bar{\lambda }}(0_{\bar{\lambda }})\) and therefore
The right-hand side of (3.20) is well defined, i.e.,
if and only if \(u_{\bar{\lambda }}(x)=O(|x|^{2s})\) if \(x\rightarrow 0_{\bar{\lambda }}\). We point out that we are in the case \(u\equiv u_{\bar{\lambda }}\) in \(\Sigma _{\bar{\lambda }}\) and therefore both u and \(u_{\bar{\lambda }}\) are \(C^\infty\) in a neighborhood of \(0_{\bar{\lambda }}\). But this is a contradiction since the solution u is positive in \({\mathbb {R}}^N\).
Therefore, to prove (3.19), we assume by contradiction that there exists a point \({\bar{x}}\) in \(\Sigma _{\bar{\lambda }}\setminus \{0_{\bar{\lambda }}\}\) where
We fix now \(r>0\) such that \(0_{\bar{\lambda }} \notin \overline{B_r}({\bar{x}} )\) and \(0 \notin \overline{B_r}({\bar{x}}_{\bar{\lambda }})\). Then using Proposition 2.1 we have that for some \(0<\beta <2s\) it holds
for some positive constant C. As a consequence, using also (2.1), we can write the pointwise formulation of the problems (\({\mathcal {P}}\)) and (\({\mathcal {P}}_\lambda\)) for u and \(u_{\bar{\lambda }}\), respectively, at the point \(x={\bar{x}}\). Therefore,
It is worth noticing that, by (3.21) and the fact that \(|{\bar{x}}_\lambda |<|{\bar{x}}|\) for \(\lambda <0\), from (3.22) it follows that
On the other hand, by (1.1), (3.1)-(3.4), (3.21) and the fact that the function \(u_{\bar{\lambda }}(y)-u(y)\) is odd with respect to the hyperplane \(\partial \Sigma _{\bar{\lambda }}=T_{\bar{\lambda }}\), it follows that
Since \(|{\bar{x}}-y|<|{\bar{x}} -y_{\bar{\lambda }}|\) for \({\bar{x}},y\in \Sigma _{\bar{\lambda }}\) and \(u\not \equiv u_{\bar{\lambda }}\), \(u\le u_{\bar{\lambda }}\) in \(\Sigma _{\bar{\lambda }}\), from (3.24), by continuity, we have
a contradiction with (3.23). Since \({\bar{x}}\) is an arbitrary point in \(\Sigma _{\bar{\lambda }}\setminus \{0_{\bar{\lambda }}\}\), this implies (3.19).
Now notice that, for \(\lambda ^*<0\) given, the inequality in (3.15) holds for any \(\lambda \le \lambda ^*<0\) for a constant \(C=2p\) that is independent of \(\lambda\). Then, since \(\bar{\lambda }<0\), there exists \(\varepsilon _1>0\) such that \(\bar{\lambda }+2\varepsilon <0\) for any \(\varepsilon \in (0,\varepsilon _1)\). Let us set \(\lambda ^*=\bar{\lambda }+2\varepsilon _1\). Recalling the notation introduced in (3.5) and (3.6), we consider the function \(w_{\bar{\lambda }+\varepsilon }\). Using the same notation as above let us consider \(w_{\bar{\lambda }+\varepsilon }\) so that
Exploiting the fact that \(u<u_{\bar{\lambda }}\) in \(\Sigma _{\bar{\lambda }}\) and the fact that the solution u is continuous in \(\mathbb {R}^N\setminus \{0\}\) (resp. \(u_\lambda\) is continuous in \(\mathbb {R}^N\setminus \{0_{\bar{\lambda }}\}\)), we deduce that:
given any \(R>0\) (large) and \(\delta ,\delta _1>0\) (small) we can fix \(\bar{\varepsilon }=\bar{\varepsilon }(R,\delta ,\delta _1)>0\) and \(\bar{\varepsilon }<\varepsilon _1\), such that (arguing by continuity)
We repeat now the argument above using \(w_{\bar{\lambda }+\varepsilon }\) as test function in the same fashion as we did using \(w_\lambda\) and get again
Since
for R large and \(\delta ,\delta _1\) small (recall (3.16)), choosing \(\bar{\varepsilon }(R,\delta ,\delta _1)\) as above and eventually reducing it, we can assume that
Then from (3.26) we reach that \(w_{\bar{\lambda }+\varepsilon }=0\) and this is a contradiction to (3.17). Therefore
Step 3 Finally, the symmetry (and monotonicity) in the5 \(x_1\)-direction follows as standard repeating the argument in the \((-x_1)\)-direction. The radial symmetry result (and the monotonicity of the solution) follows as well performing the Moving Plane Method in any direction \(\nu \in \mathbb {S}^{N-1}\). \(\square\)
In the second part of this section, we study the asymptotic behavior of the solutions to (\({\mathcal {P}}\)) as stated in Theorem 1.3. To do this, we are going to use a useful representation result (Lemma 3.1 here) and then we reformulate our equation (\({\mathcal {P}}\)) to an equivalent singular equation without linear singular term. Towards this, we shall recall a result by Frank, Lieb and Seringer [25] (in particular equality (4.3) proved in [25, pag. 935]).
Lemma 3.1
Let \(0<\gamma <(N-2s)/{2}\). If \(u\in C^{\infty }_0({\mathbb {R}}^N\setminus \{0\})\) and \(v(x)=|x|^{\gamma }u\), then
where
Consider, the function \(\Psi _{N,s}:\bigg [0, \frac{N-2s}{2}\bigg ]\rightarrow [0,\Lambda _{N,s}]\) defined by
Then \(\Psi _{N,s}\) is strictly increasing and surjective (see [25, Lemma 3.2]). Therefore, given \(\theta \in (0,\Lambda _{N,s})\), there exists unique \(\gamma \in \big (0, (N-2s)/2\big )\) such that
Substituting \(\Lambda _{N,s}\) from (1.2) to (3.28) yields
Let us define the space \(\dot{H}^{s,\gamma }(\mathbb {R}^N)\) as the closure of \(C^{\infty }_0({\mathbb {R}}^N)\) with respect to the norm
We also define
Note that by [21], one has that the space \(\dot{W}^{s,\gamma }({\mathbb {R}}^N)\) coincides with \(\dot{H}^{s,\gamma }({\mathbb {R}}^N)\).
As a consequence of the ground state representation given by Lemma 3.1, we will transform our problem (\({\mathcal {P}}\)) into another nonlocal problem in weighted spaces. Namely, we consider \(u\in {\dot{H}}^s(\mathbb {R}^N)\cap L^{q+1}({\mathbb {R}}^N)\) a solution of the problem (\({\mathcal {P}}\)) and set \(v=v_\gamma :=|x|^{\gamma }u\), with \(\gamma\) given by (3.29). Furthermore, we introduce the operator \((-\Delta _{\gamma })^s\), defined as duality product
for any \(v,\phi \in \dot{H}^{s,\gamma }({\mathbb {R}}^N)\). Using this representation, if u solves (\({\mathcal {P}}\)), then
is a weak solution to
In the following proposition, we prove the boundedness of the weak solutions to (3.33). This result will be the key in the proof of Theorem 1.3. We have
Proposition 3.1
Let \(p>2^{*}_s-1\),
and let v be a solution of (3.33). Then \(v\in L^{\infty }({\mathbb {R}}^N)\).
Proof
We aim to apply Moser iteration technique to prove this theorem. For \(\beta \ge 1\) and \(T>0\), we define
Since v satisfies the problem (3.33) and \(\phi _{\beta , T}\) is a Lipschitz function, it follows that
Moreover, in the weak distribution sense, we have that
see [19]. By using the weighted Sobolev inequality [1, 21] we have
On the other hand, using (3.36), we get
where we used that \(t\phi '(t)\le \beta \phi (t)\). Using (3.37) and (3.38), we obtain
where \(C=C(s,N,\gamma )\) and
and
where \(B_1^c:={\mathbb {R}}^N\setminus B^1\).
Step 1 We prove that
To prove this step, first we estimate B.
We observe that under our assumptions we have
Therefore by interpolation inequality we infer that
Next, we estimate A as below:
where
We also observe that using weighted Young’s inequality we get
Therefore
Note that since \(q>p\)
and therefore
Thus, choosing \(\varepsilon<<1\), from (3.44), we obtain
Putting together (3.42) and (3.45) into (3.39), we have
with \(C=C(p,q,s,N,\gamma )\) a positive constant. By (3.43), we can fix m such that
where C is given in (3.46). Hence
up to redefine the constant C and where we used that \(\phi (t)\le t^{\beta }\), \(\beta =2^*_s/2\) and
since (3.34) holds. By Fatou’s lemma, taking \(T\rightarrow \infty\), we obtain
Step 2 In this step we establish the iteration formula, i.e., the inequality (3.55) below.
Towards this, first we observe that from the definition of B (3.40), we can also write
Since \(p+1>2^*_s\), using (3.32) and Lemma 2.1 (see in particular (2.7)), we get the following estimate
On the other hand, from the first inequality of (3.45), we can estimate A further as follows:
Therefore ( recall \(\phi (t)\le t^{\beta }\) and that \(0<\gamma <(N-2s)/{2}\) )
Combining (3.48) and (3.49) along with (3.39), we have
where \(C=C(p,q,s,N,\gamma ,\Vert u\Vert _{L^{q+1}({\mathbb {R}}^N)})\) is a suitable positive constant. From the definition of \(\phi (t)\), we know that \(\phi (t)= t^{\beta }\) if \(t\le T\). Using this and Fatou’s lemma, we obtain
Hence
up to redefine the constant C given in (3.50). This implies
For \(k\ge 1\), let us define \(\{\beta _{k}\}\) by
Thus taking \(\beta =\beta _{k+1}\) in (3.51), we get
Moreover, (3.52) implies
Consequently, we can rewrite (3.53) as
Iterating the relation (3.54) yields
Therefore, \(\beta _{k+1}\rightarrow \infty\). If we denote
we get the recurrence formula \(A_{k+1}\le C_{k+1}A_k\), \(k\ge 1\).
Step 3 The conclusion namely we prove that \(\Vert v\Vert _{L^{\infty }(\mathbb {R}^N)}\le C\).
Arguing by induction we have
Indeed the series \(\sum_{j=2}^{+\infty }\log C_{j}< +\infty\) is convergent: use standard criterias for convergence of series, recalling that \(\beta _{k+1}=\beta _1^{k}(\beta _1-1)+1\) and (3.56). Moreover \(A_1\le C\), see (3.41).
From (3.57), it follows
and then, for \(R>0\) fixed
Since (see (3.56)) \(\beta _{k}\rightarrow +\infty\) as \(k\rightarrow \infty\), we have
with C a positive constant not depending on R. This end the proof since
and thus
where \(C=C(p,q,s,N,\gamma ,\Vert u\Vert _{L^{q+1}({\mathbb {R}}^N)})\). This end the proof. \(\square\)
Proof of Theorem 1.3
Let \(\gamma _{\theta }\) the unique solution of (3.29). Setting \(v_{\gamma _{\theta }}:=|x|^{\gamma _{\theta }}u\), by Proposition 3.1 we have that
for some positive constant \(C=C(p,q,s,N,\gamma _{\theta },\Vert u\Vert _{L^{q+1}({\mathbb {R}}^N)})\). Then
\(\square\)
Data availability
All data generated or analyzed during this study are included in this published article.
References
Abdellaoui, B., Bentiffour, R.: Caffarelli-Kohn-Nirenberg type inequalities of fractional order with applications. J. Funct. Anal. 272(10), 3998–4029 (2017)
Barrios, B., Figalli, A., Valdinoci, E.: Bootstrap regularity for integro-differential operators and its application to nonlocal minimal surfaces. Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5) 2, 609–639 (2014)
Barrios, B., Medina, M., Peral, I.: Some remarks on the solvability of non local elliptic problems with the Hardy potential. Com. Contemp. Math. 16(4), 1350046 (2014)
Barrios, B., Montoro, L., Sciunzi, B.: On the moving plane method for nonlocal problems in bounded domains. J. Anal. Math. 135(1), 37–57 (2018)
Bhakta, M., Biswas, A., Ganguly, D., Montoro, L.: Integral representation of solutions using Green function for fractional Hardy equations. J. Differ. Equ. 269(7), 5573–5594 (2020)
Bhakta, M., Chakraborty, S., Pucci, P.: Fractional Hardy-Sobolev equations with nonhomogeneous terms. Adv. Nonlinear Anal. 10(1), 1086–1116 (2021)
Bhakta, M., Mukherjee, D.: Nonlocal scalar field equations: qualitative properties, asymptotic profiles and local uniqueness of solutions. J. Differ. Equ. 266, 6985–7037 (2019)
Bhakta, M., Mukherjee, D.: Semilinear nonlocal elliptic equations with critical and supercritical exponents. Commun. Pure Appl. Anal. 16(5), 1741–1766 (2017)
Bhakta, M., Mukherjee, D., Santra, S.: Profile of solutions for nonlocal equations with critical and supercritical nonlinearities. Commun. Contemp. Math. 21(1), 1750099 (2019)
Bhakta, M., Santra, S.: On singular equations with critical and supercritical exponents. J. Differ. Equ. 263(5), 2886–2953 (2017)
Bjorland, C., Caffarelli, L., Figalli, A.: Non-Local Gradient Dependent Operators. Adv. Math. 230(4–6), 1859–1894 (2012)
Caffarelli, L., Figalli, A.: Regularity of solutions to the parabolic fractional obstacle problem. J. Reine Angew. Math. 680, 191–233 (2013)
Caffarelli, L., Silvestre, L.: Regularity theory for fully nonlinear integro-differential equations. Comm. Pure Appl. Math. 62(5), 597–638 (2009)
Caffarelli, L., Silvestre, L.: Regularity results for nonlocal equations by approximation. Arch. Rational Mech. Anal. 200(1), 59–88 (2011)
Caffarelli, L., Silvestre, L.: The Evans-Krylov theorem for non local fully non linear equations. Ann. Math. 174(2), 1163–1187 (2011)
Caffarelli, L., Silvestre, L.: An extension problem related to the fractional Laplacian. Commun. Partial Differ. Equ. 32(7–9), 1245–1260 (2007)
Chen, W., Li, C., Ou, B.: Classification of solutions for an integral equation. Comm. Pure Appl. Math. 59(3), 330–343 (2006)
Di Nezza, E., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136(5), 521–573 (2012)
Dipierro, S., Montoro, L., Peral, I., Sciunzi, B.: Qualitative properties of positive solutions to nonlocal critical problems involving the Hardy-Leray potential. Calc. Var. Partial Differ. Equ. 55(4), 29 (2016)
Dipierro, S., Soave, N., Valdinoci, E.: On fractional elliptic equations in Lipschitz sets and epigraphs: regularity, monotonicity and rigidity results. Math. Ann. 369(3–4), 1283–1326 (2017)
Dipierro, S., Valdinoci, E.: A density property for fractional weighted Sobolev Spaces. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 26(4), 397–422 (2015)
Fall, M.M.: Semilinear elliptic equations for the fractional Laplacian with Hardy potential. Nonlinear Anal. 193, 111311 (2020)
Fall, M.M., Felli, V.: Unique continuation properties for relativistic Schrödinger operators with a singular potential. Discrete Contin. Dyn. Syst. 35(12), 5827–5867 (2015)
Felmer, P., Wang, Y.: Radial symmetry of positive solutions to equations involving the fractional Laplacian. Commun. Contemp. Math. 16(1), 1350023 (2014)
Frank, R.L., Lieb, E.H., Seiringer, R.: Hardy-Lieb-Thirring inequalities for fractional Schrödinger operators. J. Am. Math. Soc. 21(4), 925–950 (2008)
Ghoussoub, N., Shakerian, S.: Borderline variational problems involving fractional Laplacians and critical singularities. Adv. Nonlinear Stud 15(3), 527–555 (2015)
Jarohs, S., Weth, T.: Asymptotic symmetry for a class of nonlinear fractional reaction-diffusion equations. Discrete Contin. Dyn. Syst. 34(6), 2581–2615 (2014)
Jarohs, S., Weth, T.: Symmetry via antisymmetric maximum principles in nonlocal problems of variable order. Ann. Mat. Pura Appl. (4) 195(1), 273–291 (2016)
Kuusi, T., Mingione, G., Sire, Y.: A fractional Gehring lemma, with applications to nonlocal equations. Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 25(4), 345–358 (2014)
Kuusi, T., Mingione, G., Sire, Y.: Nonlocal equations with measure data. Comm. Math. Phys. 337(3), 1317–1368 (2015)
Kuusi, T., Mingione, G., Sire, Y.: Nonlocal self-improving properties. Anal. PDE 8(1), 57–114 (2015)
Merle, F., Peletier, L.: Asymptotic behaviour of positive solutions of elliptic equations with critical and supercritical growth. II. The non-radial case. J. Funct. Anal 105(1), 1–41 (1992)
Montoro, L., Punzo, F., Sciunzi, B.: Qualitative properties of singular solutions to nonlocal problems. Ann. Mat. Pura Appl. (4) 197(3), 941–964 (2018)
Palatucci, G.: The Dirichlet problem for the \(p\)-fractional Laplace equation. Nonlinear Anal. 177(part B), 699–732 (2018)
Park, Y.J.: Fractional Polya-Szegö Inequality. J. Chungcheong Math. Soc. 24(2), 267–271 (2011)
Ros-Oton, X., Serra, J.: The Dirichlet problem for the fractional Laplacian: regularity up to the boundary. J. Math. Pures Appl. 101, 275–302 (2014)
Serra, J.: \(C^{ +\alpha }\) regularity for concave nonlocal fully nonlinear elliptic equations with rough kernels. Calc. Var. Partial Differ. Equ. 54(4), 3571–3601 (2015)
Servadei, R., Valdinoci, E.: Mountain Pass solutions for non-local elliptic operators. J. Math. Anal. Appl. 389(2), 887–898 (2012)
Servadei, R., Valdinoci, E.: Variational methods for non-local operators of elliptic type. Discrete Contin. Dyn. Syst. 33(5), 2105–2137 (2013)
Soave, N., Valdinoci, E.: Overdetermined problems for the fractional Laplacian in exterior and annular sets. J. Anal. Math. 137(1), 101–134 (2019)
Terracini, S.: On positive entire solutions to a class of equations with a singular coefficient and critical exponent. Adv. Differ. Equ. 1(2), 241–264 (1996)
Acknowledgements
The authors would like to thank the anonymous referee for his/her useful suggestions and comments. The research of M. Bhakta is partially supported by the SERB WEA grant (WEA/2020/000005) and DST Swarnajaynti fellowship (SB/SJF/2021-22/09). D. Ganguly is partially supported by INSPIRE faculty fellowship (IFA17-MA98). L. Montoro is partially supported by PRIN project 2017JPCAPN (Italy): Qualitative and quantitative aspects of nonlinear PDEs and Project PDI2019-110712GB-100, 2020-2023, MICINN (Spain): Ecuaciones con perturbaciones de potencias del Laplaciano.
Funding
Open access funding provided by Università della Calabria within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
On behalf of all authors, the corresponding author states that there is no conflict of interest.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Bhakta, M., Ganguly, D. & Montoro, L. Fractional Hardy equations with critical and supercritical exponents. Annali di Matematica 202, 397–430 (2023). https://doi.org/10.1007/s10231-022-01246-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10231-022-01246-2