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Bifurcation results for the critical Choquard problem involving fractional p-Laplacian operator

Boundary Value Problems20182018:132

https://doi.org/10.1186/s13661-018-1050-7

  • Received: 14 March 2018
  • Accepted: 15 August 2018
  • Published:

Abstract

By using an abstract critical point theorem based on a pseudo-index related to the cohomological index, we prove the bifurcation results for the critical Choquard problems involving fractional p-Laplacian operator:
$$(- \Delta)_{p}^{s} u = \lambda \vert u \vert ^{p-2} u + \biggl( \int_{\Omega}\frac{ \vert u \vert ^{p_{\mu,s}^{*}}}{ \vert x-y \vert ^{\mu}}\,dy \biggr) \vert u \vert ^{p_{\mu,s}^{*}-2}u \quad \text{in } \Omega,\qquad u = 0\quad \text{in } {\mathbb {R}}^{N} \setminus \Omega, $$
where Ω is a bounded domain in \({\mathbb {R}}^{N}\) with Lipschitz boundary, λ is a real parameter, \(p\in(1,\infty)\), \(s\in (0,1)\), \(N>sp\), and \(p_{\mu,s}^{*}=\frac{(N-\frac{\mu}{2})p}{N-sp}\) is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality. These extend results in the literature for the fractional Choquard problems, and they are still new for a p-Laplacian case.

Keywords

  • Fractional p-Laplacian
  • Choquard equation
  • Critical exponent
  • \({\mathbb {Z}}_{2}\)-cohomological index
  • Pseudo-index

1 Introduction and main results

Given \(p \in(1,\infty)\), \(s \in(0,1)\), \(\mu\in(0,N)\), \(N > sp\), we consider the problem
$$ \textstyle\begin{cases} (- \Delta)_{p}^{s} u = \lambda \vert u \vert ^{p-2} u + (\int_{\Omega}\frac{ \vert u \vert ^{p_{\mu,s}^{*}}}{ \vert x-y \vert ^{\mu}}\,dy ) \vert u \vert ^{p_{\mu,s}^{*}-2}u, & \text{in } \Omega,\\ u = 0, & \text{in } \mathbb {R}^{N} \setminus\Omega, \end{cases} $$
(1)
where \(p_{\mu,s}^{\ast}= \frac{(N-\frac{\mu}{2})p}{N-sp}\) and Ω is a bounded domain in \(\mathbb {R}^{N}\) with Lipschitz boundary. \((- \Delta)_{p}^{s}\) is the associated fractional operator which, up to a normalization constant, is defined as
$$(- \Delta)_{p}^{s} u(x) = 2 \lim_{\varepsilon \searrow0} \int_{\mathbb {R}^{N} \setminus B_{\varepsilon }(x)} \frac{ \vert u(x) - u(y) \vert ^{p-2} (u(x) - u(y))}{ \vert x - y \vert ^{N+sp}} \,dy, \quad x \in \mathbb {R}^{N}. $$
This definition is consistent with the usual definition of the linear fractional Laplacian operator \((- \Delta)^{s}\) when \(p = 2\). Recently, much attention has been paid to the problems involving nonlocal operators, and we may refer to [2, 12, 15, 18, 32, 3638] for more details. In particular, fractional p-eigenvalue problems have been studied in [4, 6, 8, 16, 17, 19, 20, 29], and existence theories in the critical case in [1, 13, 2125, 30, 33].
On the other hand, Equation (1) is also related to the so-called Choquard equation which is inspired by the following Hardy–Littlewood–Sobolev inequality:
$$ \int_{\mathbb {R}^{N}} \int_{\mathbb {R}^{N}}\frac{ \vert u(x) \vert ^{p_{\mu,s}^{\ast}} \vert u(y) \vert ^{p_{\mu,s}^{\ast}}}{ \vert x-y \vert ^{\mu}} \,dx \,dy\leq\widetilde{C}(N,\mu , s, p) \biggl( \int_{\mathbb {R}^{N}} \vert u \vert ^{p_{s}^{\ast}}\,dx \biggr)^{\frac{2p_{\mu ,s}^{\ast}}{p_{s}^{\ast}}}, $$
where \(p_{s}^{\ast}= \frac{Np}{N-sp}\). For more details about fractional Choquard equations, we refer the readers to [5, 9, 11, 24, 34, 35]. Very recently, Mukherjee and Sreenadh [24] have obtained the following multiplicity result, extending the results in [10] in the local case, for the Brezis–Nirenberg type problem of semilinear fractional Choquard equation:
$$ \textstyle\begin{cases} (- \Delta)^{s} u = (\int_{\Omega}\frac{ \vert u \vert ^{2_{\mu ,s}^{*}}}{ \vert x-y \vert ^{\mu}}\,dy ) \vert u \vert ^{2_{\mu,s}^{*}-2}u+\lambda u, & \text{in } \Omega,\\ u = 0, & \text{in } \mathbb {R}^{N} \setminus\Omega, \end{cases} $$
(2)
where Ω is a bounded domain with Lipschitz boundary in \(\mathbb {R}^{N}\), \(N>2s\), \(0< s<1\), \(2_{\mu,s}^{*}= \frac{2N-\mu}{N-2s}\), and \(\lambda >0\) is a parameter. Let \(0 < \lambda_{1} < \lambda_{2} \le\lambda_{3} \le\cdots\to+ \infty \) be the eigenvalues of the problem
$$\textstyle\begin{cases} (- \Delta)^{s} u = \lambda u, & \text{in } \Omega,\\ u = 0, & \text{in } \mathbb {R}^{N} \setminus\Omega. \end{cases} $$
There exists a constant \(\lambda^{*}\) such that if there are q number of eigenvalues lying between λ and \(\lambda+\lambda^{*}\), then problem (2) has q distinct pairs of solutions (see [24, Theorem 2.6]).

In this paper, we use a more general construction based on sublevel sets as in Perera and Szulkin [31] (see also Perera et al. [28, Proposition 3.23]) and extend the above multiplicity result to the quasilinear nonlocal problem (1). However, the standard sequence of variational eigenvalues of \((- \Delta )_{p}^{s}\) based on the genus does not provide enough information about the structure of the sublevel sets to implement the linking construction. Therefore we use a different sequence of eigenvalues as in Iannizzotto et al. [14] that is based on the \(\mathbb {Z}_{2}\)-cohomological index of Fadell and Rabinowitz [7], which is defined as follows.

Definition 1.1

Let W be a Banach space, and let \({\mathcal {A}}\) denote the class of symmetric subsets of \(W \setminus \{{0} \}\). For \(A \in {\mathcal {A}}\), let \(\overline{A} = A/\mathbb {Z}_{2}\) be the quotient space of A with each u andu identified, let \(f : \overline{A} \to \mathbb {R}\mathrm{P}^{\infty}\) be the classifying map of , and let \(f^{\ast}: H^{\ast}(\mathbb {R}\mathrm{P}^{\infty}) \to H^{\ast}(\overline{A})\) be the induced homomorphism of the Alexander–Spanier cohomology rings. The cohomological index of A is defined by
$$i(A) = \textstyle\begin{cases} \sup \{{m \ge1 : f^{\ast}(\omega^{m-1}) \ne0} \}, & A \ne \emptyset ,\\ 0, & A = \emptyset, \end{cases} $$
where \(\omega\in H^{1}(\mathbb {R}\mathrm{P}^{\infty})\) is the generator of the polynomial ring \(H^{\ast}(\mathbb {R}\mathrm{P}^{\infty}) = \mathbb {Z}_{2}[\omega]\).

For instance, the classifying map of the unit sphere \(S^{m-1}\) in \(\mathbb {R}^{m}\), \(m \ge1\) is the inclusion \(\mathbb {R}\mathrm{P}^{m-1} \subset \mathbb {R}\mathrm{P}^{\infty}\), which induces isomorphisms on \(H^{q}\) for \(q \le m - 1\), so \(i(S^{m-1}) = m\).

Define the so-called Gagliardo seminorm by
$$[u]_{s,p} = \biggl( \int_{\mathbb {R}^{2N}} \frac{ \vert u(x) - u(y) \vert ^{p}}{ \vert x - y \vert ^{N+sp}} \,dx \,dy \biggr)^{1/p}, $$
where \(u : \mathbb {R}^{N} \to \mathbb {R}\) is a measurable function, and give the fractional Sobolev space by
$$W^{s,p}\bigl(\mathbb {R}^{N}\bigr) = \bigl\{ {u \in L^{p} \bigl(\mathbb {R}^{N}\bigr) : [u]_{s,p} < \infty} \bigr\} , $$
which can be equipped with the norm
$$\Vert u \Vert _{s,p} = \bigl( \vert u \vert _{p}^{p} + [u]_{s,p}^{p} \bigr)^{1/p}, $$
where \(|\cdot|_{p}\) is the norm in \(L^{p}(\mathbb {R}^{N})\) (see Di Nezza et al. [26] for details). We work in the closed linear subspace
$$X_{p}^{s}(\Omega) = \bigl\{ {u \in W^{s,p}\bigl( \mathbb {R}^{N}\bigr) : u = 0 \text{ a.e. in } \mathbb {R}^{N} \setminus\Omega} \bigr\} , $$
equivalently renormed by setting \(\|\cdot\|_{s,p} = [\cdot]_{s,p}\) (see [26, Theorem 7.1]), which is a uniformly convex Banach space. By [26, Theorem 6.5 and Corollary 7.2], the embedding \(X_{p}^{s}(\Omega) \hookrightarrow L^{r}(\Omega)\) is continuous for \(r \in[1,p_{s}^{\ast}]\) and compact for \(r \in[1,p_{s}^{\ast})\).
The Dirichlet spectrum of \((- \Delta)_{p}^{s}\) in Ω consists of those \(\lambda\in \mathbb {R}\) for which the problem
$$ \textstyle\begin{cases} (- \Delta)_{p}^{s} u = \lambda \vert u \vert ^{p-2} u, & \text{in } \Omega ,\\ u = 0, & \text{in } \mathbb {R}^{N} \setminus\Omega, \end{cases} $$
(3)
has a nontrivial solution. We will use a suitable minimax scheme to define an incremental and unbounded sequence of variational eigenvalues. The standard scheme based on the genus is not applicable here, so we will use the following scheme based on the cohomological index introduced in Iannizzotto et al. [14] (see also Perera [27]). Let
$$\Psi(u) = \frac{1}{ \vert u \vert _{p}^{p}}, \quad u \in {\mathcal {M}}= \bigl\{ {u \in X_{p}^{s}( \Omega) : \Vert u \Vert _{s,p} = 1} \bigr\} . $$
Then eigenvalues of problem (3) are consistent with critical values of Ψ. We use the standard notation
$$\Psi^{a} = \bigl\{ {u \in {\mathcal {M}}: \Psi(u) \le a} \bigr\} , \qquad \Psi_{a} = \bigl\{ {u \in {\mathcal {M}}: \Psi(u) \ge a} \bigr\} , \quad a \in \mathbb {R}$$
for the sublevel sets and superlevel sets, respectively. Let \({\mathcal {F}}\) denote the class of symmetric subsets of \({\mathcal {M}}\), and let
$$\lambda_{k} := \inf_{M \in {\mathcal {F}}, i(M) \ge k} \sup_{u \in M} \Psi (u), \quad k \in \mathbb {N}^{+}. $$
Then \(0 < \lambda_{1} < \lambda_{2} \le\lambda_{3} \le\cdots\to+ \infty\) is a sequence of eigenvalues of problem (3), and
$$ \lambda_{k} < \lambda_{k+1}\quad \implies\quad i\bigl( \Psi^{\lambda_{k}}\bigr) = i({\mathcal {M}}\setminus\Psi_{\lambda_{k+1}}) = k $$
(4)
(see Iannizzotto et al. [14, Proposition 2.4]). Making essential use of the index information in (4), we will prove the following theorem:
Let
$$C_{H,L}:=\sup_{u \in X_{p}^{s}(\Omega) \setminus \{{0} \} }\frac { \vert u \vert _{p_{s}^{*}}}{ (\int_{\Omega}\int_{\Omega}\frac{ \vert u(x) \vert ^{p_{\mu ,s}^{\ast}} \vert u(y) \vert ^{p_{\mu,s}^{\ast}} }{ \vert x-y \vert ^{\mu}}\,dx \,dy )^{\frac{1}{2p_{\mu,s}^{\ast}}}} \quad (\mbox{see Lemma 2.2 below}), $$
and
$$ S_{H,L}:= \inf_{u \in X_{p}^{s}(\Omega) \setminus \{{0} \}} \frac { \Vert u \Vert _{s,p}^{p}}{ (\int_{\Omega}\int_{\Omega}\frac { \vert u(x) \vert ^{p_{\mu,s}^{\ast}} \vert u(y) \vert ^{p_{\mu,s}^{\ast}} }{ \vert x-y \vert ^{\mu}}\,dx \,dy )^{\frac{p}{2p_{\mu,s}^{\ast}}}}. $$

Theorem 1.2

  1. (i)
    If
    $$\lambda_{1} - \frac{S_{H,L}}{ \vert {\Omega} \vert ^{\frac {sp}{N}}C_{H,L}^{p}}< \lambda< \lambda_{1}, $$
    then problem (1) has a pair of nontrivial solutions \(\pm u^{\lambda}\) such that \(u^{\lambda}\to0\) as \(\lambda\nearrow\lambda_{1}\).
     
  2. (ii)
    If \(\lambda_{k} \le\lambda< \lambda_{k+1} = \cdots= \lambda_{k+m} < \lambda_{k+m+1}\) for some \(k, m \in \mathbb {N}^{+}\) and
    $$ \lambda> \lambda_{k+1} - \frac{S_{H,L}}{ \vert {\Omega} \vert ^{\frac {sp}{N}}C_{H,L}^{p}}, $$
    (5)
    then problem (1) has m distinct pairs of nontrivial solutions \(\pm u^{\lambda}_{j}\), \(j = 1,\dots,m\), such that \(u^{\lambda}_{j} \to0\) as \(\lambda\nearrow\lambda_{k+1}\).
     

Corollary 1.3

Problem (1) has a nontrivial solution for all \(\lambda\in \bigcup_{k=1}^{\infty}(\lambda_{k} - \frac{S_{H,L}}{ \vert {\Omega} \vert ^{\frac{sp}{N}}C_{H,L}^{p}},\lambda_{k} )\).

We note that \(\lambda_{1} \ge\frac{S_{H,L}}{ \vert {\Omega} \vert ^{\frac {sp}{N}}C_{H,L}^{p}}\). Indeed, if \(\varphi_{1}\) is an eigenfunction associated with \(\lambda_{1}\),
$$\lambda_{1} = \frac{ \Vert \varphi_{1} \Vert _{s,p}^{p}}{ \vert \varphi_{1} \vert _{p}^{p}} \ge\frac{S_{H,L} (\int_{\Omega}\int_{\Omega}\frac { \vert \varphi_{1}(x) \vert ^{p_{\mu,s}^{\ast}} \vert \varphi_{1}(y) \vert ^{p_{\mu ,s}^{\ast}} }{ \vert x-y \vert ^{\mu}}\,dx \,dy )^{\frac{p}{2p_{\mu,s}^{\ast}}}}{ \vert \varphi_{1} \vert _{p}^{p}} \ge \frac{S_{H,L}}{ \vert {\Omega} \vert ^{\frac {sp}{N}}C_{H,L}^{p}} $$
by the Hölder inequality.

2 Preliminaries and some known results

The following Hardy–Littlewood–Sobolev inequality is essential to applying the variational approach for problem (1).

Lemma 2.1

([19, Theorem 4.3])

Assume \(1< r\), \(t<\infty\), and \(0<\mu<N\) with \(\frac{1}{r}+\frac {1}{t}+\frac{\mu}{N}=2\). If \(u\in L^{r}(\mathbb {R}^{N})\) and \(v\in L^{t}(\mathbb {R}^{N})\), then there exists \(C(N,\mu,r,t)>0\) such that
$$ \int_{\mathbb {R}^{N}} \int_{\mathbb {R}^{N}}\frac{ \vert u(x) \vert \vert v(y) \vert }{ \vert x-y \vert ^{\mu}} \,dx \,dy\leq C(N,\mu,r,t) \vert u \vert _{r} \vert v \vert _{t}. $$
By the Hardy–Littlewood–Sobolev inequality, there exists \(\widetilde {C}(N,\mu, s, p)>0\) such that
$$ \int_{\mathbb {R}^{N}} \int_{\mathbb {R}^{N}}\frac{ \vert u(x) \vert ^{p_{\mu,s}^{\ast}} \vert u(y) \vert ^{p_{\mu,s}^{\ast}}}{ \vert x-y \vert ^{\mu}} \,dx \,dy\leq\widetilde{C}(N,\mu , s, p) \vert u \vert _{{p_{s}^{\ast}}}^{2p_{\mu,s}^{\ast}} $$
(6)
for all \(u\in X_{p}^{s}(\Omega)\). Hence, by the Sobolev embedding,
$$ \int_{\mathbb {R}^{N}} \int_{\mathbb {R}^{N}}\frac{ \vert u(x) \vert ^{p_{\mu,s}^{\ast}} \vert u(y) \vert ^{p_{\mu,s}^{\ast}}}{ \vert x-y \vert ^{\mu}} \,dx \,dy\leq C(N,\mu,s,p) \Vert u \Vert _{s,p}^{2p_{\mu,s}^{\ast}}, $$
for some \(C(N,\mu,s,p)>0\). Define
$$ S_{H,L} = \inf_{u \in X_{p}^{s}(\Omega) \setminus \{{0} \}} \frac { \Vert u \Vert _{s,p}^{p}}{ (\int_{\Omega}\int_{\Omega}\frac { \vert u(x) \vert ^{p_{\mu,s}^{\ast}} \vert u(y) \vert ^{p_{\mu,s}^{\ast}} }{ \vert x-y \vert ^{\mu}}\,dx \,dy )^{\frac{p}{2p_{\mu,s}^{\ast}}}}. $$
(7)
Clearly, \(S_{H,L}>0\).
Now we can introduce the \(C^{1}\)- functional associated with problem (1) by
$$I_{\lambda}(u) = \frac{1}{p} \Vert u \Vert _{s,p}^{p} - \frac{\lambda}{p} \vert u \vert _{p}^{p} - \frac{1}{2p_{\mu,s}^{\ast}} \int_{\Omega}\int _{\Omega}\frac{ \vert u(x) \vert ^{p_{\mu,s}^{\ast}} \vert u(y) \vert ^{p_{\mu,s}^{\ast }} }{ \vert x-y \vert ^{\mu}}\,dx \,dy, \quad u \in X_{p}^{s}(\Omega). $$

Moreover, u is a weak solution of problem (1) if and only if u is a critical point of functional \(I_{\lambda}\).

Let
$$ \Vert u \Vert _{o}:= \biggl( \int_{\Omega} \int_{\Omega}\frac { \vert u(x) \vert ^{p_{\mu,s}^{*}} \vert u(y) \vert ^{p_{\mu,s}^{*}} }{ \vert x-y \vert ^{\mu}} \,dx \,dy \biggr)^{\frac{1}{2p_{\mu,s}^{\ast}}} $$
for \(u\in L^{p_{s}^{*}}(\Omega)\), we have the following lemma.

Lemma 2.2

There exists \(C_{H,L}>0\) such that \(|u|_{p_{s}^{*}}\leq C_{H,L}\|u\|_{o}\).

Proof

Let u, \(v\in L^{p_{s}^{*}}(\Omega)\), then using the Hölder inequality and the semigroup property of Riesz potential, we get
$$\begin{aligned} & \int_{\Omega} \int_{\Omega}\frac{ \vert (u+v)(x) \vert ^{p_{\mu,s}^{\ast}} \vert (u+v)(y) \vert ^{p_{\mu,s}^{\ast}} }{ \vert x-y \vert ^{\mu}} \,dx \,dy\\ &\quad = \int_{\Omega } \int_{\Omega}\frac{ \vert (u+v)(x) \vert ^{2p_{\mu,s}^{\ast}} }{ \vert x-y \vert ^{\mu }} \,dx \,dy \\ &\quad \leq \int_{\Omega} \int_{\Omega}\frac{ \vert u(x) \vert \vert (u+v)(x) \vert ^{2p_{\mu,s}^{\ast}-1} }{ \vert x-y \vert ^{\mu}} \,dx \,dy+ \int_{\Omega } \int_{\Omega}\frac{ \vert v(x) \vert \vert (u+v)(x) \vert ^{2p_{\mu,s}^{\ast}-1} }{ \vert x-y \vert ^{\mu}} \,dx \,dy \\ &\quad \leq \biggl( \int_{\Omega} \int_{\Omega}\frac{ \vert u(x) \vert ^{2p_{\mu ,s}^{\ast}} }{ \vert x-y \vert ^{\mu}} \,dx \,dy \biggr)^{\frac{1}{2p_{\mu ,s}^{\ast}}} \biggl( \int_{\Omega} \int_{\Omega}\frac{ \vert (u+v)(x) \vert ^{2p_{\mu ,s}^{\ast}} }{ \vert x-y \vert ^{\mu}} \,dx \,dy \biggr)^{1-\frac{1}{2p_{\mu ,s}^{\ast}}} \\ &\qquad {}+ \biggl( \int_{\Omega} \int_{\Omega}\frac{ \vert v(x) \vert ^{2p_{\mu ,s}^{\ast}} }{ \vert x-y \vert ^{\mu}} \,dx \,dy \biggr)^{\frac{1}{2p_{\mu ,s}^{\ast}}} \biggl( \int_{\Omega} \int_{\Omega}\frac{ \vert (u+v)(x) \vert ^{2p_{\mu ,s}^{\ast}} }{ \vert x-y \vert ^{\mu}} \,dx \,dy \biggr)^{1-\frac{1}{2p_{\mu ,s}^{\ast}}} \\ &\quad = \biggl( \biggl( \int_{\Omega} \int_{\Omega}\frac{ \vert u(x) \vert ^{2p_{\mu ,s}^{\ast}} }{ \vert x-y \vert ^{\mu}} \,dx \,dy \biggr)^{\frac{1}{2p_{\mu ,s}^{\ast}}} + \biggl( \int_{\Omega} \int_{\Omega}\frac{ \vert v(x) \vert ^{2p_{\mu,s}^{\ast }} }{ \vert x-y \vert ^{\mu}} \,dx \,dy \biggr)^{\frac{1}{2p_{\mu,s}^{\ast }}} \biggr) \\ &\qquad {}\times \biggl( \int_{\Omega} \int_{\Omega}\frac{ \vert (u+v)(x) \vert ^{2p_{\mu ,s}^{\ast}} }{ \vert x-y \vert ^{\mu}} \,dx \,dy \biggr)^{1-\frac{1}{2p_{\mu ,s}^{\ast}}}. \end{aligned}$$
So, we have \(\|u+v\|_{o}\leq\|u\|_{o}+\|v\|_{o}\) and other properties of the norm are also satisfied by \(\|\cdot\|_{o}\). So, \(\|\cdot\|_{o}\) is a norm on \(L^{p_{s}^{*}}(\Omega)\) and \(L^{p_{s}^{*}}(\Omega)\) is a Banach space under this norm. From (6), we have
$$ \Vert u \Vert _{o}\leq\widetilde{C}(N,\mu, s, p)^{\frac{1}{2p_{\mu ,s}^{\ast}}} \vert u \vert _{p_{s}^{*}}. $$
So, the identity map from \((L^{p_{s}^{*}}(\Omega),\|\cdot\|_{o})\) to \((L^{p_{s}^{*}}(\Omega),|\cdot|_{p_{s}^{\ast}})\) is linear and bounded. Thus, by open mapping theorem, we obtain \(\|\cdot\|_{o}\) is a norm equivalent to the standard norm \(|\cdot|_{p_{s}^{*}}\) on \(L^{p_{s}^{*}}(\Omega )\). So we have \(|u|_{p_{s}^{*}}\leq C_{H,L}\|u\|_{o}\). □

Definition 2.3

([3])

Let \({\mathcal {A}}^{\ast}\) denote the class of symmetric subsets of W, let \(r > 0\), let \(S_{r} = \{{u \in W : \|u\|_{s,p} = r} \}\), let \(0 < b \le+ \infty\), and let Γ denote the group of odd homeomorphisms of W that are the identity outside \(Ij^{-1}(0,b)\). The pseudo-index of \(M \in {\mathcal {A}}^{\ast}\) related to i, \(S_{r}\), and Γ is defined by
$$i^{\ast}(M) = \min_{\gamma\in\Gamma} i\bigl(\gamma(M) \cap S_{r}\bigr), $$
where i is in Definition 1.1.

The following is a general critical point theorem based on a pseudo-index related to the cohomological index which we apply here (see also Perera et al. [28, Proposition 3.44]).

Lemma 2.4

([30])

Let \(A_{0}\), \(B_{0}\) be symmetric subsets of \(S_{1}\) such that \(A_{0}\) is compact, \(B_{0}\) is closed, and
$$i(A_{0}) \ge k + m, \qquad i(S_{1} \setminus B_{0}) \le k $$
for some integers \(k \ge0\) and \(m \ge1\). Assume that there exists \(R > r\) such that
$$\sup I(A) \le0 < \inf I(B), \qquad\sup I(X) < b, $$
where \(A = \{{Ru : u \in A_{0}} \}\), \(B = \{{ru : u \in B_{0}} \}\), and \(X = \{{tu : u \in A, 0 \le t \le1} \}\). For \(j = k + 1,\dots,k + m\), let
$${\mathcal {A}}_{j}^{\ast}= \bigl\{ {M \in {\mathcal {A}}^{\ast}: M \textit{ is compact and } i^{\ast}(M) \ge j} \bigr\} $$
and set
$$c_{j}^{\ast}:= \inf_{M \in {\mathcal {A}}_{j}^{\ast}} \max _{u \in M} I(u). $$
Then
$$\inf I(B) \le c_{k+1}^{\ast}\le\dotsb\le c_{k+m}^{\ast}\le\sup I(X), $$
in particular, \(0 < c_{j}^{\ast}< b\). If, in addition, I satisfies the \((\mathrm{PS})_{c}\) condition for all \(c \in(0,b)\), that is, if every sequence \(\{u_{j}\} \subset W\) such that \(I(u_{j}) \to c\) and \(I'(u_{j}) \to 0\) possesses a convergent subsequence, then each \(c_{j}^{\ast}\) is a critical value of I and there are m distinct pairs of associated critical points.

3 Proof of Theorem 1.2

We only prove (ii) because the proof of (i) is similar and relatively simpler. First, \(I_{\lambda}\) satisfies the \((\mathrm{PS})_{c}\) condition for all \(c < \frac {2p_{\mu,s}^{*}-p}{2pp_{\mu,s}^{*}} S_{H,L}^{\frac{2p_{\mu ,s}^{*}}{2p_{\mu,s}^{*}-p}}\). In fact, let \(c <\frac{2p_{\mu,s}^{*}-p}{2pp_{\mu,s}^{*}} S_{H,L}^{\frac{2p_{\mu,s}^{*}}{2p_{\mu,s}^{*}-p}} \), and let \(\{u_{j}\}\) be a sequence in \(X_{p}^{s}(\Omega)\) such that
$$\begin{aligned}& I_{\lambda}(u_{j}) = \frac{1}{p} \Vert u_{j} \Vert _{s,p}^{p} - \frac{\lambda}{p} \vert u_{j} \vert _{p}^{p} - \frac{1}{2p_{\mu,s}^{\ast}} \Vert u_{j} \Vert _{o}^{2p_{\mu,s}^{\ast}} = c + \mathrm {o}(1), \end{aligned}$$
(8)
$$\begin{aligned}& \begin{aligned}[b] I_{\lambda}'(u_{j}) v &= \int_{\mathbb {R}^{2N}} \frac{ \vert u_{j}(x) - u_{j}(y) \vert ^{p-2} (u_{j}(x) - u_{j}(y)) (v(x) - v(y))}{ \vert x - y \vert ^{N+sp}} \,dx \,dy \\ & \quad {}- \lambda \int_{\Omega} \vert u_{j} \vert ^{p-2} u_{j} v \,dx - \int_{\Omega}\int _{\Omega}\frac{ \vert u_{j}(y) \vert ^{p_{\mu,s}^{\ast}} \vert u_{j}(x) \vert ^{p_{\mu,s}^{\ast}-2} u_{j}(x) v(x) }{ \vert x-y \vert ^{N+sp}} \,dx \\ &= \mathrm {o}\bigl( \Vert v \Vert _{s,p}\bigr), \quad\forall v \in X_{p}^{s}(\Omega), \end{aligned} \end{aligned}$$
(9)
as \(j \to\infty\). Then
$$\biggl(\frac{1}{p}-\frac{1}{2p_{\mu,s}^{*}}\biggr) \Vert u_{j} \Vert _{o}^{2p_{\mu ,s}^{\ast}} = I_{\lambda}(u_{j}) - \frac{1}{p} I_{\lambda}'(u_{j}) u_{j} = \mathrm {o}\bigl( \Vert u_{j} \Vert _{s,p} \bigr) + O(1), $$
which together with (8), Lemma 2.2, and the Hölder inequality shows that \(\{u_{j}\}\) is bounded in \(X_{p}^{s}(\Omega)\). Then there exists a subsequence of \(\{u_{j}\}\) still denoted by \(\{u_{j}\}\) and \(u\in X_{p}^{s}(\Omega)\) such that
$$u_{j}\rightharpoonup u \quad \mbox{weakly in } X_{p}^{s}( \Omega),\mbox{ and in } L^{p_{s}^{*}}(\Omega), $$
Denoting by \(p' = p/(p - 1)\) the Hölder conjugate of p, \(|u_{j}(x) - u_{j}(y)|^{p-2} (u_{j}(x) - u_{j}(y))/|x - y|^{(N+sp)/p'}\) is bounded in \(L^{p'}(\mathbb {R}^{2N})\) and converges to \(|u(x) - u(y)|^{p-2} (u(x) - u(y))/|x - y|^{(N+sp)/p'}\) a.e. in \(\mathbb {R}^{2N}\), and \((v(x) - v(y))/|x - y|^{(N+sp)/p} \in L^{p}(\mathbb {R}^{2N})\), so the first integral in (9) converges to
$$\int_{\mathbb {R}^{2N}} \frac{ \vert u(x) - u(y) \vert ^{p-2} (u(x) - u(y)) (v(x) - v(y))}{ \vert x - y \vert ^{N+sp}} \,dx \,dy $$
for a further subsequence. In addition,
$$\int_{\Omega} \vert u_{j} \vert ^{p-2} u_{j} v \,dx \to \int_{\Omega} \vert u \vert ^{p-2} uv \,dx, $$
and
$$ \begin{aligned}[b] &\int_{\Omega}\int_{\Omega}\frac{ \vert u_{j}(y) \vert ^{p_{\mu,s}^{\ast}} \vert u_{j}(x) \vert ^{p_{\mu,s}^{\ast}-2} u_{j}(x) v(x) }{ \vert x-y \vert ^{\mu}} \,dx \,dy\\ &\quad \to \int_{\Omega}\int_{\Omega}\frac{ \vert u(y) \vert ^{p_{\mu,s}^{\ast}} \vert u(x) \vert ^{p_{\mu,s}^{\ast}-2} u(x) v(x) }{ \vert x-y \vert ^{\mu}} \,dx \,dy. \end{aligned} $$
(10)
Indeed, the Riesz potential defines a linear continuous map from \(L^{\frac{2N}{2N-\mu}}(\Omega)\) to \(L^{\frac{2N}{\mu}}(\Omega)\) by the Hardy–Littlewood–Sobolev inequality. Then
$$ \frac{ \vert u_{j} \vert ^{p_{\mu,s}^{*}}}{ \vert x-y \vert ^{\mu}}\rightharpoonup\frac { \vert u \vert ^{p_{\mu,s}^{*}}}{ \vert x-y \vert ^{\mu}}\quad \mbox{in }L^{\frac{2N}{\mu}}(\Omega), $$
(11)
as \(n\rightarrow\infty\). Moreover,
$$ \vert u_{j} \vert ^{p_{\mu,s}^{*}-2}u_{j} \rightharpoonup \vert u \vert ^{p_{\mu,s}^{*}-2}u\quad \mbox{in } L^{\frac{p_{s}^{*}}{p_{\mu,s}^{*}}}(\Omega). $$
(12)
Combining (11) with (12) and \(\frac{p_{s}^{*}}{p_{\mu ,s}^{*}}=\frac{2N}{2N-\mu}\), we get the desired result (10). So passing to the limit in (9) shows that \(u \in X_{p}^{s}(\Omega)\) is a weak solution of (1).
Let \(\widetilde{u}_{j} = u_{j} - u\) and we will give the proof of \(\widetilde{u}_{j} \to0\) in \(X_{p}^{s}(\Omega)\). From [30, Lemma 3.2], we get
$$ \Vert \widetilde{u}_{j} \Vert _{s,p}^{p} = \Vert u_{j} \Vert _{s,p}^{p} - \Vert u \Vert _{s,p}^{p} + \mathrm {o}(1). $$
(13)
We discuss as in Lemma 2.2 in [10] and have
$$ \Vert \widetilde{u}_{j} \Vert _{o}^{2p_{\mu,s}^{\ast}} = \Vert u_{j} \Vert _{o}^{2p_{\mu,s}^{\ast}} - \Vert u \Vert _{o}^{2p_{\mu,s}^{\ast}} + \mathrm {o}(1). $$
(14)
Taking \(v = u_{j}\) in (9) gives
$$ \Vert u_{j} \Vert _{s,p}^{p} = \lambda \vert u \vert _{p}^{p} + \Vert u_{j} \Vert _{o}^{2p_{\mu ,s}^{\ast}} + \mathrm {o}(1). $$
(15)
Since \(\{u_{j}\}\) is bounded in \(X_{p}^{s}(\Omega)\) and converges to u in \(L^{p}(\Omega)\), and testing (9) with \(v = u\) gives
$$ \Vert u \Vert _{s,p}^{p} = \lambda \vert u \vert _{p}^{p} + \Vert u \Vert _{o}^{2p_{\mu ,s}^{\ast}}. $$
(16)
Combining (13)–(16) and (7) gives
$$\Vert \widetilde{u}_{j} \Vert _{s,p}^{p} = \Vert \widetilde{u}_{j} \Vert _{o}^{2p_{\mu ,s}^{\ast}} + \mathrm {o}(1) \le\frac{ \Vert \widetilde{u}_{j} \Vert _{s,p}^{2p_{\mu ,s}^{\ast}}}{S_{H,L}^{2p_{\mu,s}^{\ast}/p}} + \mathrm {o}(1), $$
so
$$ \Vert \widetilde{u}_{j} \Vert _{s,p}^{p} \bigl(S_{H,L}^{2p_{\mu,s}^{\ast}/p} - \Vert \widetilde{u}_{j} \Vert _{s,p}^{2p_{\mu,s}^{\ast}- p} \bigr) \le \mathrm {o}(1). $$
(17)
On the other hand,
$$\begin{aligned} c & = \frac{1}{p} \Vert u_{j} \Vert _{s,p}^{p} - \frac{\lambda}{p} \vert u \vert _{p}^{p} - \frac{1}{2p_{\mu,s}^{\ast}} \Vert u_{j} \Vert _{o}^{2p_{\mu ,s}^{\ast}} + \mathrm {o}(1) \quad \text{by (8)} \\ & = \biggl(\frac{1}{p}-\frac{1}{2p_{\mu,s}^{\ast}}\biggr) \bigl( \Vert u_{j} \Vert _{s,p}^{p} - \lambda \vert u \vert _{p}^{p} \bigr) + \mathrm {o}(1) \quad \text{by (15)} \\ & = \biggl(\frac{1}{p}-\frac{1}{2p_{\mu,s}^{\ast}}\biggr) \bigl( \Vert \widetilde{u}_{j} \Vert _{s,p}^{p} + \Vert u \Vert _{s,p}^{p} - \lambda \vert u \vert _{p}^{p} \bigr) + \mathrm {o}(1) \quad \text{by (13)} \\ & = \biggl(\frac{1}{p}-\frac{1}{2p_{\mu,s}^{\ast}}\biggr) \bigl( \Vert \widetilde{u}_{j} \Vert _{s,p}^{p} + \Vert u \Vert _{o}^{2p_{\mu,s}^{\ast}} \bigr) + \mathrm {o}(1) \quad \text{by (16)} \\ & \ge\biggl(\frac{1}{p}-\frac{1}{2p_{\mu,s}^{\ast}}\biggr) \Vert \widetilde{u}_{j} \Vert _{s,p}^{p} + \mathrm {o}(1), \end{aligned}$$
so
$$ \limsup_{j \to\infty} \Vert \widetilde{u}_{j} \Vert _{s,p}^{p} \le\frac {2pp_{\mu,s}^{\ast}c}{2p_{\mu,s}^{\ast}-p} < S_{H,L}^{\frac{2p_{\mu ,s}^{\ast}}{2p_{\mu,s}^{\ast}-p}}. $$
(18)
We conclude from (17) and (18) that \(\|\widetilde{u}_{j}\|_{s,p} \to0\).
Now we can apply Lemma 2.4 with \(b = \frac{2p_{\mu,s}^{*}-p}{2pp_{\mu,s}^{*}} S_{H,L}^{\frac{2p_{\mu,s}^{*}}{2p_{\mu,s}^{*}-p}}\). By [30, Proposition 3.5], \(\Psi^{\lambda_{k+m}}\) has a compact symmetric subset \(A_{0}\) with
$$i(A_{0}) = k + m. $$
We take \(B_{0} = \Psi_{\lambda_{k+1}}\), so that
$$i(S_{1} \setminus B_{0}) = k $$
by (4). Let \(R > r > 0\) and let A, B, and X be as in Lemma 2.4. For \(u \in B_{0}\),
$$I_{\lambda}(ru) \ge\frac{r^{p}}{p} \biggl(1 - \frac{\lambda}{\lambda _{k+1}} \biggr) - \frac{r^{2p_{\mu,s}^{\ast}}}{2p_{\mu,s}^{\ast}S_{H,L}^{2p_{\mu,s}^{\ast}/p}} $$
by (7). It follows that \(\inf I_{\lambda}(B) > 0\) if r is sufficiently small since \(\lambda< \lambda_{k+1}\) and \(2p_{\mu ,s}^{\ast}> p\). For \(u \in A_{0} \subset\Psi^{\lambda_{k+1}}\),
$$I_{\lambda}(Ru) \le\frac{R^{p}}{p} \biggl(1 - \frac{\lambda}{\lambda _{k+1}} \biggr) - \frac{R^{2p_{\mu,s}^{\ast}}}{2p_{\mu,s}^{\ast}} \biggl(\frac{1}{\lambda_{k+1} \vert \Omega \vert ^{{(p_{s}^{*}-p)}/{p_{s}^{*}}}C_{H,L}^{p}} \biggr)^{\frac{2p_{\mu,s}^{*}}{p}}, $$
so there exists \(R > r\) such that \(I_{\lambda}\le0\) on A. For \(u \in X\),
$$\begin{aligned} I_{\lambda}(u) & \le\frac{\lambda_{k+1} - \lambda}{p} \vert \Omega \vert ^{\frac{p_{s}^{*}-p}{p_{s}^{*}}} \vert u \vert _{p_{s}^{*}}^{p} - \frac{ \vert u \vert _{p_{s}^{*}}^{2p_{\mu ,s}^{*}}}{2p_{\mu,s}^{\ast}(C_{H,L})^{2p_{\mu,s}^{\ast}}} \\ & \le\sup_{\rho\ge0} \biggl[\frac{ \vert \Omega \vert ^{\frac {p_{s}^{*}-p}{p_{s}^{*}}}(\lambda_{k+1} - \lambda) \rho}{p} - \frac{\rho ^{2p_{\mu,s}^{\ast}/p}}{2p_{\mu,s}^{\ast}(C_{H,L})^{2p_{\mu,s}^{\ast}}} \biggr] \\ & = \biggl(\frac{1}{p}-\frac{1}{2p_{\mu,s}^{\ast}}\biggr) \vert {\Omega } \vert ^{\frac {2p_{\mu,s}^{\ast}(p_{s}^{*}-p)}{p_{s}^{*}(2p_{\mu,s}^{*}-p)}} (C_{H,L})^{\frac {2pp_{\mu,s}^{*}}{2p_{\mu,s}^{*}-p}} (\lambda_{k+1} - \lambda)^{2p_{\mu ,s}^{\ast}/({2p_{\mu,s}^{\ast}}-p)}. \end{aligned}$$
So
$$\begin{aligned} \sup I_{\lambda}(X) &\le\biggl(\frac{1}{p}-\frac{1}{2p_{\mu,s}^{\ast}}\biggr) \vert {\Omega} \vert ^{\frac{2p_{\mu,s}^{\ast}(p_{s}^{*}-p)}{p_{s}^{*}(2p_{\mu ,s}^{*}-p)}} (C_{H,L})^{\frac{2pp_{\mu,s}^{*}}{2p_{\mu,s}^{*}-p}} ( \lambda_{k+1} - \lambda)^{2p_{\mu,s}^{\ast}/{(2p_{\mu,s}^{\ast}}-p)} \\ &< \frac{2p_{\mu,s}^{\ast}-p}{2pp_{\mu,s}^{\ast}} S_{H,L}^{2p_{\mu ,s}^{\ast}/{(2p_{\mu,s}^{\ast}-p)}} \end{aligned} $$
by (5). Thus, functional \(I_{\lambda}\) has m distinct pairs of (nontrivial) critical points \(\pm u^{\lambda}_{j}\), \(j = 1,\dots,m\), such that
$$ \begin{aligned}[b] &0 < I_{\lambda}\bigl(u^{\lambda}_{j}\bigr) \le\biggl(\frac{1}{p}-\frac{1}{2p_{\mu ,s}^{\ast}}\biggr) \vert {\Omega} \vert ^{\frac{2p_{\mu,s}^{\ast}(p_{s}^{*}-p)}{p_{s}^{*}(2p_{\mu,s}^{*}-p)}} (C_{H,L})^{\frac{2pp_{\mu ,s}^{*}}{2p_{\mu,s}^{*}-p}} (\lambda_{k+1} - \lambda)^{2p_{\mu,s}^{\ast}/{(2p_{\mu,s}^{\ast}}-p)} \to0\\ &\quad \text{as } \lambda\nearrow\lambda_{k+1}. \end{aligned} $$
(19)
Then
$$\bigl\Vert u^{\lambda}_{j} \bigr\Vert _{o}^{2p_{\mu,s}^{\ast}} = \frac{2pp_{\mu,s}^{\ast}}{2p_{\mu,s}^{\ast}-p} \biggl[I_{\lambda}\bigl(u^{\lambda}_{j} \bigr) - \frac{1}{p} I_{\lambda}'\bigl(u^{\lambda}_{j} \bigr) u^{\lambda}_{j} \biggr] = \frac{2pp_{\mu ,s}^{\ast}}{2p_{\mu,s}^{\ast}-p} I_{\lambda}\bigl(u^{\lambda}_{j}\bigr) \to0 $$
and hence \(u^{\lambda}_{j} \to0\) in \(L^{p}(\Omega)\). Applying the Hölder inequality and Lemma 2.2, we deduce that
$$\bigl\Vert u^{\lambda}_{j} \bigr\Vert _{s,p}^{p} = p I_{\lambda}\bigl(u^{\lambda}_{j}\bigr) + \lambda \bigl\vert u^{\lambda}_{j} \bigr\vert _{p}^{p} + \frac{p}{2p_{\mu,s}^{\ast}} \bigl\Vert u^{\lambda}_{j} \bigr\Vert _{o}^{2p_{\mu,s}^{\ast}} \to0. $$

Declarations

Acknowledgements

The authors want to thank the editors and reviewers sincerely.

Availability of data and materials

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Authors’ information

All authors come from Jiangnan University of China.

Funding

The authors are supported financially by the National Natural Science Foundation of China(11501252,11571176).

Authors’ contributions

The authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
School of Science, Jiangnan University, Wuxi, People’s Republic of China

References

  1. Autuori, G., Fiscella, A., Pucci, P.: Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 125, 699–714 (2015) MathSciNetView ArticleMATHGoogle Scholar
  2. Autuori, G., Pucci, P.: Elliptic problems involving the fractional Laplacian in \({\mathbb {R}}^{N}\). J. Differ. Equ. 255(8), 2340–2362 (2013) View ArticleMATHGoogle Scholar
  3. Benci, V.: On critical point theory for indefinite functionals in the presence of symmetries. Trans. Am. Math. Soc. 274, 533–572 (1982) MathSciNetView ArticleMATHGoogle Scholar
  4. Brasco, L., Parini, E., Squassina, M.: Stability of variational eigenvalues for the fractional p-Laplacian. Discrete Contin. Dyn. Syst., Ser. A 36(4), 1813–1845 (2017) MathSciNetView ArticleMATHGoogle Scholar
  5. d’Avenia, P., Siciliano, G., Squassina, M.: On fractional Choquard equations. Math. Models Methods Appl. Sci. 25(8), 1447–1476 (2014) MathSciNetView ArticleMATHGoogle Scholar
  6. Del Pezzo, L.M., Salort, A.M.: The first non-zero Neumann p-fractional eigenvalue. Nonlinear Anal., Theory Methods Appl. 118, 130–143 (2015) MathSciNetView ArticleMATHGoogle Scholar
  7. Fadell, E.R., Rabinowitz, P.H.: Generalized cohomological index theories for Lie group actions with an application to bifurcation questions for Hamiltonian systems. Invent. Math. 45, 139–174 (1978) MathSciNetView ArticleMATHGoogle Scholar
  8. Franzina, G., Palatucci, G.: Fractional p-eigenvalues. Riv. Mat. Univ. Parma 5(2), 373–386 (2014) MathSciNetMATHGoogle Scholar
  9. Gao, F., Shen, Z., Yang, M.: On the critical Choquard equation with potential well. arXiv preprint (2017). arXiv:1703.01737
  10. Gao, F., Yang, M.: On the Brezis–Nirenberg type critical problem for nonlinear Choquard equation. Sci. China Math. 61(7), 1219–1242 (2018) MathSciNetView ArticleMATHGoogle Scholar
  11. Guo, L., Hu, T.: Existence and asymptotic behavior of the least energy solutions for fractional Choquard equations with potential well. arXiv preprint (2017) arXiv:1703.08028
  12. Ho, K., Perera, K., Sim, I., Squassina, M.: A note on fractional p-Laplacian problems with singular weights. J. Fixed Point Theory Appl. 19(1), 157–173 (2017) MathSciNetView ArticleMATHGoogle Scholar
  13. Huang, L., Yang, Y.: Asymmetric critical fractional p-Laplacian problems. Electron. J. Differ. Equ. 2017, 103 (2017) MathSciNetView ArticleMATHGoogle Scholar
  14. Iannizzotto, A., Liu, S., Perera, K., Squassina, M.: Existence results for fractional p-Laplacian problems via Morse theory. Adv. Calc. Var. 9(2), 101–125 (2016) MathSciNetView ArticleMATHGoogle Scholar
  15. Iannizzotto, A., Mosconi, S., Squassina, M.: A note on global regularity for the weak solutions of fractional p-Laplacian equations. Rend. Lincei Mat. Appl. 27, 15–24 (2016) MathSciNetMATHGoogle Scholar
  16. Iannizzotto, A., Mosconi, S., Squassina, M.: Global Hölder regularity for the fractional p-Laplacian. Rev. Mat. Iberoam. 32, 1355–1394 (2016) View ArticleMATHGoogle Scholar
  17. Iannizzotto, A., Squassina, M.: Weyl-type laws for fractional p-eigenvalue problems. Asymptot. Anal. 88(4), 233–245 (2014) MathSciNetMATHGoogle Scholar
  18. Lehrer, R., Maia, L.A., Squassina, M.: On fractional p-Laplacian problems with weight. Differ. Integral Equ. 28, 1–2 (2014) MathSciNetMATHGoogle Scholar
  19. Lieb, E.H., Loss, M.: Analysis. Graduate Studies in Mathematics, vol. 14. Am. Math. Soc., Providence (2001) MATHGoogle Scholar
  20. Lindgren, E., Lindqvist, P.: Fractional eigenvalues. Calc. Var. Partial Differ. Equ. 49(1–2), 795–826 (2014) MathSciNetView ArticleMATHGoogle Scholar
  21. Mosconi, S., Perera, K., Squassina, M., Yang, Y.: The Brezis–Nirenberg problem for the fractional p-Laplacian. Calc. Var. Partial Differ. Equ. 55(4), 105 (2016) MathSciNetView ArticleMATHGoogle Scholar
  22. Mosconi, S., Squassina, M.: Nonlocal problems at nearly critical growth. Nonlinear Anal. 136, 84–101 (2016) MathSciNetView ArticleMATHGoogle Scholar
  23. Mosconi, S., Squassina, M.: Recent progresses in the theory of nonlinear nonlocal problems. Bruno Pini Math. Anal. Semin. 7, 147–164 (2016) MathSciNetMATHGoogle Scholar
  24. Mukherjee, T., Sreenadh, K.: Fractional Choquard equation with critical nonlinearities. arXiv preprint (2016). arXiv:1605.06805
  25. Mukherjee, T., Sreenadh, K.: On Dirichlet problem for fractional p-Laplacian with singular nonlinearity. Adv. Nonlinear Anal. (2016). https://doi.org/10.1515/anona-2016-0100 Google Scholar
  26. Nezza, E.D., Palatucci, G., Valdinoci, E.: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521–573 (2012) MathSciNetView ArticleMATHGoogle Scholar
  27. Perera, K.: Nontrivial critical groups in p-Laplacian problems via the Yang index. Topol. Methods Nonlinear Anal. 21(2), 301–309 (2003) MathSciNetView ArticleMATHGoogle Scholar
  28. Perera, K., Agarwal, R.P., O’Regan, D.: Morse Theoretic Aspects of p-Laplacian Type Operators. Mathematical Surveys and Monographs, vol. 161. Am. Math. Soc., Providence (2010) View ArticleMATHGoogle Scholar
  29. Perera, K., Squassina, M., Yang, Y.: A note on the Dancer–Fuc̆ík spectra of the fractional p-Laplacian and Laplacian operators. Adv. Nonlinear Anal. 4(1), 13–23 (2015) MathSciNetMATHGoogle Scholar
  30. Perera, K., Squassina, M., Yang, Y.: Bifurcation and multiplicity results for critical fractional p-Laplacian problems. Math. Nachr. 289(2–3), 332–342 (2016) MathSciNetView ArticleMATHGoogle Scholar
  31. Perera, K., Szulkin, A.: p-Laplacian problems where the nonlinearity crosses an eigenvalue. Discrete Contin. Dyn. Syst. 13(3), 743–753 (2005) MathSciNetView ArticleMATHGoogle Scholar
  32. Pucci, P., Xiang, M., Zhang, B.: Existence and multiplicity of entire solutions for fractional p-Kirchhoff equations. Adv. Nonlinear Anal. 5(1), 27–55 (2016) MathSciNetMATHGoogle Scholar
  33. Shang, X., Zhang, J., Yang, Y.: Positive solutions of nonhomogeneous fractional Laplacian problem with critical exponent. Commun. Pure Appl. Anal. 13(2), 567–584 (2014) MathSciNetMATHGoogle Scholar
  34. Singh, G.: Nonlocal perturbations of fractional Choquard equation. arXiv preprint (2017). arXiv:1705.05775
  35. Wang, F., Xiang, M.: Multiplicity of solutions for a class of fractional Choquard–Kirchhoff equations involving critical nonlinearity. Anal. Math. Phys. (2017). https://doi.org/10.1007/s13324-017-0174-8 Google Scholar
  36. Wang, L., Zhang, B.: Infinitely many solutions for Schrödinger–Kirchhoff type equations involving the fractional p-Laplacian and critical exponent. Electron. J. Differ. Equ. 2016, 339 (2016) MATHGoogle Scholar
  37. Xiang, M., Zhang, B.: Degenerate Kirchhoff problems involving the fractional p-Laplacian without the (AR) condition. Complex Var. Elliptic Equ. 60(9), 1–11 (2015) MathSciNetView ArticleMATHGoogle Scholar
  38. Xiang, M., Zhang, B., Rǎdulescu, V.D.: Existence of solutions for perturbed fractional p-Laplacian equations. J. Differ. Equ. 260(2), 1392–1413 (2016) MathSciNetView ArticleMATHGoogle Scholar

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