- Research
- Open Access
- Published:
Multiplicity results for a fractional Kirchhoff equation involving sign-changing weight function
Boundary Value Problems volume 2016, Article number: 212 (2016)
Abstract
In this paper, we prove the existence and multiplicity of solutions for a fractional Kirchhoff equation involving a sign-changing weight function which generalizes the corresponding result of Tsung-fang Wu (Rocky Mt. J. Math. 39:995-1011, 2009). Our main results are based on the method of a Nehari manifold.
1 Introduction
In this paper, we consider the following fractional elliptic equation with sign-changing weight functions:
where Ω is a smooth bounded domain in \(\mathbb{R}^{N}\), \(N > 2s\), \(0 < s < 1\), \(0 \le q < 1 < r < p_{s}^{*} - 1\) (\(p_{s}^{*} = \frac{pN}{N-ps}\)); \(\lambda > 0\), \(M(t) = a + b t^{p-1}\), \((-\Delta)_{p}^{s}\) is the fractional p-Laplacian operator defined as
We may assume that the weight functions \(f(x)\) and \(g(x)\) are as follows:
-
(H1)
\(f^{+} = \max \{f, 0\} \not \equiv 0\), and \(f \in L^{\mu_{q}}(\Omega)\) where \(\mu_{q} = \frac{\mu}{\mu - (q+1)}\) for some \(\mu \in (q+1, p_{s}^{*})\), with in addition \(f(x) \ge 0\) a.e. in Ω in the case \(q = 0\);
-
(H2)
\(g^{+} = \max \{g, 0\} \not \equiv 0\), and \(g \in L^{\nu_{r}}(\Omega)\) where \(\nu_{r} = \frac{\nu}{\nu - (r+1)}\) for some \(\nu \in (r+1, p_{s}^{*})\).
The fractional Kirchhoff type problems have been studied by many authors in recent years; see [2–6] and references therein. In the subcritical case, Pucci and Saldi in [5] studied the following Kirchhoff type problem in \(\mathbb{R}^{N}\):
with \(n > ps\), \(s \in (0, 1)\), and they established the existence and multiplicity of entire solutions using variational methods and topological degree theory for the above problem with a real parameter λ under the suitable integrability assumptions of the weights V, w, and h. In [7], Mishra and Sreenadh have studied the following Kirchhoff problem with sign-changing weights:
and they obtained the multiplicity of non-negative solutions in the subcritical case \(\alpha < p_{s}^{*}\) by minimizing the energy functional over non-empty decompositions of Nehari manifold.
When \(p=2\), \(s=1\), \(a=1\) and \(b=0\), problem (1.1) is reduced to the following semilinear elliptic equation:
In [1], Wu proved that equation (1.2) involving a sign-changing weight function has at least two solutions by using the Nehari manifold.
Motivated by the above work, in this paper, we investigate the existence and multiplicity of solutions for a fractional Kirchhoff equation (1.1) and extend the main results of Wu [1].
This article is organized as follows. In Section 2, we give some notations and preliminaries. Section 3 is devoted to the proof that problem (1.1) has at least two solutions for λ sufficiently small.
2 Preliminaries
For any \(s \in (0, 1)\), \(1 < p < \infty\), we define
where \(Q = \mathbb{R}^{2N} \setminus (\mathcal{C} \Omega \times \mathcal{C} \Omega)\) with \(\mathcal{C} \Omega = \mathbb{R}^{N} \setminus \Omega\). The space X is endowed with the norm defined by
The functional space \(X_{0}\) denotes the closure of \(C_{0}^{\infty}(\Omega)\) in X. By [8], the space \(X_{0}\) is a Hilbert space with scalar product
and the norm
For further details on X and \(X_{0}\) and also for their properties, we refer to [8] and the references therein.
Throughout this section, we denote the best Sobolev constant by \(S_{l}\) for the embedding of \(X_{0}\) into \(L^{l}(\Omega)\), which is defined as
where \(l \in [p, p_{s}^{*}]\).
A function \(u \in X_{0}\) is a weak solution of problem (1.1) if
Associated with equation (1.1), we consider the energy functional \(\mathcal{J}_{\lambda, M}\) in \(X_{0}\)
where \(\hat{M}(t) = \int_{0}^{t} M(\mu) \,d \mu\).
It is easy to see that the solutions of equation (1.1) are the critical points of the energy functional \(\mathcal{J}_{\lambda, M}\).
The Nehari manifold for \(\mathcal{J}_{\lambda, M}\) is defined as
The Nehari manifold \(\mathcal{N}_{\lambda, M}(\Omega)\) is closely linked to the behavior of functions of the form \(h_{\lambda, M} : t \to \mathcal{J}_{\lambda, M}(tu) \) for \(t > 0\), named fibering maps [9]. If \(u \in X_{0}\), we have
and
Obviously,
which implies that for \(u \in X_{0} \setminus \{0\}\) and \(t > 0\), \(h_{\lambda, M}(t) = 0\) if and only if \(tu \in \mathcal{N}_{\lambda, M}(\Omega)\), i.e., positive critical points of \(h_{\lambda, M}\) correspond to points on the Nehari manifold. In particular, \(h_{\lambda, M}(1) = 0\) if and only if \(u \in \mathcal{N}_{\lambda, M}(\Omega)\). Hence, we define
For each \(u \in \mathcal{N}_{\lambda, M}(\Omega)\), we have
Let \(M(t) = a + b t^{p-1}\), where \(a > 0\), \(b \ge 0\) and \(p > 1\). If \(u \in \mathcal{N}_{\lambda, M}^{0}(\Omega)\), then \(h^{\prime \prime}_{\lambda, M}(1) = 0\), and we have by (2.1) and (2.2)
For convenience, we let
-
(H3)
\(0 < q < 1\), \(p > 1+q\) and \(p_{s}^{*} - 1 > r \begin{cases} > p^{2} - 1, & b \neq 0, \\ > p-1, & b=0. \end{cases}\)
Lemma 2.1
If (H1) and (H3) hold, then the energy functional \(\mathcal{J}_{\lambda, M}\) is coercive and bounded below on \(\mathcal{N}_{\lambda, M}(\Omega)\).
Proof
For \(u \in \mathcal{N}_{\lambda, M}(\Omega)\), we have by the Hölder and Sobolev inequalities
where \(\mu_{q} = \frac{\mu}{\mu-(q+1)}\), \(\mu \in (q+1, p_{s}^{*})\). Thus \(\mathcal{J}_{\lambda, M}\) is coercive and bounded below on \(\mathcal{N}_{\lambda, M}(\Omega)\). □
Lemma 2.2
Let (H1)-(H3) hold. There exists \(\lambda_{1} > 0\) such that for any \(\lambda \in (0, \lambda_{1})\), we have \(\mathcal{N}_{\lambda, M}^{0}(\Omega) = \emptyset\).
Proof
If not, that is, \(\mathcal{N}_{\lambda, M}^{0} (\Omega) \neq \emptyset\) for each \(\lambda > 0\), then by (2.3) and the Hölder and Sobolev inequalities, we have for \(u_{0} \in \mathcal{N}_{\lambda, M}^{0}(\Omega)\)
which implies that
and so
Similarly, we obtain by (2.4) and the Hölder and Sobolev inequalities
which implies that
But (2.5) contradicts (2.6) if λ is sufficiently small. Hence, we conclude that there exists \(\lambda_{1} > 0\) such that \(\mathcal{N}_{\lambda, M}^{0}(\Omega) = \emptyset\) for \(\lambda \in (0, \lambda_{1})\). □
Let
From Lemma 2.2, for \(\lambda \in (0, \lambda_{1})\), we write \(\mathcal{N}_{\lambda, M}(\Omega) = \mathcal{N}_{\lambda, M}^{+}(\Omega) \cup \mathcal{N}_{\lambda, M}^{-}(\Omega)\) and define
Lemma 2.3
(i) If \(u \in \mathcal{N}_{\lambda, M}^{+}(\Omega)\), then \(\int_{\Omega} f |u|^{q+1} \,dx > 0\).
(ii) If \(u \in \mathcal{N}_{\lambda, M}^{-}(\Omega)\), then \(\int_{\Omega} g |u|^{r+1} \,dx > 0\).
The proof is immediate from (2.3) and (2.4).
Define the function \(k_{u} : \mathbb{R}^{+} \to \mathbb{R}\) as follows:
Obviously, \(t u \in \mathcal{N}_{\lambda, M}(\Omega)\) if and only if \(k_{u}(t) = \lambda \int_{\Omega} f |u|^{q+1} \,dx\). Moreover,
which implies that \(t^{q} k_{u}^{\prime}(t) = h_{\lambda, M}^{\prime \prime}(t)\) for \(t u \in \mathcal{N}_{\lambda, M}(\Omega)\). That is, \(u \in \mathcal{N}_{\lambda, M}^{+}(\Omega)\) (or \(\mathcal{N}_{\lambda, M}^{-}(\Omega)\)) if and only if \(k_{u}^{\prime}(t) > 0\) (or <0).
Set
Lemma 2.4
Assume that (H1)-(H3) hold. Let \(\lambda_{2} = \frac{A}{\|f\|_{L^{\mu_{q}}} S_{\mu}^{q+1}}\). Then, for each \(u \in X_{0} \setminus \{0\}\) and \(\lambda \in (0, \lambda_{2})\), we have:
-
(1)
If \(\int_{\Omega} f |u|^{q+1} \,dx \le 0\), then there exists a unique \(t^{-} = t^{-}(u) > t_{\max}(u)\) such that \(t^{-} u \in \mathcal{N}_{\lambda, M}^{-}(\Omega)\) and
$$ \mathcal{J}_{\lambda, M}\bigl(t^{-} u\bigr) = \sup_{t \ge 0} \mathcal{J}_{\lambda, M}(t u) > 0. $$(2.10) -
(2)
If \(\int_{\Omega} f |u|^{q+1} \,dx > 0\), then there exists a unique \(0 < t^{+} = t^{+}(u) < t_{\max}(u) < t^{-}\) such that \(t^{+} u \in \mathcal{N}_{\lambda, M}^{+}(\Omega)\), \(t^{-} u \in \mathcal{N}_{\lambda, M}^{-}(\Omega)\) and
$$ \mathcal{J}_{\lambda, M}\bigl(t^{+} u\bigr) = \inf _{0 \le t \le t_{\max}(u)} \mathcal{J}_{\lambda, M}(t u), \qquad \mathcal{J}_{\lambda, M}\bigl(t^{-} u\bigr) = \sup_{t \ge 0} \mathcal{J}_{\lambda, M}(t u). $$(2.11)
Proof
and
which implies that \(k_{u}(0) = 0\), \(k_{u}(t) \to - \infty\) as \(t \to \infty\), \(\lim_{t \to 0^{+}} k_{u}^{\prime}(t) > 0\) and \(\lim_{t \to \infty} k_{u}^{\prime}(t) < 0\). Thus there exists a unique \(t_{\max}(u) := t_{\max} > 0\) such that \(k_{u}(t)\) is increasing on \((0, t_{\max})\), decreasing on \((t_{\max}, \infty)\) and \(k_{u}^{\prime}(t_{\max})=0\). Moreover, \(t_{\max}\) is the root of
From (2.12), we obtain
Hence, we have by (2.12), (2.13), and the Hölder and Sobolev inequalities
Case (1): \(\int_{\Omega} f |u|^{q+1} \,dx \le 0\). Then \(k_{u}(t) = \lambda \int_{\Omega} f |u|^{q+1} \,dx \) has unique solution \(t^{-} > t_{\max}\) and \(k_{u}^{\prime}(t^{-}) < 0\). On the other hand, we have
and
Hence, \(t^{-} u \in \mathcal{N}_{\lambda, M}^{-}(\Omega)\) or \(t^{-} = 1\). For \(t > t_{\max}\), we obtain
for \(t = t^{-}\). Thus, \(\mathcal{J}_{\lambda, M}(u) = \sup_{t \ge 0} \mathcal{J}_{\lambda, M}(t u)\). Furthermore, we have
Let
Similar to the argument in the function \(k_{u}(t)\), we see that \(h_{u}(t)\) achieves its maximum at \(t_{m} \ge (\frac{a \|u\|_{X_{0}}^{p}}{\int_{\Omega} g |u|^{r+1} \,dx} )^{\frac{1}{r-p+1} } \). Thus, we have
Case (2): \(\int_{\Omega} f |u|^{q+1} \,dx > 0\). By (2.14) and
Then there exist \(t^{+}\) and \(t^{-}\) such that \(0 < t^{+} < t_{\max} < t^{-}\),
Moreover, we have \(k_{u}^{\prime}(t^{+}) > 0\) and \(k_{u}^{\prime}(t^{-}) < 0\). Thus, there are two multiples of u lying in \(\mathcal{N}_{\lambda, M}(\Omega)\), that is, \(t^{+} u \in \mathcal{N}_{\lambda, M}^{+}(\Omega)\) and \(t^{-} u \in \mathcal{N}_{\lambda, M}^{-}(\Omega)\), and \(\mathcal{J}_{\lambda, M}(t^{-} u) \ge \mathcal{J}_{\lambda, M}(t u) \ge \mathcal{J}_{\lambda, M}(t^{+} u)\) for each \(t \in [t^{+}, t^{-}]\) and \(\mathcal{J}_{\lambda, M}(t^{+} u) \le \mathcal{J}_{\lambda, M}(t u)\) for each \(t \in [0, t^{+}]\). Hence, \(t^{-} = 1\) and
□
Lemma 2.5
If (H3) holds, then we have \(c_{\lambda} \le c_{\lambda}^{+} < 0\).
Proof
For \(u \in \mathcal{N}_{\lambda, M}^{+}\), we get
Thus, we have
which implies that \(c_{\lambda} \le c_{\lambda}^{+} < 0\). □
3 Main results
Using the idea of Ni-Takagi [10], we have the following.
Lemma 3.1
For each \(u \in \mathcal{N}_{\lambda, M}(\Omega)\), there exist \(\epsilon > 0\) and a differentiable function \(\xi : B(0; \epsilon) \subset X_{0} \to \mathbb{R}^{+}\) such that \(\xi(0) = 1\), the function \(\xi(v)(u-v) \in \mathcal{N}_{\lambda, M}(\Omega)\) and
for all \(v \in X_{0}\), where
Proof
For \(u \in \mathcal{N}_{\lambda, M}(\Omega)\), we define a function \(\mathcal{F} : \mathbb{R} \times X_{0} \to \mathbb{R}\) by
Then \(\mathcal{F}_{u}(1, 0) = \langle \mathcal{J}_{\lambda, M}^{\prime}(u), u \rangle = 0\) and
From the implicit function theorem, we know that there exist \(\epsilon > 0\) and a differentiable function \(\xi : B(0; \epsilon) \subset X_{0} \to \mathbb{R}\) such that \(\xi(0) = 1\),
where W is as in (3.2), and
which is equivalent to
which implies that \(\xi(v)(u-v) \in \mathcal{N}_{\lambda, M}(\Omega)\). □
Similar to the argument in Lemma 3.1, we can obtain the following lemma.
Lemma 3.2
For each \(u \in \mathcal{N}_{\lambda, M}^{-}(\Omega)\), there exist \(\epsilon > 0\) and a differentiable function \(\xi^{-} : B(0; \epsilon) \subset X_{0} \to \mathbb{R}^{+}\) such that \(\xi^{-}(0) = 1\), the function \(\xi^{-}(v)(u-v) \in \mathcal{N}_{\lambda, M}^{-}(\Omega)\) and
for all \(v \in X_{0}\), where W is as in (3.2).
Let
-
(H4)
\(p < 2 + \frac{(r-1)q}{r}\).
Moreover, we let
and
Remark 3.1
By (H4) we know that \(p^{*} < 0\).
Lemma 3.3
Assume that (H1)-(H4) hold. Let \(\Gamma_{0} = \min \{\lambda_{1}, \lambda_{2}, \lambda_{3}\}\), then for \(\lambda \in (0, \Gamma_{0})\):
-
(i)
There exists a minimizing sequence \(\{u_{n}\} \subset \mathcal{N}_{\lambda, M}(\Omega)\) such that
$$\mathcal{J}_{\lambda, M}(u_{n}) = c_{\lambda} + o(1), \qquad \mathcal{J}_{\lambda, M}^{\prime}(u_{n}) = o(1) \quad \textit{in } (X_{0})^{*}. $$ -
(ii)
There exists a minimizing sequence \(\{u_{n}\} \subset \mathcal{N}_{\lambda, M}^{-}(\Omega)\) such that
$$\mathcal{J}_{\lambda, M}(u_{n}) = c_{\lambda}^{-} + o(1), \qquad \mathcal{J}_{\lambda, M}^{\prime}(u_{n}) = o(1) \quad \textit{in } (X_{0})^{*}. $$
Proof
By the Ekeland variational principle [11] and Lemma 2.2, there exists a minimizing sequence \(\{u_{n}\} \subset \mathcal{N}_{\lambda, M}(\Omega)\) such that
and
Let n large enough, by Lemma 2.5, we obtain
which implies that
This implies \(u_{n} \neq 0\) and by using (3.4), (3.5), and the Hölder inequality, we get
and
In the following, we will prove that
By using Lemma 3.1 with \(u_{n}\) we get the functions \(\xi_{n} : B(0; \epsilon_{n}) \to \mathbb{R}^{+}\) for some \(\epsilon_{n} > 0\), such that \(\xi_{n}(w) (u_{n} - w) \in \mathcal{N}_{\lambda, M}(\Omega)\). For fixed \(n \in \mathbb{N}\), we choose \(0 < \rho < \epsilon_{n}\). Let \(u \in X_{0}\) with \(u \neq 0\) and let \(w_{\rho} = \frac{\rho u}{\|u\|_{X_{0}}}\). Set \(\eta_{\rho} = \xi_{n}(w_{\rho}) (u_{n} - w_{\rho})\), since \(\eta_{\rho} \in \mathcal{N}_{\lambda, M}(\Omega)\), we deduce from (3.4) that
and by the mean value theorem, we obtain
Hence,
By \(\xi_{n}(w_{\rho}) (u_{n} - w_{\rho}) \in \mathcal{N}_{\lambda, M}(\Omega)\) and (3.8) it follows that
Thus,
Since
and
taking the limit \(\rho \to 0\) in (3.9), we obtain
for some constant \(C > 0\), independent of ρ. In the following, we will show that \(\|\xi_{n}^{\prime}(0)\|\) is uniformly bounded in n. From (3.1), (3.7), and the Hölder inequality, we obtain for some \(\kappa > 0\)
We only need to prove that
for some \(c > 0\) and n large enough. If (3.10) is fails, then there exists a subsequence \(\{u_{n}\}\) such that
Combining (3.11) with (3.6), we may find a suitable constant \(d > 0\) such that
By (3.11) and \(u_{n} \in \mathcal{N}_{\lambda, M}(\Omega)\), we have
Moreover, we have by (3.11) and (3.13)
which implies that
Let
where
From (3.11), it is easy to see that
Thus,
But, by (3.12), (3.14), and \(\lambda \in \Gamma_{0}\),
which contradicts (3.16), where \(p^{*} = \frac{(p-2)r}{r-1} - q < 0\).
Hence, we obtain
This completes the proof of (i). Similarly, we can prove (ii) by using Lemma 3.2. □
Theorem 3.4
Assume that (H1)-(H4) hold. For each \(0 < \lambda < \Gamma_{0}\) (\(\Gamma_{0}\) is as in Lemma 3.3), the functional \(\mathcal{J}_{\lambda, M}\) has a minimizer \(u_{\lambda}^{+}\) in \(\mathcal{N}_{\lambda, M}^{+}(\Omega)\) satisfying:
-
(1)
\(\mathcal{J}_{\lambda, M}(u_{\lambda}^{+}) = c_{\lambda}^{+} = c_{\lambda} \);
-
(2)
\(u_{\lambda}^{+}\) is a solution of (1.1).
Proof
By Lemma 3.3(i), there exists a minimizing sequence \(\{u_{n}\} \subset \mathcal{N}_{\lambda, M}(\Omega) \) for \(\mathcal{J}_{\lambda, M}\) on \(\mathcal{N}_{\lambda, M}(\Omega)\) such that
From Lemma 2.5 and the compact embedding theorem, we see that there exist a subsequence \(\{u_{n}\}\) and \(u_{\lambda}^{+} \in X_{0}\) such that
and
In the following we will prove that \(\int_{\Omega} f |u_{\lambda}^{+}|^{q+1} \,dx \neq 0\). In fact, if not, by (3.17) and the Hölder inequality we can obtain
as \(n \to \infty\). Hence,
and
which contradicts \(\mathcal{J}_{\lambda, M}(u_{n}) \to c_{\lambda} < 0\) as \(n \to \infty\). Furthermore,
Thus, \(u_{\lambda}^{+} \in \mathcal{N}_{\lambda, M}(\Omega)\) is a nonzero solution of (1.1) and \(\mathcal{J}_{\lambda, M}(u_{\lambda}^{+}) \ge c_{\lambda}\). Next, we will prove that \(\mathcal{J}_{\lambda, M}(u_{\lambda}^{+}) = c_{\lambda}\). Since
Hence, \(\mathcal{J}_{\lambda, M}(u_{\lambda}^{+}) = c_{\lambda}\). Moreover, we have \(u_{\lambda}^{+} \in \mathcal{N}_{\lambda, M}^{+}(\Omega)\). In fact, if \(u_{\lambda}^{+} \in \mathcal{N}_{\lambda, M}^{-}(\Omega)\), by Lemma 2.4, there are unique \(t^{+}\) and \(t^{-}\) such that \(t^{+} u_{\lambda}^{+} \in \mathcal{N}_{\lambda, M}^{+}(\Omega)\) and \(t^{-} u_{\lambda}^{+} \in \mathcal{N}_{\lambda, M}^{-}(\Omega)\), we have \(t_{\lambda}^{+} < t_{\lambda}^{-} = 1\). Since
there exists \(t_{\lambda}^{+} < t^{*} \le t_{\lambda}^{-} \) such that \(\mathcal{J}_{\lambda, M}(t_{\lambda}^{+} u_{\lambda}^{+}) < \mathcal{J}_{\lambda, M}(t^{*} u_{\lambda}^{+})\). By Lemma 2.4, we get
which is a contradiction. Since \(\mathcal{J}_{\lambda, M}(u_{\lambda}^{+}) = \mathcal{J}_{\lambda, M}(|u_{\lambda}^{+}|)\) and \(|u_{\lambda}^{+}| \in \mathcal{N}_{\lambda, M}^{+}(\Omega)\), we see that \(u_{\lambda}^{+}\) is a solution of (1.1) by Lemma 2.3. □
Similarly, we can obtain the theorem of existence of a local minimum for \(\mathcal{J}_{\lambda, M}\) on \(\mathcal{N}_{\lambda, M}^{-}(\Omega)\) as follows.
Theorem 3.5
Assume that (H1)-(H4) hold. For each \(0 < \lambda < \Gamma_{0}\) (\(\Gamma_{0}\) is as in Lemma 3.3), the functional \(\mathcal{J}_{\lambda, M}\) has a minimizer \(u_{\lambda}^{-}\) in \(\mathcal{N}_{\lambda, M}^{-}(\Omega)\) satisfying:
-
(1)
\(\mathcal{J}_{\lambda, M}(u_{\lambda}^{-}) = c_{\lambda}^{-} \);
-
(2)
\(u_{\lambda}^{-}\) is a solution of (1.1).
Finally, we give the main result of this paper as follows.
Theorem 3.6
Suppose that the conditions (H1)-(H4) hold. Then there exists \(\Gamma_{0} > 0\) such that for \(\lambda \in (0, \Gamma_{0})\), (1.1) has at least two solutions.
Proof
From Theorems 3.4, 3.5, we see that (1.1) has two solutions \(u_{\lambda}^{+}\) and \(u_{\lambda}^{-}\) such that \(u_{\lambda}^{+} \in \mathcal{N}_{\lambda, M}^{+}(\Omega)\), \(u_{\lambda}^{-} \in \mathcal{N}_{\lambda, M}^{-}(\Omega)\). Since \(\mathcal{N}_{\lambda, M}^{+}(\Omega) \cap \mathcal{N}_{\lambda, M}^{-}(\Omega) = \emptyset\), we see that \(u_{\lambda}^{+}\) and \(u_{\lambda}^{-}\) are different. □
Remark 3.2
Obviously, if \(p=2\), then (H3) and (H4) hold. Moreover, if \(p=2\), \(s=1\), \(a=1\), and \(b=0\), then Theorem 3.6 is in agreement with Theorem 1.2 in [1].
References
Wu, TF: Multiplicity results for a semilinear elliptic equation involving sign-changing weight function. Rocky Mt. J. Math. 39, 995-1011 (2009)
Autuori, G, Fiscella, A, Pucci, P: Stationary Kirchhoff problems involving a fractional elliptic operator and a critical nonlinearity. Nonlinear Anal. 125, 699-714 (2015)
Chen, CY, Kuo, YC, Wu, TF: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions. J. Differ. Equ. 250, 1876-1908 (2011)
Fiscella, A, Valdinoci, E: A critical Kirchhoff type problem involving a nonlocal operator. Nonlinear Anal. 94, 156-170 (2014)
Pucci, P, Saldi, S: Critical stationary Kirchhoff equations in \(\mathbb{R}^{N}\) involving nonlocal operators. Rev. Mat. Iberoam. 32, 1-22 (2016)
Pucci, P, Xiang, M, Zhang, B: Multiple solutions for nonhomogeneous Schrödinger-Kirchhoff type equations involving the fractional p-Laplacian in \(\mathbb{R}^{N}\). Calc. Var. Partial Differ. Equ. 54(3), 2785-2806 (2015)
Mishra, PK, Sreenadh, K: Existence and multiplicity results for fractional p-Kirchhoff equation with sign changing nonlinearities. Adv. Pure Appl. Math. (2015). doi:10.1515/apam-2015-0018
Di Nezza, E, Palatucci, G, Valdinoci, E: Hitchhiker’s guide to the fractional Sobolev spaces. Bull. Sci. Math. 136, 521-573 (2012)
Drabek, P, Pohozaev, SI: Positive solutions for the p-Laplacian: application of the fibering method. Proc. R. Soc. Edinb. A 127, 703-726 (1997)
Ni, WM, Takagi, I: On the shape of least energy solution to a Neumann problem. Commun. Pure Appl. Math. 44, 819-851 (1991)
Ekeland, I: On the variational principle. J. Math. Anal. Appl. 17, 324-353 (1974)
Acknowledgements
This work is supported by Natural Science Foundation of China (11571136 and 11271364).
Author information
Authors and Affiliations
Corresponding author
Additional information
Competing interests
The author declares that he has no competing interests.
Author’s contributions
All results belong to CB.
Rights and permissions
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.
About this article
Cite this article
Bai, C. Multiplicity results for a fractional Kirchhoff equation involving sign-changing weight function. Bound Value Probl 2016, 212 (2016). https://doi.org/10.1186/s13661-016-0715-3
Received:
Accepted:
Published:
DOI: https://doi.org/10.1186/s13661-016-0715-3
MSC
- 35J50
- 35J60
- 47G20
Keywords
- fractional p-Laplacian
- Kirchhoff type problem
- sign-changing weight
- Nehari manifold