Open Access

Multiplicity results for a fractional Kirchhoff equation involving sign-changing weight function

Boundary Value Problems20162016:212

https://doi.org/10.1186/s13661-016-0715-3

Received: 5 August 2016

Accepted: 11 November 2016

Published: 25 November 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.

Keywords

fractional p-Laplacian Kirchhoff type problem sign-changing weight Nehari manifold

MSC

35J50 35J60 47G20

1 Introduction

In this paper, we consider the following fractional elliptic equation with sign-changing weight functions:
$$ \textstyle\begin{cases} M ( \int_{\mathbb{R}^{2N}} \frac{|u(x) - u(y)|^{p}}{|x-y|^{N+sp}} \,dx\,dy ) (-\Delta)_{p}^{s} u = \lambda f(x) u^{q} + g(x) u^{r}, & x \in \Omega, \\ u=0, & x \in \mathbb{R}^{N} \setminus \Omega, \end{cases} $$
(1.1)
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
$$(-\Delta)_{p}^{s} u(x) = 2 \lim_{\varepsilon \searrow 0} \int_{B_{\varepsilon}(x)^{c}} \frac{|u(x) - u(y)|^{p-2} (u(x) - u(y))}{|x - y|^{N +sp}} \,dy, \quad x \in \mathbb{R}^{N}. $$
We may assume that the weight functions \(f(x)\) and \(g(x)\) are as follows:
  1. (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 \(q = 0\);

     
  2. (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 [26] and references therein. In the subcritical case, Pucci and Saldi in [5] studied the following Kirchhoff type problem in \(\mathbb{R}^{N}\):
$$\textstyle\begin{cases} M ( \int_{\mathbb{R}^{2N}} \frac{|u(x) - u(y)|^{p}}{|x-y|^{N+sp}} \,dx\,dy ) (-\Delta)_{p}^{s} u + V(x) \vert u\vert ^{p-2} u \\ \quad= \lambda w(x) |u|^{q-2} u - h(x) |u|^{r-2} u, & x \in \Omega, \\ u=0, & x \in \mathbb{R}^{N} \setminus \Omega, \end{cases} $$
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:
$$\textstyle\begin{cases} M ( \int_{\mathbb{R}^{2N}} \frac{|u(x) - u(y)|^{p}}{|x-y|^{N+sp}} \,dx\,dy ) (-\Delta)_{p}^{s} u = \lambda f(x) |u|^{q-2} u + |u|^{\alpha-2} u, & x \in \Omega, \\ u=0, & x \in \mathbb{R}^{N} \setminus \Omega, \end{cases} $$
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:
$$ \textstyle\begin{cases} -\Delta u = \lambda f(x) u^{q} + g(x) u^{r}, & x \in \Omega, \\ u=0, & x \in \partial \Omega. \end{cases} $$
(1.2)
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
$$X = \biggl\{ u | u : \mathbb{R}^{N} \to \mathbb{R} \text{ is measurable}, u|_{\Omega} \in L^{p}(\Omega), \text{ and } \int_{Q} \frac{|u(x) - u(y)|^{p}}{|x-y|^{n+ps}} \,dx \,dy < \infty \biggr\} , $$
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
$$\|u\|_{X} = \|u\|_{L^{p}(\Omega)} + \biggl( \int_{Q} \frac{|u(x) - u(y)|^{p}}{|x-y|^{n+ps}} \,dx \,dy \biggr)^{1/p}. $$
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
$$\langle u, v\rangle_{X_{0}} = \int_{Q} \frac{|u(x) - u(y)|^{p-1} (v(x) - v(y))}{|x-y|^{n+ps}} \,dx \,dy, \quad \forall u, v \in X_{0}, $$
and the norm
$$\|u\|_{X_{0}} = \biggl( \int_{Q} \frac{|u(x) - u(y)|^{p}}{|x-y|^{n+ps}} \,dx \,dy \biggr)^{1/p}. $$

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
$$S_{l} = \inf_{X_{0} \setminus \{0\} } \frac{\int_{\mathbb{R}^{2N}} \frac{|u(x) - u(y)|^{p}}{|x-y|^{N+sp} } \,dx\,dy}{ (\int_{\mathbb{R}^{N}} |u|^{l} \,dx )^{\frac{p}{l}} } > 0, $$
where \(l \in [p, p_{s}^{*}]\).
A function \(u \in X_{0}\) is a weak solution of problem (1.1) if
$$\begin{aligned}& M \biggl( \int_{Q} \frac{|u(x) - u(y)|^{p}}{|x-y|^{N+sp}} \,dx\,dy \biggr) \int_{Q} \frac{|u(x) - u(y)|^{p-2} (u(x) - u(y))(v(x) - v(y))}{|x-y|^{N+sp}} \,dx\,dy \\& \quad= \lambda \int_{\Omega} f(x) |u|^{q-1} u v \,dx + \int_{\Omega} g(x) |u|^{r-1} u v \,dx, \quad \forall v \in X_{0}. \end{aligned}$$
Associated with equation (1.1), we consider the energy functional \(\mathcal{J}_{\lambda, M}\) in \(X_{0}\)
$$\mathcal{J}_{\lambda, M}(u) = \frac{1}{p} \hat{M} \bigl(\|u \|_{X_{0}}^{p} \bigr) - \frac{\lambda}{q+1} \int_{\Omega} f |u|^{q+1} \,dx - \frac{1}{r+1} \int_{\Omega} g |u|^{r+1} \,dx, $$
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
$$\begin{aligned} \mathcal{N}_{\lambda, M}(\Omega) =& \bigl\{ u \in X_{0} \setminus \{0\} : \bigl\langle \mathcal{J}^{\prime}_{\lambda, M}(u), u \bigr\rangle = 0\bigr\} \\ =& \biggl\{ u \in X_{0} \setminus \{0\} | M \bigl(\|u \|_{X_{0}}^{p} \bigr) \|u\|_{X_{0}}^{p} - \lambda \int_{\Omega} f |u|^{q+1} \,dx - \int_{\Omega} g |u|^{r+1} \,dx = 0 \biggr\} . \end{aligned}$$
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
$$\begin{aligned}& h_{\lambda, M}(t) = \frac{1}{p} \hat{M} \bigl(t^{p} \|u \|_{X_{0}}^{p} \bigr) - \lambda \frac{t^{q+1}}{q+1} \int_{\Omega} f |u|^{q+1} \,dx - \frac{t^{r+1}}{r+1} \int_{\Omega} g |u|^{r+1} \,dx, \\& h^{\prime}_{\lambda, M}(t) = t^{p-1} M \bigl(t^{p} \|u\|_{X_{0}}^{p} \bigr) \|u\|_{X_{0}}^{p} - \lambda t^{q} \int_{\Omega} f |u|^{q+1} \,dx - t^{r} \int_{\Omega} g |u|^{r+1} \,dx, \end{aligned}$$
and
$$\begin{aligned} h^{\prime \prime}_{\lambda, M}(t) =& (p-1) t^{p-2} M \bigl(t^{p} \|u\|_{X_{0}}^{p} \bigr) \|u \|_{X_{0}}^{p} + p t^{2p-2} M^{\prime} \bigl(t^{p} \|u\|_{X_{0}}^{p} \bigr) \|u \|_{X_{0}}^{2p} \\ &{}- q \lambda t^{q-1} \int_{\Omega} f |u|^{q+1} \,dx - r t^{r-1} \int_{\Omega} g |u|^{r+1} \,dx. \end{aligned}$$
Obviously,
$$\begin{aligned} t h^{\prime}_{\lambda, M}(t) =& M \bigl(t^{p} \|u \|_{X_{0}}^{p} \bigr) \|t u\|_{X_{0}}^{p} - \lambda \int_{\Omega} f |t u|^{q+1} \,dx - \int_{\Omega} g |t u|^{r+1} \,dx \\ =& \bigl\langle \mathcal{J}_{\lambda, M}(tu), tu\bigr\rangle , \end{aligned}$$
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
$$\begin{aligned}& \mathcal{N}_{\lambda, M}^{+}(\Omega) = \bigl\{ u \in \mathcal{N}_{\lambda, M}( \Omega) : h^{\prime \prime}_{u, M}(1) > 0\bigr\} , \\& \mathcal{N}_{\lambda, M}^{0}(\Omega) = \bigl\{ u \in \mathcal{N}_{\lambda, M}(\Omega) : h^{\prime \prime}_{u, M}(1) = 0\bigr\} , \\& \mathcal{N}_{\lambda, M}^{-}(\Omega) = \bigl\{ u \in \mathcal{N}_{\lambda, M}( \Omega) : h^{\prime \prime}_{u, M}(1) < 0\bigr\} . \end{aligned}$$
For each \(u \in \mathcal{N}_{\lambda, M}(\Omega)\), we have
$$\begin{aligned} h^{\prime \prime}_{\lambda, M}(1) =& (p-1) M \bigl(\|u\|_{X_{0}}^{p} \bigr) \|u\|_{X_{0}}^{p} + p M^{\prime} \bigl(\|u \|_{X_{0}}^{p} \bigr) \|u\|_{X_{0}}^{2p} \\ &{}- q \lambda \int_{\Omega} f |u|^{q+1} \,dx - r \int_{\Omega} g |u|^{r+1} \,dx \\ =& (p-r-1) M \bigl(\|u\|_{X_{0}}^{p} \bigr) \|u \|_{X_{0}}^{p} + p M^{\prime} \bigl(\|u \|_{X_{0}}^{p} \bigr) \|u\|_{X_{0}}^{2p} - \lambda(q-r) \int_{\Omega} f |u|^{q+1} \,dx \end{aligned}$$
(2.1)
$$\begin{aligned} =& (p-q-1) M \bigl(\|u\|_{X_{0}}^{p} \bigr) \|u \|_{X_{0}}^{p} + p M^{\prime} \bigl(\|u \|_{X_{0}}^{p} \bigr) \|u\|_{X_{0}}^{2p} - (r - q) \int_{\Omega} g |u|^{r+1} \,dx. \end{aligned}$$
(2.2)
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)
$$\begin{aligned}& a (p-r-1) \|u\|_{X_{0}}^{p} + b \bigl(p^{2}-r-1\bigr) \|u\|_{X_{0}}^{p^{2}} - \lambda (q-r) \int_{\Omega} f |u|^{q+1} \,dx = 0, \end{aligned}$$
(2.3)
$$\begin{aligned}& a (p-q-1) \|u\|_{X_{0}}^{p} + b \bigl(p^{2}-q-1\bigr) \|u\|_{X_{0}}^{p^{2}} - (r-q) \int_{\Omega} g |u|^{r+1} \,dx = 0. \end{aligned}$$
(2.4)
For convenience, we let
  1. (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
$$\begin{aligned} \mathcal{J}_{\lambda, M}(u) =& a \biggl(\frac{1}{p} - \frac{1}{r+1} \biggr) \|u\|_{X_{0}}^{p} + b \biggl( \frac{1}{p^{2}} - \frac{1}{r+1} \biggr) \|u\|_{X_{0}}^{p^{2}} \\ & {}- \lambda \biggl(\frac{1}{q+1} - \frac{1}{r+1} \biggr) \int_{\Omega} f|u|^{q+1} \,dx \\ = & a \biggl(\frac{1}{p} - \frac{1}{r+1} \biggr) \|u \|_{X_{0}}^{p} + b \biggl(\frac{1}{p^{2}} - \frac{1}{r+1} \biggr) \|u\|_{X_{0}}^{p^{2}} \\ &{}- \lambda \frac{r-q}{(q+1)(r+1)} \int_{\Omega} f|u|^{q+1} \,dx \\ \ge & a \biggl(\frac{1}{p} - \frac{1}{r+1} \biggr) \|u \|_{X_{0}}^{p} + b \biggl(\frac{1}{p^{2}} - \frac{1}{r+1} \biggr) \|u\|_{X_{0}}^{p^{2}} \\ &{}- \lambda \frac{r-q}{(q+1)(r+1)} \|f\|_{L^{\mu_{q}}} S_{\mu}^{q+1} \|u\|_{X_{0}}^{q+1}, \end{aligned}$$
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 \(u_{0} \in \mathcal{N}_{\lambda, M}^{0}(\Omega)\)
$$\begin{aligned} a (r-p+1) \|u_{0}\|_{X_{0}}^{p} \le & a (r-p+1) \|u_{0}\|_{X_{0}}^{p} + b\bigl(r-p^{2}+1 \bigr) \|u_{0}\|_{X_{0}}^{p^{2}} \\ =& \lambda (r-q) \int_{\Omega} f |u_{0}|^{q+1} \,dx, \end{aligned}$$
which implies that
$$\begin{aligned} \|u_{0}\|_{X_{0}}^{p} \le & \frac{ \lambda (r-q)}{a (r-p+1)} \int_{\Omega} f |u_{0}|^{q+1} \,dx \\ \le & \frac{ \lambda (r-q)}{a (r-p+1)} \|f\|_{L^{\mu_{q}} } S_{\mu}^{q+1} \|u_{0}\|_{X_{0}}^{q+1} \end{aligned}$$
and so
$$ \|u_{0}\|_{X_{0}} \le \biggl(\frac{ \lambda (r-q)}{a (r-p+1)} \|f\|_{L^{\mu_{q}} } S_{\mu}^{q+1} \biggr)^{\frac{1}{p-q-1}}. $$
(2.5)
Similarly, we obtain by (2.4) and the Hölder and Sobolev inequalities
$$\|u_{0}\|_{X_{0}}^{p} \le \frac{ r-q}{a (p-q+1)} \|g \|_{L^{\nu_{r}} } S_{\nu}^{r+1} \|u_{0} \|_{X_{0}}^{r+1}, $$
which implies that
$$ \|u_{0}\|_{X_{0}} \ge \biggl(\frac{a (p-q+1)}{ r-q} \|g\|_{L^{\nu_{r}} }^{-1} S_{\nu}^{-(r+1)} \biggr)^{\frac{1}{r-p+1}}. $$
(2.6)
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
$$c_{\lambda} = \inf_{u \in \mathcal{N}_{\lambda, M}(\Omega)} \mathcal{J}_{\lambda, M}(u). $$
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
$$c_{\lambda}^{+} = \inf_{u \in \mathcal{N}_{\lambda, M}^{+}(\Omega)} \mathcal{J}_{\lambda, M}(u) \quad \text{and} \quad c_{\lambda}^{-} = \inf_{u \in \mathcal{N}_{\lambda, M}^{-}(\Omega)} \mathcal{J}_{\lambda, M}(u). $$

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:
$$ k_{u}(t) = t^{p-q-1} M \bigl(t^{p} \|u \|_{X_{0}}^{p} \bigr) \|u\|_{X_{0}}^{p} - t^{r - q} \int_{\Omega} g |u|^{r+1} \,dx \quad t > 0. $$
(2.7)
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,
$$\begin{aligned} k_{u}^{\prime}(t) =& (p-q-1) t^{p-q-2} M \bigl(t^{p} \|u\|_{X_{0}}^{p} \bigr) \|u \|_{X_{0}}^{p} + p t^{2p-q-2} M^{\prime} \bigl(t^{p} \|u\|_{X_{0}}^{p} \bigr) \|u \|_{X_{0}}^{2p} \\ &{}- (r - q) t^{r - q - 1} \int_{\Omega} g |u|^{r+1} \,dx, \end{aligned}$$
(2.8)
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
$$\begin{aligned} A =& \frac{a(r-p+1)}{r - q} \biggl(\frac{a(p-q-1)}{(r - q) \|g\|_{L^{\nu_{r}} } S_{\nu}^{r+1}} \biggr)^{\frac{p-q-1}{r-p+1}} \\ &{}+ \frac{b (r - p^{2}+1)}{r - q} \biggl(\frac{a(p-q-1)}{(r - q) \|g\|_{L^{\nu_{r}} } S_{\nu}^{r+1}} \biggr)^{\frac{p^{2}-q-1}{r-p+1}}. \end{aligned}$$
(2.9)

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. (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. (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

From (2.7) and (2.8), we have
$$k_{u}(t) = a t^{p-q-1} \|u\|_{X_{0}}^{p} + b t^{p^{2} - q - 1} \|u\|_{X_{0}}^{p^{2}} - t^{r - q} \int_{\Omega} g |u|^{r+1} \,dx \quad t \ge 0, $$
and
$$k_{u}^{\prime}(t) = t^{-q-1} \biggl[a(p-q-1) t^{p-1}\|u\|_{X_{0}}^{p} + b \bigl(p^{2} - q - 1\bigr) t^{p^{2} - 1} \|u\|_{X_{0}}^{p^{2}} - (r - q) t^{r} \int_{\Omega} g |u|^{r+1} \,dx \biggr], $$
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
$$ a(p-q-1) t_{\max}^{p-1}\|u\|_{X_{0}}^{p} + b \bigl(p^{2} - q - 1\bigr) t_{\max}^{p^{2} - 1} \|u \|_{X_{0}}^{p^{2}} - (r - q) t_{\max}^{r} \int_{\Omega} g |u|^{r+1} \,dx = 0. $$
(2.12)
From (2.12), we obtain
$$ t_{\max} \ge \biggl(\frac{a(p-q-1) \|u\|_{X_{0}}^{p}}{(r - q) \int_{\Omega} g |u|^{r+1} \,dx} \biggr)^{\frac{1}{r-p+1}} \ge \frac{1}{\|u\|_{X_{0}}} \biggl(\frac{a(p-q-1)}{(r - q) \|g\|_{L^{\nu_{r}}} S_{\nu}^{r+1}} \biggr)^{\frac{1}{r-p+1}} : = t_{*}. $$
(2.13)
Hence, we have by (2.12), (2.13), and the Hölder and Sobolev inequalities
$$\begin{aligned} k_{u}(t_{\max}) =& t_{\max}^{p-q-1} \biggl[a \|u\|_{X_{0}}^{p} + b t_{\max}^{p(p-1)} \|u\|_{X_{0}}^{p^{2}} - t_{\max}^{r - p+1} \int_{\Omega} g |u|^{r+1} \,dx \biggr] \\ =& \frac{a(r-p+1)}{r - q} t_{\max}^{p-q-1} \|u\|_{X_{0}}^{p} + \frac{b (r - p^{2}+1)}{r - q} t_{\max}^{p^{2}-q-1} \|u\|_{X_{0}}^{p^{2}} \\ \ge & \frac{a(r-p+1)}{r - q} t_{*}^{p-q-1} \|u\|_{X_{0}}^{p} + \frac{b (r - p^{2}+1)}{r - q} t_{*}^{p^{2}-q-1} \|u\|_{X_{0}}^{p^{2}} \\ \ge & \frac{a(r-p+1)}{r - q} \biggl(\frac{a(p-q-1)}{(r - q) \|g\|_{L^{\nu_{r}} } S_{\nu}^{r+1}} \biggr)^{\frac{p-q-1}{r-p+1}} \|u \|_{X_{0}}^{q+1} \\ &{}+ \frac{b (r - p^{2}+1)}{r - q} \biggl(\frac{a(p-q-1)}{(r - q) \|g\|_{L^{\nu_{r}} } S_{\nu}^{r+1}} \biggr)^{\frac{p^{2}-q-1}{r-p+1}} \|u \|_{X_{0}}^{q+1} \\ = & A \|u\|_{X_{0}}^{q+1}. \end{aligned}$$
(2.14)
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
$$\begin{aligned}& a (p-q-1) \bigl\| t^{-} u\bigr\| _{X_{0}}^{p} + b\bigl(p^{2}-q-1 \bigr) \bigl\| t^{-} u\bigr\| _{X_{0}}^{p^{2}} - (r-q) \int_{\Omega} g \bigl|t^{-} u\bigr|^{r+1} \,dx \\& \quad =\bigl(t^{-}\bigr)^{2+q} \biggl[ a (p-q-1) \bigl(t^{-} \bigr)^{p-q-2} \|u\|_{X_{0}}^{p} + b\bigl(p^{2}-q-1 \bigr) \bigl(t^{-}\bigr)^{p^{2}-q-2} \|u\|_{X_{0}}^{p^{2}} \\& \qquad {}- (r-q) \bigl(t^{-}\bigr)^{r-q-1} \int_{\Omega} g |u|^{r+1} \,dx \biggr] \\& \quad = \bigl(t^{-}\bigr)^{2+q} k_{u}^{\prime}\bigl(t^{-} \bigr) < 0 \end{aligned}$$
and
$$\begin{aligned}& \bigl\langle \mathcal{J}^{\prime}_{\lambda, M}\bigl(t^{-} u\bigr), t^{-} u \bigr\rangle \\& \quad = a \bigl(t^{-}\bigr)^{p} \|u\|_{X_{0}}^{p} + b \bigl(t^{-}\bigr)^{p^{2}} \|u\|_{X_{0}}^{p^{2}} - \lambda \bigl(t^{-}\bigr)^{q+1} \int_{\Omega} f |u|^{q+1} \,dx - \bigl(t^{-} \bigr)^{r+1} \int_{\Omega} g |u|^{r+1} \,dx \\& \quad = \bigl(t^{-}\bigr)^{q+1} \biggl[k_{u}\bigl(t^{-}\bigr) - \lambda \int_{\Omega} f |u|^{q+1} \,dx \biggr] = 0. \end{aligned}$$
Hence, \(t^{-} u \in \mathcal{N}_{\lambda, M}^{-}(\Omega)\) or \(t^{-} = 1\). For \(t > t_{\max}\), we obtain
$$\begin{aligned}& a (p-q-1) \|t u\|_{X_{0}}^{p} + b\bigl(p^{2}-q-1 \bigr) \|t u\|_{X_{0}}^{p^{2}} - (r-q) \int_{\Omega} g |t u|^{r+1} \,dx < 0, \\& \frac{d^{2}}{dt^{2}} \mathcal{J}_{\lambda, M} (tu) < 0, \\& \frac{d}{dt} \mathcal{J}_{\lambda, M} (tu) = a t^{p-1} \|u \|_{X_{0}}^{p} + b t^{p^{2}-1} \|u\|_{X_{0}}^{p^{2}} - \lambda t^{q} \int_{\Omega} f |u|^{q+1} \,dx - t^{r} \int_{\Omega} g |u|^{r+1} \,dx = 0, \end{aligned}$$
for \(t = t^{-}\). Thus, \(\mathcal{J}_{\lambda, M}(u) = \sup_{t \ge 0} \mathcal{J}_{\lambda, M}(t u)\). Furthermore, we have
$$\mathcal{J}_{\lambda, M}(u) \ge \mathcal{J}_{\lambda, M}(tu) \ge \frac{a}{p} t^{p} \|u\|_{X_{0}}^{p} + \frac{b}{p^{2}} t^{p^{2}} \|u\|_{X_{0}}^{p^{2}} - \frac{1}{r+1} t^{r+1} \int_{\Omega} g |u|^{r+1} \,dx, \quad t \ge 0. $$
Let
$$h_{u}(t) = \frac{a}{p} t^{p} \|u \|_{X_{0}}^{p} + \frac{b}{p^{2}} t^{p^{2}} \|u \|_{X_{0}}^{p^{2}} - \frac{1}{r+1} t^{r+1} \int_{\Omega} g |u|^{r+1} \,dx, \quad t \ge 0. $$
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
$$\mathcal{J}_{\lambda, M}(u) \ge h_{u}(t_{m}) \ge \frac{ap(r+1-p) + b(r+1-p^{2})}{p^{2}(r+1)} \biggl(\frac{a \|u\|_{X_{0}}^{r+1}}{\int_{\Omega} g |u|^{r+1} \,dx} \biggr)^{\frac{p}{r-p+1} } > 0. $$
Case (2): \(\int_{\Omega} f |u|^{q+1} \,dx > 0\). By (2.14) and
$$\begin{aligned} k_{u}(0) =& 0 < \lambda \int_{\Omega} f |u|^{q+1} \,dx \le \lambda \|f \|_{L^{\mu_{q}}} S_{\mu}^{q+1} \|u\|_{X_{0}}^{q+1} \\ < & \lambda_{2} \|f\|_{L^{\mu_{q}}} S_{\mu}^{q+1} \|u\|_{X_{0}}^{q+1} = A \|u\|_{X_{0}}^{q+1} \le k_{u}(t_{\max}), \quad \text{for } \lambda \in (0, \lambda_{2}). \end{aligned}$$
Then there exist \(t^{+}\) and \(t^{-}\) such that \(0 < t^{+} < t_{\max} < t^{-}\),
$$k_{u}\bigl(t^{+}\bigr) = \lambda \int_{\Omega} f |u|^{q+1} \,dx = k_{u}\bigl(t^{-} \bigr). $$
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
$$\mathcal{J}_{\lambda, M}(u) = \sup_{t \ge 0} \mathcal{J}_{\lambda, M}(t u),\qquad \mathcal{J}_{\lambda, M}\bigl(t^{+} u\bigr) = \inf_{0 \le t \le t_{\max}} \mathcal{J}_{\lambda, M}(t u). $$
 □

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
$$(r - q) \lambda \int_{\Omega} f |u|^{q+1} \,dx > a (r - p + 1) \|u \|_{X_{0}}^{p} + b\bigl(r - p^{2} + 1\bigr) \|u \|_{X_{0}}^{p^{2}}. $$
Thus, we have
$$\begin{aligned} J_{\lambda, M}(u) =& \frac{a(r-p+1)}{p(r+1)} \|u\|_{X_{0}}^{p} + \frac{b(r-p^{2}+1)}{p^{2}(r+1)} \|u\|_{X_{0}}^{p^{2}} - \frac{\lambda(r-q)}{(q+1)(r+1)} \int_{\Omega} f |u|^{q+1} \,dx \\ < & \frac{a(r-p+1)}{r+1} \biggl[\frac{1}{p} - \frac{1}{q+1} \biggr] \|u\|_{X_{0}}^{p} + \frac{b(r-p^{2}+1)}{r+1} \biggl[ \frac{1}{p^{2}} - \frac{1}{q+1} \biggr] \|u\|_{X_{0}}^{p^{2}} < 0, \end{aligned}$$
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
$$ \bigl\langle \xi^{\prime}(0), v \bigr\rangle = \frac{W}{a (p-q-1) \|u\|_{X_{0}}^{p} + b(p^{2}-q-1) \|u\|_{X_{0}}^{p^{2}} - (r-q) \int_{\Omega} g |u|^{r+1} \,dx}, $$
(3.1)
for all \(v \in X_{0}\), where
$$\begin{aligned} W =& a p \int_{Q} \frac{|u(x) - u(y)|^{p-2} (u(x) - u(y))(v(x) - v(y))}{|x-y|^{N+sp}} \,dx\,dy \\ &{}+ b p^{2} \int_{Q} \frac{|u(x) - u(y)|^{p^{2}-2} (u(x) - u(y))(v(x) - v(y))}{|x-y|^{N+s p^{2}}} \,dx\,dy \\ &{}- (q+1) \lambda \int_{\Omega} f |u|^{q-1} u v \,dx -(r+1) \int_{\Omega} g |u|^{r-1}uv \,dx. \end{aligned}$$
(3.2)

Proof

For \(u \in \mathcal{N}_{\lambda, M}(\Omega)\), we define a function \(\mathcal{F} : \mathbb{R} \times X_{0} \to \mathbb{R}\) by
$$\begin{aligned} \mathcal{F}_{u}(\xi, w) =& \bigl\langle \mathcal{J}_{\lambda, M}^{\prime} \bigl(\xi(u-w)\bigr), \xi(u-w) \bigr\rangle \\ =& \xi^{p} M\bigl(\xi^{p} \|u-w\|_{X_{0}}^{p} \bigr) \|u-w\|_{X_{0}}^{p} \\ &{}- \xi^{q+1} \lambda \int_{\Omega} f |u-w|^{q+1} \,dx - \xi^{r+1} \int_{\Omega} g |u-w|^{r+1} \,dx \\ =& a \xi^{p} \|u-w\|_{X_{0}}^{p} + b \xi^{p^{2}} \|u-w\|_{X_{0}}^{p^{2}} \\ &{}- \xi^{q+1} \lambda \int_{\Omega} f |u-w|^{q+1} \,dx - \xi^{r+1} \int_{\Omega} g |u-w|^{r+1} \,dx. \end{aligned}$$
Then \(\mathcal{F}_{u}(1, 0) = \langle \mathcal{J}_{\lambda, M}^{\prime}(u), u \rangle = 0\) and
$$\begin{aligned} \frac{d}{d\xi} \mathcal{F}_{u}(1, 0) =& a p \|u \|_{X_{0}}^{p} + b p^{2} \|u\|_{X_{0}}^{p^{2}} -(q+1)\lambda \int_{\Omega} f |u|^{q+1} \,dx -(r+1) \int_{\Omega} g |u|^{r+1} \,dx \\ =& a (p-q-1) \|u\|_{X_{0}}^{p} + b\bigl(p^{2}-q-1 \bigr) \|u\|_{X_{0}}^{p^{2}} - (r-q) \int_{\Omega} g |u|^{r+1} \,dx \neq 0. \end{aligned}$$
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 \(\xi(0) = 1\),
$$\bigl\langle \xi^{\prime}(0), v \bigr\rangle = \frac{W}{a (p-q-1) \|u\|_{X_{0}}^{p} + b(p^{2}-q-1) \|u\|_{X_{0}}^{p^{2}} - (r-q) \int_{\Omega} g |u|^{r+1} \,dx}, $$
where W is as in (3.2), and
$$\mathcal{F}_{u}\bigl(\xi(v), v\bigr) = 0 \quad\text{for all } v \in B(0; \epsilon) $$
which is equivalent to
$$\bigl\langle \mathcal{J}_{\lambda, M}^{\prime}\bigl(\xi(v) (u-v)\bigr), \xi(v) (u-v) \bigr\rangle = 0 \quad \text{for all } v \in B(0; \epsilon), $$
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
$$\bigl\langle \bigl(\xi^{-}\bigr)^{\prime}(0), v \bigr\rangle = \frac{W}{a (p-q-1) \|u\|_{X_{0}}^{p} + b(p^{2}-q-1) \|u\|_{X_{0}}^{p^{2}} - (r-q) \int_{\Omega} g |u|^{r+1} \,dx}, $$
for all \(v \in X_{0}\), where W is as in (3.2).
Let
  1. (H4)

    \(p < 2 + \frac{(r-1)q}{r}\).

     
Moreover, we let
$$p^{*} = \frac{(p-2)r}{r-1} - q $$
and
$$\begin{aligned} \lambda_{3} =& \biggl(\frac{a(p-q-1)(r- p^{2} + 1)}{(r-q)(p^{2} - q -1)} \biggr) \biggl( \frac{a(p-q-1)}{r-q} \biggr)^{\frac{(p-q-1)}{(p-q-1-p^{*})(r-1)}} \\ & {}\times \biggl(\frac{1}{\|f\|_{L^{\mu_{q}}} S_{\mu}^{q+1}} \biggr) \biggl(\frac{1}{\|g\|_{L^{\nu_{r}}} S_{\nu}^{r+1}} \biggr)^{\frac{(p-q-1)}{(r-1)(p-q-1-p^{*})}}. \end{aligned}$$

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 \(\lambda \in (0, \Gamma_{0})\):
  1. (i)
    There exists a minimizing sequence \(\{u_{n}\} \subset \mathcal{N}_{\lambda, M}(\Omega)\) such that
    $$\mathcal{J}_{\lambda, M}(u_{n}) = c_{\lambda} + o(1), \qquad \mathcal{J}_{\lambda, M}^{\prime}(u_{n}) = o(1) \quad \textit{in } (X_{0})^{*}. $$
     
  2. (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
$$ \mathcal{J}_{\lambda, M}(u_{n}) < c_{\lambda} + \frac{1}{n} $$
(3.3)
and
$$ \mathcal{J}_{\lambda, M}(u_{n}) < \mathcal{J}_{\lambda, M}(w) + \frac{1}{n} \|w-u_{n} \|_{X_{0}} \quad \forall w \in \mathcal{N}_{\lambda, M}(\Omega). $$
(3.4)
Let n large enough, by Lemma 2.5, we obtain
$$\begin{aligned} \mathcal{J}_{\lambda, M}(u_{n}) =& \frac{a(r-p+1)}{p(r+1)} \|u_{n}\|_{X_{0}}^{p} + \frac{b(r-p^{2}+1)}{p^{2}(r+1)} \|u_{n}\|_{X_{0}}^{p^{2}} - \frac{\lambda(r-q)}{(q+1)(r+1)} \int_{\Omega} f |u_{n}|^{q+1} \,dx \\ < & c_{\lambda} + \frac{1}{n} < \frac{c_{\lambda}}{2}, \end{aligned}$$
which implies that
$$ \|f\|_{L^{\mu_{q}} } S_{\mu}^{q+1} \|u_{n}\|_{X_{0}}^{q+1} \ge \int_{\Omega} f |u_{n}|^{q+1} \,dx > - \frac{(q+1)(r+1)}{\lambda(r-q)} \frac{c_{\lambda}}{2} > 0. $$
(3.5)
This implies \(u_{n} \neq 0\) and by using (3.4), (3.5), and the Hölder inequality, we get
$$ \|u_{n}\|_{X_{0}} > \biggl[ - \frac{(q+1)(r+1)}{\lambda(r-q)} \frac{c_{\lambda}}{2} \|f\|_{L^{\mu_{q}} }^{-1} S_{\mu}^{-(q+1)} \biggr]^{\frac{1}{q+1}} $$
(3.6)
and
$$ \|u_{n}\|_{X_{0}} < \biggl[\frac{\lambda p(r-q)(r+1)}{a(q+1)(r+1)(r-p+1)} \|f\|_{L^{\mu_{q}} } S_{\mu}^{q+1} \biggr]^{\frac{1}{p-q-1}}. $$
(3.7)
In the following, we will prove that
$$\bigl\Vert \mathcal{J}_{\lambda, M}^{\prime}(u_{n})\bigr\Vert _{(X_{0})^{*}} \to 0 \quad \text{as } n \to \infty. $$
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
$$\mathcal{J}_{\lambda, M}(\eta_{\rho}) - J_{\lambda, M}(u_{n}) \ge - \frac{1}{n} \|\eta_{\rho}-u_{n}\|_{X_{0}} \quad \forall w \in \mathcal{N}_{\lambda, M}(\Omega), $$
and by the mean value theorem, we obtain
$$\bigl\langle \mathcal{J}_{\lambda, M}^{\prime}(u_{n}), \eta_{\rho} - u_{n}\bigr\rangle + o\bigl(\Vert \eta_{\rho} - u_{n}\Vert _{X_{0}}\bigr) \ge - \frac{1}{n} \|\eta_{\rho}-u_{n}\|_{X_{0}}. $$
Hence,
$$\begin{aligned}& \bigl\langle \mathcal{J}_{\lambda, M}^{\prime}(u_{n}), - w_{\rho}\bigr\rangle + \bigl(\xi_{n}(w_{\rho}) - 1 \bigr) \bigl\langle \mathcal{J}_{\lambda, M}^{\prime}(u_{n}), u_{n} - w_{\rho}\bigr\rangle \\& \quad \ge - \frac{1}{n} \|\eta_{\rho}-u_{n} \|_{X_{0}} + o \bigl(\|\eta_{\rho} - u_{n}\|_{X_{0}} \bigr). \end{aligned}$$
(3.8)
By \(\xi_{n}(w_{\rho}) (u_{n} - w_{\rho}) \in \mathcal{N}_{\lambda, M}(\Omega)\) and (3.8) it follows that
$$\begin{aligned}& - \rho \biggl\langle \mathcal{J}_{\lambda, M}^{\prime}(u_{n}), \frac{u}{\|u\|_{X_{0}}} \biggr\rangle + \bigl(\xi_{n}(w_{\rho}) - 1\bigr) \bigl\langle \mathcal{J}_{\lambda, M}^{\prime}(u_{n}) - \mathcal{J}_{\lambda, M}^{\prime}(\eta_{\rho}), u_{n} - w_{\rho}\bigr\rangle \\& \quad \ge - \frac{1}{n} \|\eta_{\rho}-u_{n} \|_{X_{0}} + o\bigl(\|\eta_{\rho} - u_{n}\|_{X_{0}}\bigr). \end{aligned}$$
Thus,
$$\begin{aligned} \biggl\langle \mathcal{J}_{\lambda, M}^{\prime}(u_{n}), \frac{u}{\|u\|_{X_{0}}} \biggr\rangle \le & \frac{1}{n \rho} \|\eta_{\rho}-u_{n} \|_{X_{0}} + \frac{1}{\rho} o \bigl(\|\eta_{\rho} - u_{n}\|_{X_{0}} \bigr) \\ &{}+ \frac{(\xi_{n}(w_{\rho}) - 1)}{\rho} \bigl\langle \mathcal{J}_{\lambda, M}^{\prime}(u_{n}) - \mathcal{J}_{\lambda, M}^{\prime}(\eta_{\rho}), u_{n} - w_{\rho}\bigr\rangle . \end{aligned}$$
(3.9)
Since
$$\|\eta_{\rho}-u_{n}\|_{X_{0}} \le \rho \bigl\vert \xi_{n}(w_{\rho})\bigr\vert + \bigl\vert \xi_{n}(w_{\rho}) - 1\bigr\vert \|u_{n} \|_{X_{0}} $$
and
$$\lim_{n \to \infty} \frac{|\xi_{n}(w_{\rho}) - 1|}{\rho} \le \bigl\Vert \xi_{n}^{\prime}(0)\bigr\Vert , $$
taking the limit \(\rho \to 0\) in (3.9), we obtain
$$\biggl\langle \mathcal{J}_{\lambda, M}^{\prime}(u_{n}), \frac{u}{\|u\|_{X_{0}}} \biggr\rangle \le \frac{C}{n} \bigl(1 + \bigl\| \xi_{n}^{\prime}(0)\bigr\| \bigr) $$
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 \(\kappa > 0\)
$$\bigl\langle \xi_{n}^{\prime}(0), v\bigr\rangle \le \frac{\kappa \|v\|_{X_{0}} }{a (p-q-1) \|u_{n}\|_{X_{0}}^{p} + b(p^{2}-q-1) \|u_{n}\|_{X_{0}}^{p^{2}} - (r-q) \int_{\Omega} g |u_{n}|^{r+1} \,dx}. $$
We only need to prove that
$$ \biggl\vert a (p-q-1) \|u_{n}\|_{X_{0}}^{p} + b\bigl(p^{2}-q-1\bigr) \|u_{n}\|_{X_{0}}^{p^{2}} - (r-q) \int_{\Omega} g |u_{n}|^{r+1} \,dx \biggr\vert > c $$
(3.10)
for some \(c > 0\) and n large enough. If (3.10) is fails, then there exists a subsequence \(\{u_{n}\}\) such that
$$ a (p-q-1) \|u_{n}\|_{X_{0}}^{p} + b \bigl(p^{2}-q-1\bigr) \|u_{n}\|_{X_{0}}^{p^{2}} - (r-q) \int_{\Omega} g |u_{n}|^{r+1} \,dx = o(1). $$
(3.11)
Combining (3.11) with (3.6), we may find a suitable constant \(d > 0\) such that
$$ \int_{\Omega} g |u_{n}|^{r+1} \,dx \ge d \quad \text{for } n \text{ sufficiently large}. $$
(3.12)
By (3.11) and \(u_{n} \in \mathcal{N}_{\lambda, M}(\Omega)\), we have
$$\begin{aligned}& \lambda \int_{\Omega} f |u_{n}|^{q+1} \,dx \\& \quad= a \|u_{n}\|_{X_{0}}^{p} + b \|u_{n} \|_{X_{0}}^{p^{2}} - \int_{\Omega} g |u_{n}|^{r+1} \,dx \\& \quad= \frac{1}{p^{2}-q-1} \bigl( a \bigl(p^{2}-q-1\bigr) \|u_{n}\|_{X_{0}}^{p} + b\bigl(p^{2}-q-1 \bigr) \|u_{n}\|_{X_{0}}^{p^{2}} \bigr) - \int_{\Omega} g |u_{n}|^{r+1} \,dx \\& \quad\ge \frac{1}{p^{2}-q-1} \bigl( a (p-q-1) \|u_{n} \|_{X_{0}}^{p} + b\bigl(p^{2}-q-1\bigr) \|u_{n}\|_{X_{0}}^{p^{2}} \bigr) - \int_{\Omega} g |u_{n}|^{r+1} \,dx \\& \quad= \frac{r-q}{p^{2}-q-1} \int_{\Omega} g |u_{n}|^{r+1} \,dx - \int_{\Omega} g |u_{n}|^{r+1} \,dx + o(1) \\& \quad= \frac{r-p^{2}+1}{p^{2}-q-1} \int_{\Omega} g |u_{n}|^{r+1} \,dx + o(1). \end{aligned}$$
(3.13)
Moreover, we have by (3.11) and (3.13)
$$\begin{aligned} a (p-q-1) \|u_{n}\|_{X_{0}}^{p} \le& a (p-q-1) \|u_{n}\|_{X_{0}}^{p} + b\bigl(p^{2}-q-1 \bigr) \|u_{n}\|_{X_{0}}^{p^{2}} \\ =& (r-q) \int_{\Omega} g |u_{n}|^{r+1} \,dx + o(1) \\ \le & \lambda \frac{(p^{2}-q-1)(r-q)}{r-p^{2}+1} \int_{\Omega} f |u_{n}|^{q+1} \,dx + o(1) \\ \le & \lambda \frac{(p^{2}-q-1)(r-q)}{r-p^{2}+1} \|f\|_{L^{\mu_{q}} } S_{\mu}^{q+1} \|u_{n}\|_{X_{0}}^{q+1} + o(1), \end{aligned}$$
which implies that
$$ \|u_{n}\|_{X_{0}} \le \biggl(\lambda \frac{(p^{2}-q-1)(r-q)}{a(p-q-1)(r-p^{2}+1)} \|f\|_{L^{\mu_{q}} } S_{\mu}^{q+1} \biggr)^{\frac{1}{p-q-1}} + o(1). $$
(3.14)
Let
$$\mathcal{I}_{\lambda, M}(u) = K(p,q,r) \biggl(\frac{\|u\|_{X_{0}}^{p r}}{\int_{\Omega} g |u_{n}|^{r+1} \,dx} \biggr)^{\frac{1}{r-1}} - \lambda \int_{\Omega} f |u|^{q+1} \,dx, $$
where
$$K(p,q,r) = \biggl(\frac{a(p-q-1)}{r - q} \biggr)^{\frac{r}{r-1}} \frac{r-p^{2}+1}{p^{2}-q-1}. $$
From (3.11), it is easy to see that
$$ \|u_{n}\|_{X_{0}}^{p} \le \frac{r-q}{a(p-q-1)} \int_{\Omega} g |u_{n}|^{r+1} \,dx. $$
(3.15)
Thus,
$$\begin{aligned} \mathcal{I}_{\lambda, M}(u_{n}) \le & \biggl(\frac{a(p-q-1)}{r - q} \biggr)^{\frac{r}{r-1}} \frac{r-p^{2}+1}{p^{2}-q-1} \biggl(\frac{ (\frac{r-q}{a(p-q-1)} )^{r} (\int_{\Omega} g |u_{n}|^{r+1} \,dx )^{r} }{\int_{\Omega} g |u_{n}|^{r+1}\,dx } \biggr)^{\frac{1}{r-1}} \\ &{}- \frac{r-p^{2}+1}{p^{2}-q-1} \int_{\Omega} g |u_{n}|^{r+1} \,dx + o(1) \\ =& o(1). \end{aligned}$$
(3.16)
But, by (3.12), (3.14), and \(\lambda \in \Gamma_{0}\),
$$\begin{aligned} \mathcal{I}_{\lambda, M}(u_{n}) \ge & K(p,q,r) \biggl( \frac{\|u_{n}\|_{X_{0}}^{p r}}{\|g\|_{L^{\nu_{r}} } S_{\nu}^{r+1} \|u_{n}\|_{X_{0}}^{r+1}} \biggr)^{\frac{1}{r-1}} - \lambda \|f\|_{L^{\mu_{q}} } S_{\mu}^{q+1} \|u_{n}\|_{X_{0}}^{q+1} \\ =& \|u_{n}\|_{X_{0}}^{q+1} \bigl( K(p,q,r) \|g \|_{L^{\nu_{r}} }^{\frac{1}{1-r}} S_{\nu}^{\frac{r+1}{1-r}} \|u_{n}\|_{X_{0}}^{p^{*}} - \lambda \|f\|_{L^{\mu_{q}} } S_{\mu}^{q+1} \bigr) \\ \ge & \|u_{n}\|_{X_{0}}^{q+1} \biggl\{ K(p,q,r) \|g \|_{L^{\nu_{r}} }^{\frac{1}{1-r}} S_{\nu}^{\frac{r+1}{1-r}} \biggl[\lambda \frac{(p^{2}-q-1)(r-q)}{a(p-q-1)(r-p^{2}+1)} \|f\|_{L^{\mu_{q}} } S_{\mu}^{q+1} \biggr]^{\frac{p^{*}}{p-q-1}} \\ &{}- \lambda \|f\|_{L^{\mu_{q}} } S_{\mu}^{q+1}\biggr\} , \end{aligned}$$
which contradicts (3.16), where \(p^{*} = \frac{(p-2)r}{r-1} - q < 0\).
Hence, we obtain
$$\biggl\langle \mathcal{J}_{\lambda, M}^{\prime}(u_{n}), \frac{u}{\|u\|_{X_{0}} } \biggr\rangle \le \frac{C}{n}. $$
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. (1)

    \(\mathcal{J}_{\lambda, M}(u_{\lambda}^{+}) = c_{\lambda}^{+} = c_{\lambda} \);

     
  2. (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
$$\mathcal{J}_{\lambda, M}(u_{n}) = c_{\lambda} + o(1), \qquad \mathcal{J}_{\lambda, M}^{\prime}(u_{n}) = o(1) \quad \text{in } (X_{0})^{*}. $$
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
$$u_{n} \rightharpoonup u_{\lambda}^{+} \quad \text{weakly in } X_{0} $$
and
$$ u_{n} \to u_{\lambda}^{+} \quad \text{strongly in } L^{\eta}(\Omega) \text{ for } 1 < \eta < p_{s}^{*}. $$
(3.17)
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
$$\int_{\Omega} f |u_{n}|^{q+1} \,dx \to \int_{\Omega} f |u_{\lambda}^{+}|^{q+1} \,dx = 0 $$
as \(n \to \infty\). Hence,
$$a \|u_{n}\|_{X_{0}}^{p} + b \|u_{n} \|_{X_{0}}^{p^{2}} = \int_{\Omega} g |u_{n}|^{r+1} \,dx + o(1) $$
and
$$\mathcal{J}_{\lambda, M}(u_{n}) = a \biggl(\frac{1}{p} - \frac{1}{r+1} \biggr) \|u_{n}\|_{X_{0}}^{p} + b \biggl(\frac{1}{p^{2}} - \frac{1}{r+1} \biggr) \|u_{n} \|_{X_{0}}^{p^{2}} + o(1), $$
which contradicts \(\mathcal{J}_{\lambda, M}(u_{n}) \to c_{\lambda} < 0\) as \(n \to \infty\). Furthermore,
$$o(1) = \bigl\langle \mathcal{J}_{\lambda, M}^{\prime}(u_{n}), \phi \bigr\rangle = \bigl\langle \mathcal{J}_{\lambda, M}^{\prime} \bigl(u_{\lambda}^{+}\bigr), \phi \bigr\rangle + o(1) \quad \text{for all } \phi \in X_{0}. $$
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
$$\begin{aligned} \mathcal{J}_{\lambda, M}\bigl(u_{\lambda}^{+}\bigr) =& \frac{a}{p} \bigl\Vert u_{\lambda}^{+}\bigr\Vert _{X_{0}}^{p} + \frac{b}{p^{2}} \bigl\Vert u_{\lambda}^{+}\bigr\Vert _{X_{0}}^{p^{2}} - \frac{\lambda}{q+1} \int_{\Omega} f \bigl\vert u_{\lambda}^{+}\bigr\vert ^{q+1} \,dx - \frac{1}{r+1} \int_{\Omega} g \bigl|u_{\lambda}^{+}\bigr|^{r+1} \,dx \\ =& \biggl(\frac{a}{p} - \frac{a}{r+1} \biggr) \bigl\Vert u_{\lambda}^{+}\bigr\Vert _{X_{0}}^{p} + \biggl( \frac{b}{p^{2}} - \frac{b}{r+1} \biggr) \bigl\Vert u_{\lambda}^{+} \bigr\Vert _{X_{0}}^{p^{2}} \\ &{}+ \biggl(\frac{\lambda}{r+1} - \frac{\lambda}{q+1} \biggr) \int_{\Omega} f \bigl\vert u_{\lambda}^{+} \bigr\vert ^{q+1} \,dx \\ \le & \lim \inf_{n \to \infty} \biggl[ \biggl(\frac{a}{p} - \frac{a}{r+1} \biggr) \|u_{n}\|_{X_{0}}^{p} + \biggl(\frac{b}{p^{2}} - \frac{b}{r+1} \biggr) \|u_{n} \|_{X_{0}}^{p^{2}} \\ &{}+ \biggl(\frac{\lambda}{r+1} - \frac{\lambda}{q+1} \biggr) \int_{\Omega} f |u_{n}|^{q+1} \,dx\biggr] \\ =& \lim \inf_{n \to \infty} \mathcal{J}_{\lambda, M}(u_{n}) = c_{\lambda}. \end{aligned}$$
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
$$\frac{d}{dt} \mathcal{J}_{\lambda, M}\bigl(t_{\lambda}^{+} u_{\lambda}^{+}\bigr) = 0 \quad \text{and} \quad \frac{d^{2}}{dt^{2}} \mathcal{J}_{\lambda, M}\bigl(t_{\lambda}^{+} u_{\lambda}^{+}\bigr) > 0, $$
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
$$\mathcal{J}_{\lambda, M}\bigl(t_{\lambda}^{+} u_{\lambda}^{+}\bigr) < \mathcal{J}_{\lambda, M}\bigl(t^{*} u_{\lambda}^{+}\bigr) \le \mathcal{J}_{\lambda, M}\bigl(t_{\lambda}^{-} u_{\lambda}^{+}\bigr) = \mathcal{J}_{\lambda, M}\bigl(u_{\lambda}^{+}\bigr), $$
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. (1)

    \(\mathcal{J}_{\lambda, M}(u_{\lambda}^{-}) = c_{\lambda}^{-} \);

     
  2. (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].

Declarations

Acknowledgements

This work is supported by Natural Science Foundation of China (11571136 and 11271364).

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)
Department of Mathematics, Huaiyin Normal University

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