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

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:

$$ \textstyle\begin{cases} M ( \int_{\mathbb{R}^{2N}} \frac{|u(x) - u(y)|^{p}}{|x-y|^{N+sp}} \,dx\,dy ) (-\Delta)_{p}^{s} u = \lambda f(x) u^{q} + g(x) u^{r}, & x \in \Omega, \\ u=0, & x \in \mathbb{R}^{N} \setminus \Omega, \end{cases} $$

(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:

• (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 $\mathbb{R}^{N}$:

$$\textstyle\begin{cases} M ( \int_{\mathbb{R}^{2N}} \frac{|u(x) - u(y)|^{p}}{|x-y|^{N+sp}} \,dx\,dy ) (-\Delta)_{p}^{s} u + V(x) \vert u\vert ^{p-2} u \\ \quad= \lambda w(x) |u|^{q-2} u - h(x) |u|^{r-2} u, & x \in \Omega, \\ u=0, & x \in \mathbb{R}^{N} \setminus \Omega, \end{cases} $$

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

• (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) 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

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

• (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 $\lambda \in (0, \Gamma_{0})$:

• (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

$$ \mathcal{J}_{\lambda, M}(u_{n}) < c_{\lambda} + \frac{1}{n} $$

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

• (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

$$\mathcal{J}_{\lambda, M}(u_{n}) = c_{\lambda} + o(1), \qquad \mathcal{J}_{\lambda, M}^{\prime}(u_{n}) = o(1) \quad \text{in } (X_{0})^{*}. $$

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

• (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].