## 1 Introduction and main results

Let Ω be a smooth domain (not necessarily bounded) in ℝ N (N ≥ 3) with smooth boundary ∂Ω such that 0 ∈ Ω. We will study the multiplicity of positive solutions for the following quasilinear elliptic equation

$\left\{\begin{array}{c}\hfill -{\Delta }_{p}u-\mu \frac{\mid u{\mid }^{p-2}u}{\mid x{\mid }^{p}}=\lambda f\left(x\right)\mid u{\mid }^{q-2}u+g\left(x\right)\mid u{\mid }^{p*-2}u,\hfill \\ \hfill u=0,\hfill \end{array}\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill in\Omega ,\hfill \\ \hfill on\partial \Omega ,\hfill \end{array}\right\$
(1.1)

where Δ p u = div(|∇u|p-2u), 1 < p < N, $\mu <\stackrel{̄}{\mu }={\left(\frac{N-p}{p}\right)}^{p}$, $\stackrel{̄}{\mu }$ is the best Hardy constant, λ > 0, 1 < q < p, ${p}^{*}=\frac{Np}{N-p}$ is the critical Sobolev exponent and the weight functions $f,g:\stackrel{̄}{\Omega }\to ℝ$ are continuous, which change sign on Ω.

Let ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ be the completion of ${C}_{0}^{\infty }\left(\Omega \right)$ with respect to the norm ${\left({\int }_{\Omega }\mid \nabla u{\mid }^{p}dx\right)}^{1∕p}$. The energy functional of (1.1) is defined on ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ by

${J}_{\lambda }\left(u\right)=\frac{1}{p}{\int }_{\Omega }\left(\mid \nabla u{\mid }^{p}-\mu \frac{\mid u{\mid }^{p}}{\mid x{\mid }^{p}}\right)\phantom{\rule{0.3em}{0ex}}dx-\frac{\lambda }{q}{\int }_{\Omega }f\mid u{\mid }^{q}dx-\frac{1}{{p}^{*}}{\int }_{\Omega }g\mid u{\mid }^{{p}^{*}}dx.$

Then ${J}_{\lambda }\in {C}^{1}\left({\mathcal{D}}_{0}^{1,p}\left(\Omega \right),ℝ\right)$. $u\in {\mathcal{D}}_{0}^{1,p}\left(\Omega \right)\\left\{0\right\}$ is said to be a solution of (1.1) if $〈{J}_{\lambda }^{\prime }\left(u\right),v〉=0$ for all $v\in {\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ and a solution of (1.1) is a critical point of J λ .

Problem (1.1) is related to the well-known Hardy inequality [1, 2]:

${\int }_{\Omega }\frac{\mid u{\mid }^{p}}{\mid x{\mid }^{p}}dx\le \frac{1}{\stackrel{̄}{\mu }}{\int }_{\Omega }\mid \nabla u{\mid }^{p}dx,\phantom{\rule{1em}{0ex}}\forall u\in {C}_{0}^{\infty }\left(\Omega \right).$

By the Hardy inequality, ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ has the equivalent norm ||u||μs, where

$\parallel u{\parallel }_{\mu }^{p}={\int }_{\Omega }\left(\mid \nabla u{\mid }^{p}-\mu \frac{\mid u{\mid }^{p}}{\mid x{\mid }^{p}}\right)dx,\phantom{\rule{1em}{0ex}}\mu \in \left(-\infty ,\stackrel{̄}{\mu }\right).$

Therefore, for 1 < p < N, and $\mu <\stackrel{̄}{\mu }$, we can define the best Sobolev constant:

${S}_{\mu }\left(\Omega \right)=\underset{u\in {\mathcal{D}}_{0}^{1,p}\left(\Omega \right)\\left\{0\right\}}{inf}\frac{{\int }_{\Omega }\left(\mid \nabla u{\mid }^{p}-\mu \frac{\mid u{\mid }^{p}}{\mid x{\mid }^{p}}\right)dx}{{\left({\int }_{\Omega }\mid u{\mid }^{p*}dx\right)}^{\frac{p}{p*}}}.$
(1.2)

It is well known that S μ (Ω) = S μ (ℝ N ) = S μ . Note that S μ = S0 when μ ≤ 0 [3].

Such kind of problem with critical exponents and nonnegative weight functions has been extensively studied by many authors. We refer, e.g., in bounded domains and for p = 2 to [46] and for p > 1 to [711], while in ℝ N and for p = 2 to [12, 13], and for p > 1 to [3, 1417], and the references therein.

In the present paper, our research is mainly related to (1.1) with 1 < q < p < N, the critical exponent and weight functions f, g that change sign on Ω. When p = 2, 1 < q < 2, $\mu \in \left[0,\stackrel{̄}{\mu }\right)$, f, g are sign changing and Ω is bounded, [18] studied (1.1) and obtained that there exists Λ > 0 such that (1.1) has at least two positive solutions for all λ ∈ (0, Λ). For the case p ≠ 2, [19] studied (1.1) and obtained the multiplicity of positive solutions when 1 < q < p < N, μ = 0, f, g are sign changing and Ω is bounded. However, little has been done for this type of problem (1.1). Recently, Wang et al. [11] have studied (1.1) in a bounded domain Ω under the assumptions 1 < q < p < N, N > p2, $-\infty <\mu <\stackrel{̄}{\mu }$ and f, g are nonnegative. They also proved that there existence of Λ0> 0 such that for λ ∈ (0, Λ0), (1.1) possesses at least two positive solutions. In this paper, we study (1.1) and extend the results of [11, 18, 19] to the more general case 1 < q < p < N, $-\infty <\mu <\stackrel{̄}{\mu }$, f, g are sign changing and Ω is a smooth domain (not necessarily bounded) in ℝ N (N ≥ 3). By extracting the Palais-Smale sequence in the Nehari manifold, the existence of at least two positive solutions of (1.1) is verified.

The following assumptions are used in this paper:

$\left(\mathcal{H}\right)$$\mu <\stackrel{̄}{\mu }$, λ > 0, 1 < q < p < N, N ≥ 3.

(f1) $f\in C\left(\stackrel{̄}{\Omega }\right)\cap {L}^{q*}\left(\Omega \right)\phantom{\rule{2.77695pt}{0ex}}\phantom{\rule{2.77695pt}{0ex}}\left({q}^{*}=\frac{{p}^{*}}{{p}^{*}-q}\right)$f+ = max{f, 0} ≢ 0 in Ω.

(f2) There exist β0 and ρ0> 0 such that B(x0; 2ρ0) ⊂ Ω and f (x) ≥ β0 for all xB(x0; 2ρ0)

(g1) $g\in C\left(\stackrel{̄}{\Omega }\right)\cap {L}^{\infty }\left(\Omega \right)$ and g+ = max{g, 0} ≢ 0 in Ω.

(g2) There exist x0 ∈ Ω and β > 0 such that

where | · | denotes the L(Ω) norm.

Set

${\Lambda }_{1}={\Lambda }_{1}\left(\mu \right)={\left(\frac{p-q}{\left(p*-q\right)\mid {g}^{+}{\mid }_{\infty }}\right)}^{\frac{p-q}{p*-p}}\left(\frac{p*-p}{\left(p*-q\right)\mid {f}^{+}{\mid }_{q*}}\right){S}_{\mu }^{\frac{N}{{p}^{2}}\left(p-q\right)+\frac{q}{p}}.$
(1.3)

The main results of this paper are concluded in the following theorems. When Ω is an unbounded domain, the conclusions are new to the best of our knowledge.

Theorem 1.1 Suppose$\left(\mathcal{H}\right)$, (f1) and (g1) hold. Then, (1.1) has at least one positive solution for all λ ∈ (0, Λ1).

Theorem 1.2 Suppose$\left(\mathcal{H}\right)$, (f1) - (g2) hold, and γ is the constant defined as in Lemma 2.2. If$0\le \mu <\stackrel{̄}{\mu }$, x0 = 0 and βpγ, then (1.1) has at least two positive solutions for all$\lambda \in \left(0,\frac{q}{p}{\Lambda }_{1}\right)$.

Theorem 1.3 Suppose$\left(\mathcal{H}\right)$, (f1) - (g2) hold. If μ < 0, x0 ≠ 0, $\beta \ge \frac{N-p}{p-1}$and Np2, then (1.1) has at least two positive solutions for all$\lambda \in \left(0,\frac{q}{p}{\Lambda }_{1}\left(0\right)\right)$.

Remark 1.4 As Ω is a bounded smooth domain and p = 2, the results of Theorems 1.1, 1.2 are improvements of the main results of[18].

Remark 1.5 As Ω is a bounded smooth domain and p ≠ 2, μ = 0, then the results of Theorems 1.1, 1.2 in this case are the same as the known results in[19].

Remark 1.6 In this remark, we consider that Ω is a bounded domain. In[11], Wang et al. considered (1.1) with$\mu <\stackrel{̄}{\mu }$, λ > 0 and 1 < q < p < p2 < N. As$0\le \mu <\stackrel{̄}{\mu }$and 1 w< q < p < N, the results of Theorems 1.1, 1.2 are improvements of the main results of[11]. As μ < 0 and 1 < q < p < Np2, Theorem 1.3 is the complement to the results in [[11], Theorem 1.3].

This paper is organized as follows. Some preliminaries and properties of the Nehari manifold are established in Sections 2 and 3, and Theorems 1.1-1.3 are proved in Sections 4-6, respectively. Before ending this section, we explain some notations employed in this paper. In the following argument, we always employ C and C i to denote various positive constants and omit dx in integral for convenience. B(x0; R) is the ball centered at x0 ∈ ℝ N with the radius R > 0, ${\left({\mathcal{D}}_{0}^{1,p}\left(\Omega \right)\right)}^{-1}$ denotes the dual space of ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$, the norm in Lp (Ω) is denoted by |·| p , the quantity O(εt ) denotes |O(εt )/εt | ≤ C, o(εt ) means |o(εt )/εt | → 0 as ε → 0 and o(1) is a generic infinitesimal value. In particular, the quantity O1(εt ) means that there exist C1, C2> 0 such that C1εtO1(εt ) ≤ C2εt as ε is small enough.

## 2 Preliminaries

Throughout this paper, (f1) and (g1) will be assumed. In this section, we will establish several preliminary lemmas. To this end, we first recall a result on the extremal functions of Sμ,s.

Lemma 2.1[16]Assume that 1 < p < N and$0\le \mu <\stackrel{̄}{\mu }$. Then, the limiting problem

$\left\{\begin{array}{cc}\hfill -{\Delta }_{p}u-\mu \frac{{u}^{p-1}}{\mid x{\mid }^{p}}={u}^{p*-1},\hfill & \hfill in{ℝ}^{N}\\left\{0\right\},\hfill \\ \hfill u\in {\mathcal{D}}^{1,p}\left({ℝ}^{N}\right),\phantom{\rule{1em}{0ex}}u>0,\hfill & \hfill in{ℝ}^{N}\\left\{0\right\},\hfill \end{array}\right\$
(2.1)

${V}_{p,\mu ,\epsilon }\left(x\right)={\epsilon }^{-\frac{N-p}{p}}{U}_{p,\mu }\left(\frac{x}{\epsilon }\right)={\epsilon }^{-\frac{N-p}{p}}{U}_{p,\mu }\left(\frac{\mid x\mid }{\epsilon }\right),\phantom{\rule{1em}{0ex}}forall\epsilon >0,$

that satisfy

${\int }_{{ℝ}^{N}}\left(|\nabla {V}_{p,\mu ,\epsilon }\left(x{\right)|}^{p}-\mu \frac{|{V}_{p,\mu ,\epsilon }\left(x{\right)|}^{p}}{|x{|}^{p}}\right)={\int }_{{ℝ}^{N}}|{V}_{p,\mu ,\epsilon }\left(x{\right)|}^{p*}={S}_{\mu }^{\frac{N}{p}}.$

Furthermore, Up,μ(|x|) = Up,μ(r) is decreasing and has the following properties:

$\begin{array}{c}{U}_{p,\mu }\left(1\right)={\left(\frac{N\left(\overline{\mu }-\mu \right)}{N-p}\right)}^{\frac{1}{p*-p}},\\ \underset{r\to {0}^{+}}{\text{lim}}{r}^{a\left(\mu \right)}{U}_{p,\mu }\left(r\right)={c}_{1}>0,\phantom{\rule{0.25em}{0ex}}\underset{r\to {0}^{+}}{\text{lim}}{r}^{a\left(\mu \right)+1}|{{U}^{\prime }}_{p,\mu }\left(r\right)|={c}_{1}a\left(\mu \right)\ge 0,\\ \underset{r\to +\infty }{\text{lim}}{r}^{b\left(\mu \right)}{U}_{p,\mu }\left(r\right)={c}_{2}>0,\phantom{\rule{0.25em}{0ex}}\underset{r\to +\infty }{\text{lim}}{r}^{b\left(\mu \right)+1}|{{U}^{\prime }}_{p,\mu }\left(r\right)|={c}_{2}b\left(\mu \right)>0,\\ {c}_{3}\le {U}_{p,\mu }\left(r\right){\left({r}^{\frac{a\left(\mu \right)}{\delta }}+{r}^{\frac{b\left(\mu \right)}{\delta }}\right)}^{\delta }\le {c}_{4},\phantom{\rule{0.25em}{0ex}}\delta :=\frac{N-p}{p},\end{array}$

where c i (i = 1, 2, 3, 4) are positive constants depending on N, μ and p, and a(μ) and b(μ) are the zeros of the function h(t) = (p - 1)tp - (N - p)tp-1+ μ, t ≥ 0, satisfying$0\le a\left(\mu \right)<\frac{N-p}{p}.

Take ρ > 0 small enough such that B(0; ρ) ⊂ Ω, and define the function

${u}_{\epsilon }\left(x\right)=\eta \left(x\right){V}_{p,\mu ,\epsilon }\left(x\right)={\epsilon }^{-\frac{N-p}{p}}\eta \left(x\right){U}_{p,\mu }\left(\frac{\mid x\mid }{\epsilon }\right),$
(2.2)

where $\eta \in {C}_{0}^{\infty }\left(B\left(0;\rho \right)$ is a cutoff function such that η(x) ≡ 1 in $B\left(0,\frac{\rho }{2}\right)$.

Lemma 2.2[9, 20]Suppose 1 < p < N and$0\le \mu <\stackrel{̄}{\mu }$. Then, the following estimates hold when ε → 0.

$\begin{array}{c}\phantom{\rule{1em}{0ex}}\parallel {u}_{\epsilon }{\parallel }_{\mu }^{p}={S}_{\mu }^{\frac{N}{p}}+O\left({\epsilon }^{p\gamma }\right),\\ {\int }_{\Omega }\mid {u}_{\epsilon }{\mid }^{p*}={S}_{\mu }^{\frac{N}{p}}+O\left({\epsilon }^{p*\gamma }\right),\\ {\int }_{\Omega }\mid {u}_{\epsilon }{\mid }^{q}=\left\{\begin{array}{c}\hfill {O}_{1}\left({\epsilon }^{\theta }\right),\hfill \\ \hfill {O}_{1}\left({\epsilon }^{\theta }\mid \right)ln\epsilon \mid ,\hfill \\ \hfill {O}_{1}\left({\epsilon }^{q\gamma }\right),\hfill \end{array}\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill \frac{N}{b\left(\mu \right)}

where$\delta =\frac{N-p}{p}$, $\theta =N-\frac{N-p}{p}q$and γ = b(μ) - δ.

We also recall the following known result by Ben-Naoum, Troestler and Willem, which will be employed for the energy functional.

Lemma 2.3[21]Let Ω be an domain, not necessarily bounded, inN , 1 ≤ p < N, $1\le qand$k\left(x\right)\in {L}^{\frac{p*}{p*-q}}\left(\Omega \right)$Then, the functional

${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)\to ℝ:u↦{\int }_{{ℝ}^{N}}k\left(x\right)\mid u{\mid }^{q}dx$

is well-defined and weakly continuous.

## 3 Nehari manifold

As J λ is not bounded below on ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$, we need to study J λ on the Nehari manifold

${\mathcal{N}}_{\lambda }=\left\{u\in {\mathcal{D}}_{0}^{1,p}\left(\Omega \right)\\left\{0\right\}:〈{J}_{\lambda }^{\prime }\left(u\right),u〉=0\right\}.$

Note that ${\mathcal{N}}_{\lambda }$ contains all solutions of (1.1) and $u\in {\mathcal{N}}_{\lambda }$ if and only if

$\parallel u{\parallel }_{\mu }^{p}-\lambda {\int }_{\Omega }f\mid u{\mid }^{q}-{\int }_{\Omega }g\mid u{\mid }^{p*}=0.$
(3.1)

Lemma 3.1 J λ is coercive and bounded below on${\mathcal{N}}_{\lambda }$.

Proof Suppose $u\in {\mathcal{N}}_{\lambda }$. From (f1), (3.1), the Hölder inequality and Sobolev embedding theorem, we can deduce that

$\begin{array}{ll}\hfill {J}_{\lambda }\left(u\right)& =\frac{p*-p}{pp*}\parallel u{\parallel }_{\mu }^{p}-\lambda \frac{p*-q}{p*q}{\int }_{\Omega }f\mid u{\mid }^{q}\phantom{\rule{2em}{0ex}}\\ \ge \frac{1}{N}\parallel u{\parallel }_{\mu }^{p}-\lambda \frac{p*-q}{p*q}\mid {f}^{+}{\mid }_{q*}\mid u{\mid }_{p*}^{q}\phantom{\rule{2em}{0ex}}\\ \ge \frac{1}{N}\parallel u{\parallel }_{\mu }^{p}-\lambda \frac{p*-q}{p*q}\mid {f}^{+}{\mid }_{q*}{S}_{\mu }^{-\frac{q}{p}}\parallel u{\parallel }_{\mu }^{q}.\phantom{\rule{2em}{0ex}}\end{array}$
(3.2)

Thus, J λ is coercive and bounded below on ${\mathcal{N}}_{\lambda }$. □

Define ${\psi }_{\lambda }\left(u\right)=〈{J}_{\lambda }^{\prime }\left(u\right),u〉$. Then, for $u\in {\mathcal{N}}_{\lambda }$,

$\begin{array}{ll}\hfill 〈{{\psi }^{\prime }}_{\lambda }\left(u\right),u〉& =p\parallel u{\parallel }_{\mu }^{p}-q\lambda {\int }_{\Omega }f\mid u{\mid }^{q}-p*{\int }_{\Omega }g\mid u{\mid }^{p*}\phantom{\rule{2em}{0ex}}\\ =\left(p-q\right)\parallel u{\parallel }_{\mu }^{p}-\left(p*-q\right){\int }_{\Omega }g\mid u{\mid }^{p*}\phantom{\rule{2em}{0ex}}\\ =\lambda \left({p}^{*}-q\right){\int }_{\Omega }f\mid u{\mid }^{q}-\left(p*-p\right)\parallel u{\parallel }_{\mu }^{p}.\phantom{\rule{2em}{0ex}}\end{array}$
(3.3)

Arguing as in [22], we split ${\mathcal{N}}_{\lambda }$ into three parts:

$\begin{array}{c}{\mathcal{N}}_{\lambda }^{+}=\left\{u\in {\mathcal{N}}_{\lambda }:〈{{\psi }^{\prime }}_{\lambda }\left(u\right),u〉>0\right\},\\ {\mathcal{N}}_{\lambda }^{0}=\left\{u\in {\mathcal{N}}_{\lambda }:〈{{\psi }^{\prime }}_{\lambda }\left(u\right),u〉=0\right\},\\ {\mathcal{N}}_{\lambda }^{-}=\left\{u\in {\mathcal{N}}_{\lambda }:〈{{\psi }^{\prime }}_{\lambda }\left(u\right),u〉<0\right\}.\end{array}$

Lemma 3.2 Suppose u λ is a local minimizer of J λ on${\mathcal{N}}_{\lambda }$and${u}_{\lambda }\notin {\mathcal{N}}_{\lambda }^{0}$.

Then, ${J}_{\lambda }^{\prime }\left({u}_{\lambda }\right)=0$in${\left({\mathcal{D}}_{0}^{1,p}\left(\Omega \right)\right)}^{-1}$.

Proof The proof is similar to [[23], Theorem 2.3] and is omitted. □

Lemma 3.3 ${\mathcal{N}}_{\lambda }^{0}=\varnothing$ for all λ ∈ (0, Λ1).

Proof We argue by contradiction. Suppose that there exists λ ∈ (0, Λ1) such that ${\mathcal{N}}_{\lambda }^{0}\ne \varnothing$. Then, the fact $u\in {\mathcal{N}}_{\lambda }^{0}$ and (3.3) imply that

$\parallel u{\parallel }_{\mu }^{p}=\frac{p*-q}{p-q}{\int }_{\Omega }g\mid u{\mid }^{p*},$

and

$\parallel u{\parallel }_{\mu }^{p}=\lambda \frac{{p}^{*}-q}{{p}^{*}-p}{\int }_{\Omega }f\mid u{\mid }^{q}.$

By (f1), (g1), the Hölder inequality and Sobolev embedding theorem, we have that

$\parallel u\mid {\mid }_{\mu }\ge {\left[\frac{p-q}{\left({p}^{*}-q\right)\mid {g}^{+}{\mid }_{\infty }}\right]}^{\frac{1}{{p}^{*}-p}}{S}_{\mu }^{\frac{N}{{p}^{2}}},$

and

$\parallel u\mid {\mid }_{\mu }\le {\left[\lambda \frac{{p}^{*}-q}{{p}^{*}-p}\mid {f}^{+}{\mid }_{{q}^{*}}{{S}_{\mu }}^{-\frac{q}{p}}\right]}^{\frac{1}{p-q}}.$

Consequently,

$\lambda \ge {\left(\frac{p-q}{\left({p}^{*}-q\right)\mid {g}^{+}{\mid }_{\infty }}\right)}^{\frac{p-q}{{p}^{*}-p}}\left(\frac{{p}^{*}-p}{\left({p}^{*}-q\right)\mid {f}^{+}{\mid }_{{q}^{*}}}\right){S}_{\mu }^{\frac{N}{{p}^{2}}\left(p-q\right)+\frac{q}{p}}={\Lambda }_{1},$

For each $u\in {\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ with ${\int }_{\Omega }g\mid u{\mid }^{{p}^{*}}>0$, we set

${t}_{max}={\left(\frac{\left(p-q\right)\parallel u{\parallel }_{\mu }^{p}}{\left({p}^{*}-q\right){\int }_{\Omega }g\mid u{\mid }^{{p}^{*}}}\right)}^{\frac{1}{{p}^{*}-p}}>0.$

Lemma 3.4 Suppose that λ ∈ (0, Λ1) and$u\in {\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$is a function satisfying with${\int }_{\Omega }g\mid u{\mid }^{{p}^{*}}>0$.

1. (i)

If ${\int }_{\Omega }f\mid u{\mid }^{q}\le 0$ , then there exists a unique t - > t max such that ${t}^{-}u\in {\mathcal{N}}_{\lambda }^{-}$ and

${J}_{\lambda }\left({t}^{-}u\right)=\underset{t\ge 0}{sup}{J}_{\lambda }\left(tu\right).$
2. (ii)

If ${\int }_{\Omega }f\mid u{\mid }^{q}\le 0$, then there exists a unique t ± such that 0 < t + < t max < t -, ${t}^{+}u\in {\mathcal{N}}_{\lambda }^{+}$ and ${t}^{-}u\in {\mathcal{N}}_{\lambda }^{-}$. Moreover,

${J}_{\lambda }\left({t}^{+}u\right)=\underset{0\le t\le {t}_{max}}{inf}{J}_{\lambda }\left(tu\right),\phantom{\rule{1em}{0ex}}{J}_{\lambda }\left({t}^{-}u\right)=\underset{t\ge {t}^{+}}{sup}{J}_{\lambda }\left(tu\right).$

Proof See Brown-Wu [[24], Lemma 2.6]. □

We remark that it follows Lemma 3.3, ${\mathcal{N}}_{\lambda }={\mathcal{N}}_{\lambda }^{+}\cup {\mathcal{N}}_{\lambda }^{-}$ for all λ ∈ (0, Λ1). Furthermore, by Lemma 3.4, it follows that ${\mathcal{N}}_{\lambda }^{+}$ and ${\mathcal{N}}_{\lambda }^{-}$ are nonempty, and by Lemma 3.1, we may define

${\alpha }_{\lambda }=\underset{u\in {\mathcal{N}}_{\lambda }}{inf}{J}_{\lambda }\left(u\right),\phantom{\rule{1em}{0ex}}{\alpha }_{\lambda }^{+}=\underset{u\in {\mathcal{N}}_{\lambda }^{+}}{inf}{J}_{\lambda }\left(u\right),\phantom{\rule{1em}{0ex}}{\alpha }_{\lambda }^{-}=\underset{u\in {\mathcal{N}}_{\lambda }^{-}}{inf}{J}_{\lambda }\left(u\right).$

Lemma 3.5 (i) If λ ∈ (0, Λ1), then we have${\alpha }_{\lambda }\le {\alpha }_{\lambda }^{+}<0$.

1. (ii)

If $\lambda \in \left(0,\frac{q}{p}{\Lambda }_{1}\right)$, then ${\alpha }_{\lambda }^{-}>{d}_{0}$ for some positive constant d 0.

In particular, for each$\lambda \in \left(0,\frac{q}{p}{\Lambda }_{1}\right)$, we have${\alpha }_{\lambda }={\alpha }_{\lambda }^{+}<0<{\alpha }_{\lambda }^{-}$.

Proof (i) Suppose that $u\in {\mathcal{N}}_{\lambda }^{+}$. From (3.3), it follows that

$\frac{p-q}{{p}^{*}-q}\parallel u{\parallel }_{\mu }^{p}>{\int }_{\Omega }g\mid u{\mid }^{{p}^{*}}.$
(3.4)

According to (3.1) and (3.4), we have

$\begin{array}{ll}\hfill {J}_{\lambda }\left(u\right)& =\left(\frac{1}{p}-\frac{1}{q}\right)\parallel u{\parallel }_{\mu }^{p}+\left(\frac{1}{q}-\frac{1}{{p}^{*}}\right){\int }_{\Omega }g\mid u{\mid }^{{p}^{*}}\phantom{\rule{2em}{0ex}}\\ <\left[\left(\frac{1}{p}-\frac{1}{q}\right)+\left(\frac{1}{q}-\frac{1}{{p}^{*}}\right)\left(\frac{p-q}{{p}^{*}-q}\right)\right]\parallel u{\parallel }_{\mu }^{p}\phantom{\rule{2em}{0ex}}\\ =-\frac{p-q}{qN}\parallel u{\parallel }_{\mu }^{p}<0.\phantom{\rule{2em}{0ex}}\end{array}$

By the definitions of α λ and ${\alpha }_{\lambda }^{+}$, we get that ${\alpha }_{\lambda }\le {\alpha }_{\lambda }^{+}<0$.

(ii) Suppose $\lambda \in \left(0,\frac{q}{p}{\Lambda }_{1}\right)$ and $u\in {\mathcal{N}}_{\lambda }^{-}$. Then, (3.3) implies that

$\frac{p-q}{{p}^{*}-q}\parallel u{\parallel }_{\mu }^{p}<{\int }_{\Omega }\mid u{\mid }^{{p}^{*}}.$
(3.5)

Moreover, by (g1) and the Sobolev embedding theorem, we have

${\int }_{\Omega }g\mid u{\mid }^{{p}^{*}}\le \mid {g}^{+}{\mid }_{\infty }{S}_{\mu }^{-\frac{{p}^{*}}{p}}\parallel u{\parallel }_{\mu }^{{p}^{*}}.$
(3.6)

From (3.5) and (3.6), it follows that

$\parallel u\mid {\mid }_{\mu }>{\left(\frac{p-q}{\left({p}^{*}-q\right)\mid {g}^{+}{\mid }_{\infty }}\right)}^{\frac{1}{{p}^{*}-p}}{S}_{\mu }^{\frac{N}{{p}^{2}}}\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}u\in {\mathcal{N}}_{\lambda }^{-}.$
(3.7)

By (3.2) and (3.7), we get

$\begin{array}{ll}\hfill {J}_{\lambda }\left(u\right)& \ge \parallel u{\parallel }_{\mu }^{q}\left[\frac{1}{N}\parallel u{\parallel }_{\mu }^{p-q}-\lambda \frac{{p}^{*}-q}{{p}^{*}q}\mid {f}^{+}{\mid }_{{q}^{*}}{S}_{\mu }^{-\frac{q}{p}}\right]\phantom{\rule{2em}{0ex}}\\ >{\left(\frac{p-q}{\left({p}^{*}-q\right)\mid {g}^{+}{\mid }_{\infty }}\right)}^{\frac{q}{{p}^{*}-p}}{S}_{\mu }^{\frac{qN}{{p}^{2}}}\left[\frac{1}{N}{\left(\frac{p-q}{\left({p}^{*}-q\right)\mid {g}^{+}{\mid }_{\infty }}\right)}^{\frac{p-q}{{p}^{*}-p}}{S}_{\mu }^{\frac{N\left(p-q\right)}{{p}^{2}}}\right\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\lambda \frac{{p}^{*}-q}{{p}^{*}q}\mid {f}^{+}{\mid }_{{q}^{*}}{S}_{\mu }^{-\frac{q}{p}}]\phantom{\rule{2em}{0ex}}\end{array}$

which implies that

${J}_{\lambda }\left(u\right)>{d}_{0}\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}u\in {\mathcal{N}}_{\lambda }^{-},$

for some positive constant d0. □

Remark 3.6 If$\lambda \in \left(0,\frac{q}{p}{\Lambda }_{0}\right)$, then by Lemmas 3.4 and 3.5, for each$u\in {\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$with${\int }_{\Omega }g\mid u{\mid }^{{p}^{*}}>0$, we can easily deduce that

${t}^{-}u\in {\mathcal{N}}_{\lambda }^{-}\phantom{\rule{2.77695pt}{0ex}}and\phantom{\rule{2.77695pt}{0ex}}{J}_{\lambda }\left({t}^{-}u\right)=\underset{t\ge 0}{sup}{J}_{\lambda }\left(tu\right)\ge {\alpha }_{\lambda }^{-}>0.$

## 4 Proof of Theorem 1.1

First, we define the Palais-Smale (simply by (PS)) sequences, (PS)-values and (PS)-conditions in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ for J λ as follows:

Definition 4.1 (i) For c ∈ ℝ, a sequence {u n } is a (PS) c -sequence in${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$for J λ if J λ (u n ) = c + o(1) and (J λ )'(u n ) = o(1) strongly in${\left({\mathcal{D}}_{0}^{1,p}\left(\Omega \right)\right)}^{-1}$as n → ∞.

1. (ii)

c ∈ ℝ is a (PS)-value in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ for J λ if there exists a (PS) c -sequence in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ for J λ .

2. (iii)

J λ satisfies the (PS) c -condition in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ if any (PS) c -sequence {u n } in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ for J λ contains a convergent subsequence.

Lemma 4.2 (i) If λ ∈ (0, Λ1), then J λ has a${\left(PS\right)}_{{\alpha }_{\lambda }}$-sequence$\left\{{u}_{n}\right\}\subset {\mathcal{N}}_{\lambda }$.

1. (ii)

If $\lambda \in \left(0,\frac{q}{p}{\Lambda }_{1}\right)$, then J λ has a ${\left(PS\right)}_{{\alpha }_{\lambda }}$ -sequence $\left\{{u}_{n}\right\}\subset {\mathcal{N}}_{\lambda }^{-}$.

Proof The proof is similar to [19, 25] and the details are omitted. □

Now, we establish the existence of a local minimum for J λ on ${\mathcal{N}}_{\lambda }$.

Theorem 4.3 Suppose that N ≥ 3, $\mu <\stackrel{̄}{\mu }$, 1 < q < p < N and the conditions (f1), (g1) hold. If λ ∈ (0, Λ1), then there exists ${u}_{\lambda }\in {\mathcal{N}}_{\lambda }^{+}$ such that

1. (i)

${J}_{\lambda }\left({u}_{\lambda }\right)={\alpha }_{\lambda }={\alpha }_{\lambda }^{+}$ ,

2. (ii)

u λ is a positive solution of (1.1),

3. (iii)

||u λ || μ → 0 as λ → 0+.

Proof By Lemma 4.2 (i), there exists a minimizing sequence $\left\{{u}_{n}\right\}\subset {\mathcal{N}}_{\lambda }$ such that

${J}_{\lambda }\left({u}_{n}\right)={\alpha }_{\lambda }+o\left(1\right)\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}{J}_{\lambda }^{\prime }\left({u}_{n}\right)=o\left(1\right)\phantom{\rule{1em}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}{\left({\mathcal{D}}_{0}^{1,p}\left(\Omega \right)\right)}^{-1}.$
(4.1)

Since J λ is coercive on ${\mathcal{N}}_{\lambda }$ (see Lemma 2.1), we get that (u n ) is bounded in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$. Passing to a subsequence, there exists ${u}_{\lambda }\in {\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ such that as n → ∞

$\left\{\begin{array}{c}\hfill {u}_{n}⇀{u}_{\lambda }\phantom{\rule{2.77695pt}{0ex}}weakly\phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}{\mathcal{D}}_{0}^{1,p}\left(\Omega \right),\hfill \\ \hfill {u}_{n}⇀{u}_{\lambda }\phantom{\rule{2.77695pt}{0ex}}weakly\phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}{L}^{{p}^{*}}\left(\Omega \right),\hfill \\ \hfill {u}_{n}\to {u}_{\lambda }\phantom{\rule{2.77695pt}{0ex}}strongly\phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}{L}_{loc}^{r}\left(\Omega \right)\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}all\phantom{\rule{2.77695pt}{0ex}}1\le r<{p}^{*},\hfill \\ \hfill {u}_{n}\to {u}_{\lambda }\phantom{\rule{2.77695pt}{0ex}}a.e.\phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}\Omega .\hfill \end{array}\right\$
(4.2)

By (f1) and Lemma 2.3, we obtain

$\begin{array}{c}\hfill \lambda {\int }_{\Omega }f\mid {u}_{n}{\mid }^{q}=\lambda {\int }_{\Omega }f\mid {u}_{\lambda }{\mid }^{q}+o\left(1\right)\phantom{\rule{2.77695pt}{0ex}}as\phantom{\rule{2.77695pt}{0ex}}n\to \infty .\hfill \end{array}$
(3)

From (4.1)-(4.3), a standard argument shows that u λ is a critical point of J λ . Furthermore, the fact $\left\{{u}_{n}\right\}\subset {\mathcal{N}}_{\lambda }$ implies that

$\lambda {\int }_{\Omega }f\mid {u}_{n}{\mid }^{q}=\frac{q\left({p}^{*}-p\right)}{p\left({p}^{*}-q\right)}\parallel {u}_{n}{\parallel }_{\mu }^{p}-\frac{{p}^{*}q}{{p}^{*}-q}{J}_{\lambda }\left({u}_{n}\right).$
(4.4)

Taking n → ∞ in (4.4), by (4.1), (4.3) and the fact α λ < 0, we get

$\lambda {\int }_{\Omega }f\mid {u}_{\lambda }{\mid }^{q}\ge -\frac{{p}^{*}q}{{p}^{*}-q}{\alpha }_{\lambda }>0.$
(4.5)

Thus, ${u}_{\lambda }\in {\mathcal{N}}_{\lambda }$ is a nontrivial solution of (1.1).

Next, we prove that u n u λ strongly in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ and J λ (u λ ) = α λ . From (4.3), the fact ${u}_{n},{u}_{\lambda }\in {\mathcal{N}}_{\lambda }$ and the Fatou's lemma it follows that

$\begin{array}{ll}\hfill {\alpha }_{\lambda }& \le {J}_{\lambda }\left({u}_{\lambda }\right)=\frac{1}{N}\parallel {u}_{\lambda }{\parallel }_{\mu }^{p}-\lambda \frac{{p}^{*}-q}{{p}^{*}q}{\int }_{\Omega }f\mid {u}_{\lambda }{\mid }^{q}\phantom{\rule{2em}{0ex}}\\ \le \underset{n\to \infty }{liminf}\left(\frac{1}{N}\parallel {u}_{n}{\parallel }_{\mu }^{p}-\lambda \frac{{p}^{*}-q}{{p}^{*}q}{\int }_{\Omega }f\mid {u}_{n}{\mid }^{q}\right)\phantom{\rule{2em}{0ex}}\\ =\underset{n\to \infty }{liminf}{J}_{\lambda }\left({u}_{n}\right)={\alpha }_{\lambda },\phantom{\rule{2em}{0ex}}\end{array}$

which implies that J λ (u λ ) = α λ and $\underset{n\to \infty }{lim}\parallel {u}_{n}{\parallel }_{\mu }^{p}=\parallel {u}_{\lambda }{\parallel }_{\mu }^{p}$. Standard argument shows that u n u λ strongly in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$. Moreover, ${u}_{\lambda }\in {\mathcal{N}}_{\lambda }^{+}$. Otherwise, if ${u}_{\lambda }\in {\mathcal{N}}_{\lambda }^{-}$, by Lemma 3.4, there exist unique ${t}_{\lambda }^{+}$ and ${t}_{\lambda }^{-}$ such that ${t}_{\lambda }^{+}{u}_{\lambda }\in {\mathcal{N}}_{\lambda }^{+}$, ${t}_{\lambda }^{-}{u}_{\lambda }\in {\mathcal{N}}_{\lambda }^{-}$ and ${t}_{\lambda }^{+}<{t}_{\lambda }^{-}=1$. Since

$\frac{d}{dt}{J}_{\lambda }\left({t}_{\lambda }^{+}{u}_{\lambda }\right)=0\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}\frac{{d}^{2}}{d{t}^{2}}{J}_{\lambda }\left({t}_{\lambda }^{+}{u}_{\lambda }\right)>0,$

there exists $\stackrel{̄}{t}\in \left({t}_{\lambda }^{+},{t}_{\lambda }^{-}\right)$ such that ${J}_{\lambda }\left({t}_{\lambda }^{+}{u}_{\lambda }\right)<{J}_{\lambda }\left(\stackrel{̄}{t}{u}_{\lambda }\right)$. By Lemma 3.4, we get that

${J}_{\lambda }\left({t}_{\lambda }^{+}{u}_{\lambda }\right)<{J}_{\lambda }\left(\stackrel{̄}{t}{u}_{\lambda }\right)\le {J}_{\lambda }\left({t}_{\lambda }^{-}{u}_{\lambda }\right)={J}_{\lambda }\left({u}_{\lambda }\right),$

which is a contradiction. If $u\in {\mathcal{N}}_{\lambda }^{+}$, then $\mid u\mid \in {\mathcal{N}}_{\lambda }^{+}$, and by J λ (u λ ) = J λ (|u λ |) = α λ , we get $\mid {u}_{\lambda }\mid \in {\mathcal{N}}_{\lambda }^{+}$ is a local minimum of J λ on ${\mathcal{N}}_{\lambda }$. Then, by Lemma 3.2, we may assume that u λ is a nontrivial nonnegative solution of (1.1). By Harnack inequality due to Trudinger [26], we obtain that u λ > 0 in Ω. Finally, by (3.3), the Hölder inequality and Sobolev embedding theorem, we obtain

$\parallel {u}_{\lambda }{\parallel }_{\mu }^{p-q}<\lambda \frac{{p}^{*}-q}{{p}^{*}-p}\mid {f}^{+}{\mid }_{{q}^{*}}{S}_{\mu }^{-\frac{q}{p}}.$

which implies that ||u λ || μ → 0 as λ → 0+. □

Proof of Theorem 1.1 From Theorem 4.3, it follows that the problem (1.1) has a positive solution ${u}_{\lambda }\in {\mathcal{N}}_{\lambda }^{+}$ for all λ ∈ (0, Λ0). □

## 5 Proof of Theorem 1.2

For 1 < p < N and $\mu <\stackrel{̄}{\mu }$, let

${c}^{*}=\frac{1}{N}\mid {g}^{+}{\mid }_{\infty }^{-\frac{N-p}{p}}{S}_{\mu }^{\frac{N}{p}}.$

Lemma 5.1 Suppose {u n } is a bounded sequence in${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$. If {u n } is a (PS) c -sequence for J λ with c ∈ (0, c*), then there exists a subsequence of {u n } converging weakly to a nonzero solution of (1.1).

Proof Let $\left\{{u}_{n}\right\}\subset {\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ be a (PS) c -sequence for J λ with c ∈ (0, c*). Since {u n } is bounded in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$, passing to a subsequence if necessary, we may assume that as n → ∞

$\left\{\begin{array}{c}\hfill {u}_{n}⇀{u}_{0}\phantom{\rule{2.77695pt}{0ex}}weakly\phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}{\mathcal{D}}_{0}^{1,p}\left(\Omega \right),\hfill \\ \hfill {u}_{n}⇀{u}_{0}\phantom{\rule{2.77695pt}{0ex}}weakly\phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}{L}^{{p}^{*}}\left(\Omega \right),\hfill \\ \hfill {u}_{n}\to {u}_{0}\phantom{\rule{2.77695pt}{0ex}}strongly\phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}{L}_{loc}^{r}\left(\Omega \right)\phantom{\rule{2.77695pt}{0ex}}for\phantom{\rule{2.77695pt}{0ex}}1\le r<{p}^{*},\hfill \\ \hfill {u}_{n}\to {u}_{0}\phantom{\rule{2.77695pt}{0ex}}a.e.\phantom{\rule{2.77695pt}{0ex}}in\phantom{\rule{2.77695pt}{0ex}}\Omega .\hfill \end{array}\right\$
(5.1)

By (f1), (g1), (5.1) and Lemma 2.3, we have that ${J}_{\lambda }^{\prime }\left({u}_{0}\right)=0$ and

$\lambda {\int }_{\Omega }f\mid {u}_{n}{\mid }^{q}=\lambda {\int }_{\Omega }f\mid {u}_{0}{\mid }^{q}+o\left(1\right)\phantom{\rule{2.77695pt}{0ex}}as\phantom{\rule{2.77695pt}{0ex}}n\to \infty .$
(5.2)

Next, we verify that u0 ≢ 0. Arguing by contradiction, we assume u0 ≡ 0. Since ${J}_{\lambda }^{\prime }\left({u}_{n}\right)=o\left(1\right)$ as n → ∞ and {u n } is bounded in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$, then by (5.2), we can deduce that

$0=〈\underset{n\to \infty }{lim}{J}_{\lambda }^{\prime }\left({u}_{n}\right),{u}_{n}〉=\underset{n\to \infty }{lim}\left(\parallel {u}_{n}{\parallel }_{\mu }^{p}-{\int }_{\Omega }g\mid {u}_{n}{\mid }^{{p}^{*}}\right).$

Then, we can set

$\underset{n\to \infty }{lim}\parallel {u}_{n}{\parallel }_{\mu }^{p}=\underset{n\to \infty }{lim}{\int }_{\Omega }g\mid {u}_{n}{\mid }^{{p}^{*}}=l.$
(5.3)

If l = 0, then we get c = limn→∞J λ (u n ) = 0, which is a contradiction. Thus, we conclude that l > 0. Furthermore, the Sobolev embedding theorem implies that

$\begin{array}{ll}\hfill \parallel {u}_{n}{\parallel }_{\mu }^{p}& \ge {S}_{\mu }{\left({\int }_{\Omega }g\mid {u}_{n}{\mid }^{{p}^{*}}\right)}^{\frac{p}{{p}^{*}}}\phantom{\rule{2em}{0ex}}\\ \ge {S}_{\mu }{\left({\int }_{\Omega }\frac{g}{\mid {g}^{+}{\mid }_{\infty }}\mid {u}_{n}{\mid }^{{p}^{*}}\right)}^{\frac{p}{{p}^{*}}}\phantom{\rule{2em}{0ex}}\\ ={S}_{\mu }\mid {g}^{+}{\mid }_{\infty }^{-\frac{N-p}{N}}{\left({\int }_{\Omega }g\mid {u}_{n}{\mid }^{{p}^{*}}\right)}^{\frac{p}{{p}^{*}}}.\phantom{\rule{2em}{0ex}}\end{array}$

Then, as n → ∞ we have $l=\underset{n\to \infty }{lim}\parallel {u}_{n}{\parallel }_{\mu }^{p}\ge {S}_{\mu }\mid {g}^{+}{\mid }_{\infty }^{-\frac{N-p}{N}}{l}^{\frac{p}{{p}^{*}}}$, which implies that

$l\ge \mid {g}^{+}{\mid }_{\infty }^{-\frac{N-p}{p}}{S}_{\mu }^{\frac{N}{p}}.$
(5.4)

Hence, from (5.2)-(5.4), we get

$\begin{array}{ll}\hfill c& =\underset{n\to \infty }{lim}{J}_{\lambda }\left({u}_{n}\right)\phantom{\rule{2em}{0ex}}\\ =\frac{1}{p}\underset{n\to \infty }{lim}\parallel {u}_{n}{\parallel }_{\mu }^{p}-\frac{\lambda }{q}\underset{n\to \infty }{lim}{\int }_{\Omega }f\mid {u}_{n}{\mid }^{q}-\frac{1}{{p}^{*}}\underset{n\to \infty }{lim}{\int }_{\Omega }g\mid {u}_{n}{\mid }^{{p}^{*}}\phantom{\rule{2em}{0ex}}\\ =\left(\frac{1}{p}-\frac{1}{{p}^{*}}\right)l\phantom{\rule{2em}{0ex}}\\ \ge \frac{1}{N}\mid {g}^{+}{\mid }_{\infty }^{-\frac{N-p}{p}}{S}_{\mu }^{\frac{N}{p}}.\phantom{\rule{2em}{0ex}}\end{array}$

This is contrary to c < c*. Therefore, u0 is a nontrivial solution of (1.1). □

Lemma 5.2 Suppose$\left(\mathcal{H}\right)$and (f1) - (g2) hold. If$0<\mu <\stackrel{̄}{\mu }$, x0 = 0 and βpγ, then for any λ > 0, there exists${v}_{\lambda }\in {\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$such that

$\underset{t\ge 0}{sup}{J}_{\lambda }\left(t{v}_{\lambda }\right)<{c}^{*}.$
(5.5)

In particular, ${\alpha }_{\lambda }^{-}<{c}^{*}$for all λ ∈ (0, Λ1).

Proof From [[11], Lemma 5.3], we get that if ε is small enough, there exist t ε > 0 and the positive constants C i (i = 1, 2) independent of ε, such that

$\underset{t\ge 0}{sup}{J}_{\lambda }\left(t{u}_{\epsilon }\right)={J}_{\lambda }\left({t}_{\epsilon }{u}_{\epsilon }\right)\phantom{\rule{2.77695pt}{0ex}}\text{and}\phantom{\rule{2.77695pt}{0ex}}0<{C}_{1}\le {t}_{\epsilon }\le {C}_{2}<\infty .$
(5.6)

By (g2), we conclude that

$\begin{array}{ll}\hfill \left|{\int }_{\Omega }g\left(x\right)\mid {u}_{\epsilon }{\mid }^{{p}^{*}}-{{\int }_{\Omega }g\left(0\right)\mid {u}_{\epsilon }\mid }^{{p}^{*}}\right|& \le {\int }_{\Omega }\mid g\left(x\right)-g\left(0\right)\mid \phantom{\rule{2.77695pt}{0ex}}\mid {u}_{\epsilon }{\mid }^{{p}^{*}}\phantom{\rule{2em}{0ex}}\\ =O\left({\int }_{B\left(0;\rho \right)}\mid x{\mid }^{\beta }\mid {u}_{\epsilon }{\mid }^{{p}^{*}}\right)\phantom{\rule{2em}{0ex}}\\ =O\left({\epsilon }^{\beta }\right),\phantom{\rule{2em}{0ex}}\end{array}$

which together with Lemma 2.2 implies that

${\int }_{\Omega }g\left(x\right)\mid {u}_{\epsilon }{\mid }^{{p}^{*}}=g\left(0\right){S}_{\mu }^{\frac{N}{p}}+O\left({\epsilon }^{{p}^{*}\gamma }\right)+O\left({\epsilon }^{\beta }\right).$
(5.7)

From the fact λ > 0, 1 < q < p, β and

$\underset{t\ge 0}{max}\left(\frac{{t}^{p}}{p}{B}_{1}-\frac{{t}^{{p}^{*}}}{{p}^{*}}{B}_{2}\right)=\frac{1}{N}{B}_{1}^{\frac{N}{p}}{B}_{2}^{-\frac{N-p}{p}},\phantom{\rule{1em}{0ex}}{B}_{1}>0,{B}_{2}>0,$

and by Lemma 2.2, (5.7) and (f2), we get

$\begin{array}{ll}\hfill {J}_{\lambda }\left({t}_{\epsilon }{u}_{\epsilon }\right)& =\frac{{t}_{\epsilon }^{p}}{p}\parallel {u}_{\epsilon }{\parallel }_{\mu }^{p}-\frac{{t}_{\epsilon }^{{p}^{*}}}{{p}^{*}}{\int }_{\Omega }g\mid {u}_{\epsilon }{\mid }^{{p}^{*}}-\lambda \frac{{t}_{\epsilon }^{q}}{q}{\int }_{\Omega }f\mid {u}_{\epsilon }{\mid }^{q}\phantom{\rule{2em}{0ex}}\\ \le \frac{1}{N}\parallel {u}_{\epsilon }{\parallel }_{\mu }^{\frac{N}{p}}{\left({\int }_{\Omega }g\mid {u}_{\epsilon }{\mid }^{{p}^{*}}\right)}^{-\frac{N-p}{p}}-\lambda \frac{{C}_{1}^{q}}{q}{\beta }_{0}{\int }_{\Omega }\mid {u}_{\epsilon }{\mid }^{q}\phantom{\rule{2em}{0ex}}\\ =\frac{1}{N}{\left({S}_{\mu }^{\frac{N}{p}}+O\left({\epsilon }^{p\gamma }\right)\right)}^{\frac{N}{p}}{\left(g\left(0\right){S}_{\mu }^{\frac{N}{p}}+O\left({\epsilon }^{{p}^{*}\gamma }\right)+O\left({\epsilon }^{\beta }\right)\right)}^{-\frac{N-p}{p}}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\lambda \frac{{C}_{1}^{q}}{q}{\beta }_{0}{\int }_{\Omega }\mid {u}_{\epsilon }{\mid }^{q}\phantom{\rule{2em}{0ex}}\\ =\frac{1}{N}g{\left(0\right)}^{-\frac{N-p}{p}}{S}_{\mu }^{\frac{N}{p}}+O\left({\epsilon }^{p\gamma }\right)+O\left({\epsilon }^{\beta }\right)-\lambda \frac{{C}_{1}^{q}}{q}{\beta }_{0}{\int }_{\Omega }\mid {u}_{\epsilon }{\mid }^{q}.\phantom{\rule{2em}{0ex}}\end{array}$
(5.8)

By (5.6) and (5.8), we have that

$\underset{t\ge 0}{sup}{J}_{\lambda }\left(t{u}_{\epsilon }\right)\le {c}^{*}+O\left({\epsilon }^{p\gamma }\right)+O\left({\epsilon }^{\beta }\right)-\lambda \frac{{C}_{1}^{q}}{q}{\beta }_{0}{\int }_{\Omega }\mid {u}_{\epsilon }{\mid }^{q}.$
(5.9)
1. (i)

If $1, then by Lemma 2.2 and $\gamma =b\left(\mu \right)-\delta =b\left(\mu \right)-\frac{N-p}{p}>0$ we have that

${\int }_{\Omega }\mid {u}_{\epsilon }{\mid }^{q}={O}_{1}\left({\epsilon }^{q\gamma }\right).$

Combining this with (5.9), for any λ > 0, we can choose ε λ small enough such that

$\underset{t\ge 0}{sup}{J}_{\lambda }\left(t{u}_{{\epsilon }_{\lambda }}\right)<{c}^{*}.$
1. (ii)

If $\frac{N}{b\left(\mu \right)}\le q, then by Lemma 2.2 and γ > 0 we have that

${\int }_{\Omega }\mid {u}_{\epsilon }{\mid }^{q}=\left\{\begin{array}{cc}\hfill {O}_{1}\left({\epsilon }^{\theta }\right),\hfill & \hfill \phantom{\rule{1em}{0ex}}q>\frac{N}{b\left(\mu \right)},\hfill \\ \hfill {O}_{1}\left({\epsilon }^{\theta }\mid ln\epsilon \mid \right),\hfill & \hfill \phantom{\rule{1em}{0ex}}q=\frac{N}{b\left(\mu \right)},\hfill \end{array}\right\$

and

$p\gamma =b\left(\mu \right)p+p-N>N+\left(1-\frac{N}{p}\right)q=\theta .$

Combining this with (5.9), for any λ > 0, we can choose ε λ small enough such that

$\underset{t\ge 0}{sup}{J}_{\lambda }\left(t{u}_{{\epsilon }_{\lambda }}\right)<{c}^{*}.$

From (i) and (ii), (5.5) holds by taking ${v}_{\lambda }={u}_{{\epsilon }_{\lambda }}$.

In fact, by (f2), (g2) and the definition of ${u}_{{\epsilon }_{\lambda }}$, we have that

${\int }_{\Omega }f\mid {u}_{{\epsilon }_{\lambda }}{\mid }^{q}>0\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}{\int }_{\Omega }g\mid {u}_{{\epsilon }_{\lambda }}{\mid }^{{p}^{*}}>0.$

From Lemma 3.4, the definition of ${\alpha }_{\lambda }^{-}$ and (5.5), for any λ ∈ (0, Λ0), there exists ${t}_{{\epsilon }_{\lambda }}>0$ such that ${t}_{{\epsilon }_{\lambda }}{u}_{{\epsilon }_{\lambda }}\in {\mathcal{N}}_{\lambda }^{-}$ and

${\alpha }_{\lambda }^{-}\le {J}_{\lambda }\left({t}_{{\epsilon }_{\lambda }}{u}_{{\epsilon }_{\lambda }}\right)\le \underset{t\ge 0}{sup}{J}_{\lambda }\left(t{t}_{{\epsilon }_{\lambda }}{u}_{{\epsilon }_{\lambda }}\right)<{c}^{*}.$

The proof is thus complete. □

Now, we establish the existence of a local minimum of J λ on ${\mathcal{N}}_{\lambda }^{-}$.

Theorem 5.3 Suppose$\left(\mathcal{H}\right)$and (f1) - (g2) hold. If$0<\mu <\stackrel{̄}{\mu }$, x0 = 0, βpγ and$\lambda \in \left(0,\frac{q}{p}{\Lambda }_{1}\right)$, then there exists${U}_{\lambda }\in {\mathcal{N}}_{\lambda }^{-}$such that

1. (i)

${J}_{\lambda }\left({U}_{\lambda }\right)={\alpha }_{\lambda }^{-}$,

2. (ii)

U λ is a positive solution of (1.1).

Proof If $\lambda \in \left(0,\frac{q}{p}{\Lambda }_{1}\right)$, then by Lemmas 3.5 (ii), 4.2 (ii) and 5.2, there exists a ${\left(PS\right)}_{{\alpha }_{\lambda }}$-sequence $\left\{{u}_{n}\right\}\subset {\mathcal{N}}_{\lambda }^{-}$ in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ for J λ with ${\alpha }_{\lambda }^{-}\in \left(0,{c}^{*}\right)$. Since J λ is coercive on ${\mathcal{N}}_{\lambda }^{-}$ (see Lemma 3.1), we get that {u n } is bounded in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$. From Lemma 5.1, there exists a subsequence still denoted by {u n } and a nontrivial solution ${U}_{\lambda }\in {\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ of (1.1) such that u n U λ weakly in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$.

First, we prove that ${U}_{\lambda }\in {\mathcal{N}}_{\lambda }^{-}$. On the contrary, if ${U}_{\lambda }\in {\mathcal{N}}_{\lambda }^{+}$, then by ${\mathcal{N}}_{\lambda }^{-}\cup \left\{0\right\}$ is closed in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$, we have ||U λ || μ < lim infn→∞||u n || μ . From (g2) and U λ ≢ 0 in Ω, we have ${\int }_{\Omega }g\mid {U}_{\lambda }{\mid }^{{p}^{*}}>0$. Thus, by Lemma 3.4, there exists a unique t λ such that ${t}_{\lambda }{U}_{\lambda }\in {\mathcal{N}}_{\lambda }^{-}$. If $u\in {\mathcal{N}}_{\lambda }$, then it is easy to see that

${J}_{\lambda }\left(u\right)=\frac{1}{N}\parallel u{\parallel }_{\mu }^{p}-\lambda \left(\frac{{p}^{*}-q}{{p}^{*}q}\right){\int }_{\Omega }f\mid u{\mid }^{q}.$
(5.10)

From Remark 3.6, ${u}_{n}\in {\mathcal{N}}_{\lambda }^{-}$ and (5.10), we can deduce that

${\alpha }_{\lambda }^{-}\le {J}_{\lambda }\left({t}_{\lambda }{U}_{\lambda }\right)<\underset{n\to \infty }{lim}{J}_{\lambda }\left({t}_{\lambda }{u}_{n}\right)\le \underset{n\to \infty }{lim}{J}_{\lambda }\left({u}_{n}\right)={\alpha }_{\lambda }^{-}.$

This is a contradiction. Thus, ${U}_{\lambda }\in {\mathcal{N}}_{\lambda }^{-}$.

Next, by the same argument as that in Theorem 4.3, we get that u n U λ strongly in ${\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$ and ${J}_{\lambda }\left({U}_{\lambda }\right)={\alpha }_{\lambda }^{-}>0$ for all $\lambda \in \left(0,\frac{q}{p}{\Lambda }_{1}\right)$. Since J λ (U λ ) = J λ (|U λ |) and $\mid {U}_{\lambda }\mid \in {\mathcal{N}}_{\lambda }^{-}$, by Lemma 3.2, we may assume that U λ is a nontrivial nonnegative solution of (1.1). Finally, by Harnack inequality due to Trudinger [26], we obtain that U λ is a positive solution of (1.1). □

Proof of Theorem 1.2 From Theorem 4.3, we get the first positive solution ${u}_{\lambda }\in {\mathcal{N}}_{\lambda }^{+}$ for all λ ∈ (0, Λ0). From Theorem 5.3, we get the second positive solution ${U}_{\lambda }\in {\mathcal{N}}_{\lambda }^{+}$ for all $\lambda \in \left(0,\frac{q}{p}{\Lambda }_{0}\right)$. Since ${\mathcal{N}}_{\lambda }^{-}\cap {\mathcal{N}}_{\lambda }^{-}=\varnothing$, this implies that u λ and U λ are distinct. □

## 6 Proof of Theorem 1.3

In this section, we consider the case μ ≤ 0. In this case, it is well-known S μ = S0 where S μ is defined as in (1.2). Thus, we have ${c}^{*}=\frac{1}{N}\mid {g}^{+}{\mid }_{\infty }^{-\frac{N-p}{p}}{S}_{0}^{\frac{N}{p}}$ when μ ≤ 0.

Lemma 6.1 Suppose$\left(\mathcal{H}\right)$and (f1) - (g2) hold. If Np2, μ < 0, x0 ≠ 0 and$\frac{\beta }{p}\ge \stackrel{̃}{\gamma }:=\frac{N-p}{p\left(p-1\right)}$, then for any λ > 0 and μ < 0, there exists${v}_{\lambda ,\mu }\in {\mathcal{D}}_{0}^{1,p}\left(\Omega \right)$such that

$\underset{t\ge 0}{sup}{J}_{\lambda }\left(t{v}_{\lambda ,\mu }\right)<{c}^{*}.$
(6.1)

In particular, ${\alpha }_{\lambda }^{-}<{c}^{*}$for all λ ∈ (0, Λ1).

Proof Note that S0 has the following explicit extremals [27]:

${V}_{\epsilon }\left(x\right)=\stackrel{̄}{C}{\epsilon }^{-\frac{N-p}{p}}{\left(1+{\left(\frac{\mid x-{x}_{0}\mid }{\epsilon }\right)}^{\frac{p}{p-1}}\right)}^{-\frac{N-p}{p}},\phantom{\rule{1em}{0ex}}\forall \epsilon >0,{x}_{0}\in {ℝ}^{N},$

where $\stackrel{̄}{C}>0$ is a particular constant. Take ρ > 0 small enough such that B(x0; ρ) ⊂ Ω\{0} and set ${\stackrel{˜}{u}}_{\epsilon }\left(x\right)=\phi \left(x\right){V}_{\epsilon }\left(x\right)$, where $\phi \left(x\right)\in {C}_{0}^{\infty }\left(B\left({x}_{0};\rho \right)$ is a cutoff function such that φ(x) ≡ 1 in B(x0; ρ/2). Arguing as in Lemma 2.2, we have

${\int }_{\Omega }\mid \nabla {ũ}_{\epsilon }{\mid }^{p}={S}_{0}^{\frac{N}{p}}+O\left({\epsilon }^{p\stackrel{̃}{\gamma }}\right),$
(6.2)
${\int }_{\Omega }\mid {ũ}_{\epsilon }{\mid }^{{p}^{*}}={S}_{0}^{\frac{N}{P}}++O\left({\epsilon }^{{p}^{*}\stackrel{̃}{\gamma }}\right),$
(6.3)
${\int }_{\Omega }\mid {ũ}_{\epsilon }{\mid }^{q}=\left\{\begin{array}{c}\hfill {O}_{1}\left({\epsilon }^{\theta }\right),\hfill \\ \hfill {O}_{1}\left({\epsilon }^{\theta }\mid ln\epsilon \mid \right),\hfill \\ \hfill {O}_{1}\left({\epsilon }^{q\stackrel{̃}{\gamma }}\right),\hfill \end{array}\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill \frac{N\left(p-1\right)}{N-p}
(6.4)

where $\theta =N-\frac{N-p}{p}q$. Note that $\beta \ge p\stackrel{̃}{\gamma }$, ${p}^{*}\stackrel{̃}{\gamma }>p\stackrel{̃}{\gamma }$. Arguing as in Lemma 5.2, we deduce that there exists ${\stackrel{̃}{t}}_{\epsilon }$ satisfying $0<{C}_{1}\le {\stackrel{̃}{t}}_{\epsilon }\le {C}_{2}$, such that

$\begin{array}{ll}\hfill {J}_{\lambda }\left(t{ũ}_{\epsilon }\right)& \le \underset{t\ge 0}{sup}{J}_{\lambda }\left(t{ũ}_{\epsilon }\right)={J}_{\lambda }\left({\stackrel{̃}{t}}_{\epsilon }{ũ}_{\epsilon }\right)\phantom{\rule{2em}{0ex}}\\ =\frac{{\stackrel{̃}{t}}_{\epsilon }^{p}}{p}{\int }_{\Omega }\mid \nabla {ũ}_{\epsilon }{\mid }^{p}-\frac{{\stackrel{̃}{t}}_{\epsilon }^{{p}^{*}}}{{p}^{*}}{\int }_{\Omega }g\mid {ũ}_{\epsilon }{\mid }^{{p}^{*}}-\lambda \frac{{\stackrel{̃}{t}}_{\epsilon }^{q}}{q}{\int }_{\Omega }f\mid {ũ}_{\epsilon }{\mid }^{q}-\mu \frac{{\stackrel{̃}{t}}_{\epsilon }^{p}}{p}{\int }_{\Omega }\frac{\mid {ũ}_{\epsilon }{\mid }^{p}}{\mid x{\mid }^{p}}\phantom{\rule{2em}{0ex}}\\ \le \frac{1}{N}g{\left({x}_{0}\right)}^{-\frac{N-p}{p}}{S}_{\mu }^{\frac{N}{p}}+O\left({\epsilon }^{p\stackrel{̃}{\gamma }}\right)-\lambda \frac{{C}_{1}^{q}}{q}{\beta }_{0}{\int }_{\Omega }\mid {ũ}_{\epsilon }{\mid }^{q}\phantom{\rule{2em}{0ex}}\\ \phantom{\rule{1em}{0ex}}-\mu \parallel {x}_{0}\mid -\rho {\mid }^{-p}\frac{{C}_{2}^{p}}{p}{\int }_{\Omega }\mid {ũ}_{\epsilon }{\mid }^{p}.\phantom{\rule{2em}{0ex}}\end{array}$
(6.5)

From $\left(\mathcal{H}\right)$, Np2 and (6.4), we can deduce that

$1

and

${\int }_{\Omega }\mid {ũ}_{\epsilon }{\mid }^{q}={O}_{1}\left({\epsilon }^{q\stackrel{̃}{\gamma }}\right)\phantom{\rule{2.77695pt}{0ex}}and\phantom{\rule{2.77695pt}{0ex}}{\int }_{\Omega }\mid {ũ}_{\epsilon }{\mid }^{p}=\left\{\begin{array}{c}\hfill {O}_{1}\left({\epsilon }^{p}\mid ln\epsilon \mid \right),\hfill \\ \hfill {O}_{1}\left({\epsilon }^{p\stackrel{̃}{\gamma }}\right),\hfill \end{array}\phantom{\rule{1em}{0ex}}\begin{array}{c}\hfill p=\frac{N\left(p-1\right)}{N-p},\hfill \\ \hfill 1

Combining this with (6.5), for any λ > 0 and μ < 0, we can choose ε λ,μ small enough such that

$\underset{t\ge 0}{sup}{J}_{\lambda }\left(t{ũ}_{{\epsilon }_{\lambda ,\mu }}\right)<\frac{1}{N}g{\left({x}_{0}\right)}^{-\frac{N-p}{p}}{S}_{0}^{\frac{N}{p}}={c}^{*}.$

Therefore, (6.1) holds by taking ${v}_{\lambda ,\mu }={ũ}_{{\epsilon }_{\lambda ,\mu }}$.

In fact, by (f2), (g2) and the definition of ${ũ}_{{\epsilon }_{\lambda ,\mu }}$, we have that

${\int }_{\Omega }f\mid {ũ}_{{\epsilon }_{\lambda ,\mu }}{\mid }^{q}>0\phantom{\rule{1em}{0ex}}and\phantom{\rule{1em}{0ex}}{\int }_{\Omega }g\mid {ũ}_{{\epsilon }_{\lambda ,\mu }}{\mid }^{{p}^{*}}>0.$

From Lemma 3.4, the definition of ${\alpha }_{\lambda }^{-}$ and (6.1), for any λ ∈ (0, Λ0) and μ < 0, there exists ${t}_{{\epsilon }_{\lambda ,\mu }}>0$ such that ${t}_{{\epsilon }_{\lambda ,\mu }}{ũ}_{{\epsilon }_{\lambda ,\mu }}\in {\mathcal{N}}_{\lambda }^{-}$ and

${\alpha }_{\lambda }^{-}\le {J}_{\lambda }\left({t}_{{\epsilon }_{\lambda ,\mu }}{ũ}_{{\epsilon }_{\lambda ,\mu }}\right)\le \underset{t\ge 0}{sup}{J}_{\lambda }\left(t{t}_{{\epsilon }_{\lambda ,\mu }}{ũ}_{{\epsilon }_{\lambda ,\mu }}\right)<{c}^{*}.$

The proof is thus complete. □

Proof of Theorem 1.3 Let Λ1(0) be defined as in (1.3). Arguing as in Theorems 4.3 and 5.3, we can get the first positive solution ${ũ}_{\lambda }\in {\mathcal{N}}_{\lambda }^{+}$ for all λ ∈ (0, Λ1(0)) and the second positive solution ${Ũ}_{\lambda }\in {\mathcal{N}}_{\lambda }^{-}$ for all $\lambda \in \left(0,\frac{q}{p}{\Lambda }_{1}\left(0\right)\right)$. Since ${\mathcal{N}}_{\lambda }^{+}\cap {\mathcal{N}}_{\lambda }^{-}=\varnothing$, this implies that ${ũ}_{\lambda }$ and ${Ũ}_{\lambda }$ are distinct. □