Multiple positive solutions for a class of quasi-linear elliptic equations involving concave-convex nonlinearities and Hardy terms

Abstract

In this paper, we are concerned with the following quasilinear elliptic equation

$$-{\Delta }_{p}u-\mu \frac{\mid u{\mid }^{p-2}u}{\mid x{\mid }^p}=\lambda f\left(x\right)\mid u{\mid }^{q-2}u+g\left(x\right)\mid u{\mid }^{p*-2}u\text{in }\Omega ,u=0\text{on }\partial \Omega ,$$

where Ω ⊂ ℝ ^N is a smooth domain with smooth boundary ∂Ω such that 0 ∈ Ω, Δ _p u = div(|∇u|^p-2∇u), 1 < p < N, $\mu <\bar{\mu }={\left(\frac{N-p}{p}\right)}^p$, λ > 0, 1 < q < p, sign-changing weight functions f and g are continuous functions on $\bar{\Omega }$, $\bar{\mu }={\left(\frac{N-p}{p}\right)}^p$ is the best Hardy constant and $p^{*}=\frac{Np}{N-p}$ is the critical Sobolev exponent. By extracting the Palais-Smale sequence in the Nehari manifold, the multiplicity of positive solutions to this equation is verified.

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

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

(1.1)

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

$$J_{\lambda }\left(u\right)=\frac{1}{p}{\int }_{\Omega }\left(\mid \nabla u{\mid }^p-\mu \frac{\mid u{\mid }^p}{\mid x{\mid }^p}\right)dx-\frac{\lambda }{q}{\int }_{\Omega }f\mid u{\mid }^{q}dx-\frac{1}{p^{*}}{\int }_{\Omega }g\mid u{\mid }^{p^{*}}dx.$$

Then $J_{\lambda }\in C^1\left({\mathcal{D}}_0^{1,p}\left(\Omega \right),ℝ\right)$. $u\in {\mathcal{D}}_0^{1,p}\left(\Omega \right)\backslash \left\{0\right\}$ 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]:

$${\int }_{\Omega }\frac{\mid u{\mid }^p}{\mid x{\mid }^p}dx\le \frac{1}{\bar{\mu }}{\int }_{\Omega }\mid \nabla u{\mid }^{p}dx,\forall u\in C_0^{\infty }\left(\Omega \right).$$

By the Hardy inequality, ${\mathcal{D}}_0^{1,p}\left(\Omega \right)$ has the equivalent norm ||u||_μs, where

$$\parallel u{\parallel }_{\mu }^p={\int }_{\Omega }\left(\mid \nabla u{\mid }^p-\mu \frac{\mid u{\mid }^p}{\mid x{\mid }^p}\right)dx,\mu \in \left(-\infty ,\bar{\mu }\right).$$

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

$$S_{\mu }\left(\Omega \right)=\underset{u\in {\mathcal{D}}_0^{1,p}\left(\Omega \right)\backslash \left\{0\right\}}{inf}\frac{{\int }_{\Omega }\left(\mid \nabla u{\mid }^p-\mu \frac{\mid u{\mid }^p}{\mid x{\mid }^p}\right)dx}{{\left({\int }_{\Omega }\mid u{\mid }^{p*}dx\right)}^{\frac{p}{p*}}}.$$

(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 [4–6] and for p > 1 to [7–11], while in ℝ ^N and for p = 2 to [12, 13], and for p > 1 to [3, 14–17], 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, $\mu \in \left[0,\bar{\mu }\right)$, 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, $-\infty <\mu <\bar{\mu }$ 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, $-\infty <\mu <\bar{\mu }$, 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 <\bar{\mu }$, λ > 0, 1 < q < p < N, N ≥ 3.

(f₁) $f\in C\left(\bar{\Omega }\right)\cap L^{q*}\left(\Omega \right)\left(q^{*}=\frac{p^{*}}{p^{*}-q}\right)$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 x ∈ B(x₀; 2ρ₀)

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

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

$$\begin{array}{c}\mid g{\mid }_{\infty }=g\left(x_0\right)=\underset{x\in \bar{\Omega }}{max}g\left(x\right),g\left(x\right)>0,\forall x\in \Omega ,\\ g\left(x\right)=g\left(x_0\right)+o\left(\mid x-x_0{\mid }^{\beta }\right)\text{as }x\to 0\end{array}$$

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$0\le \mu <\bar{\mu }$, x₀ = 0 and β ≥ pγ, 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 N ≤ p², 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$\mu <\bar{\mu }$, λ > 0 and 1 < q < p < p² < N. As$0\le \mu <\bar{\mu }$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 < N ≤ p², 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 L^p (Ω) is denoted by |·| _p , the quantity O(ε^t ) denotes |O(ε^t )/ε^t | ≤ C, o(ε^t ) means |o(ε^t )/ε^t | → 0 as ε → 0 and o(1) is a generic infinitesimal value. In particular, the quantity O₁(ε^t ) means that there exist C₁, C₂> 0 such that C₁ε^t ≤ O₁(ε^t ) ≤ C₂ε^t 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 S_μ,s.

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

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

(2.1)

has positive radial ground states

$$V_{p,\mu ,\epsilon }\left(x\right)={\epsilon }^{-\frac{N-p}{p}}{U}_{p,\mu }\left(\frac{x}{\epsilon }\right)={\epsilon }^{-\frac{N-p}{p}}{U}_{p,\mu }\left(\frac{\mid x\mid }{\epsilon }\right),forall\epsilon >0,$$

that satisfy

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

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

$$\begin{array}{c}{U}_{p\mathrm{,}\mu }(1)={\left(\frac{N(\overline{\mu }-\mu )}{N-p}\right)}^{\frac{1}{p*-p}}\mathrm{,}\\ \underset{r\to 0^{+}}{\text{lim}}{r}^{a(\mu )}{U}_{p\mathrm{,}\mu }(r)=c_1>\mathrm{0,}\underset{r\to 0^{+}}{\text{lim}}{r}^{a(\mu )+1}|{U^{\prime }}_{p\mathrm{,}\mu }(r)|=c_{1}a(\mu )\ge \mathrm{0,}\\ \underset{r\to +\infty }{\text{lim}}{r}^{b(\mu )}{U}_{p\mathrm{,}\mu }(r)=c_2>\mathrm{0,}\underset{r\to +\infty }{\text{lim}}{r}^{b(\mu )+1}|{U^{\prime }}_{p\mathrm{,}\mu }(r)|=c_{2}b(\mu )>\mathrm{0,}\\ c_3\le U_{p\mathrm{,}\mu }(r){(r^{\frac{a(\mu )}{\delta }}+r^{\frac{b(\mu )}{\delta }})}^{\delta }\le c_4\mathrm{,}\delta :=\frac{N-p}{p}\mathrm{,}\end{array}$$

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)t^p - (N - p)t^p-1+ μ, t ≥ 0, satisfying$0\le a\left(\mu \right)<\frac{N-p}{p}<b\left(\mu \right)\le \frac{N-p}{p-1}$.

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 $\eta \in C_0^{\infty }(B(0;\rho )$ 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$0\le \mu <\bar{\mu }$. Then, the following estimates hold when ε → 0.

$$\begin{array}{c}\parallel u_{\epsilon }{\parallel }_{\mu }^p=S_{\mu }^{\frac{N}{p}}+O\left({\epsilon }^{p\gamma }\right),\\ {\int }_{\Omega }\mid u_{\epsilon }{\mid }^{p*}=S_{\mu }^{\frac{N}{p}}+O\left({\epsilon }^{p*\gamma }\right),\\ {\int }_{\Omega }\mid u_{\epsilon }{\mid }^q=\left\{\begin{array}{c}\hfill O_1\left({\epsilon }^{\theta }\right),\hfill \\ \hfill O_1\left({\epsilon }^{\theta }\mid \right)ln\epsilon \mid ,\hfill \\ \hfill O_1\left({\epsilon }^{q\gamma }\right),\hfill \end{array}\begin{array}{c}\hfill \frac{N}{b\left(\mu \right)}<q<p*,\hfill \\ \hfill q=\frac{N}{b\left(\mu \right)},\hfill \\ \hfill 1\le q<\frac{N}{b\left(\mu \right)},\hfill \end{array}\right.\end{array}$$

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, in ℝ ^N , 1 ≤ p < N, $1\le q<p*=\frac{pN}{N-p}$and$k\left(x\right)\in L^{\frac{p*}{p*-q}}\left(\Omega \right)$Then, the functional

$${\mathcal{D}}_0^{1,p}\left(\Omega \right)\to ℝ:u\mapsto {\int }_{{ℝ}^N}k\left(x\right)\mid u{\mid }^{q}dx$$

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

$${\mathcal{N}}_{\lambda }=\left\{u\in {\mathcal{D}}_0^{1,p}\left(\Omega \right)\backslash \left\{0\right\}:〈J_{\lambda }^{\prime }\left(u\right),u〉=0\right\}.$$

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

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

(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 $u\in {\mathcal{N}}_{\lambda }$,

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

(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,

$$\lambda \ge {\left(\frac{p-q}{\left(p^{*}-q\right)\mid g^{+}{\mid }_{\infty }}\right)}^{\frac{p-q}{p^{*}-p}}\left(\frac{p^{*}-p}{\left(p^{*}-q\right)\mid f^{+}{\mid }_{q^{*}}}\right)S_{\mu }^{\frac{N}{p^2}\left(p-q\right)+\frac{q}{p}}={\Lambda }_1,$$

which is a contradiction. □

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${\int }_{\Omega }g\mid u{\mid }^{p^{*}}>0$.

• (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

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

• (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,

  $$J_{\lambda }\left(t^{+}u\right)=\underset{0\le t\le t_{max}}{inf}{J}_{\lambda }\left(tu\right),J_{\lambda }\left(t^{-}u\right)=\underset{t\ge t^{+}}{sup}{J}_{\lambda }\left(tu\right).$$

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

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

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

• (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

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

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

$$\parallel u\mid {\mid }_{\mu }>{\left(\frac{p-q}{\left(p^{*}-q\right)\mid g^{+}{\mid }_{\infty }}\right)}^{\frac{1}{p^{*}-p}}{S}_{\mu }^{\frac{N}{p^2}}forallu\in {\mathcal{N}}_{\lambda }^{-}.$$

(3.7)

By (3.2) and (3.7), we get

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

which implies that

$$J_{\lambda }\left(u\right)>d_{0}forallu\in {\mathcal{N}}_{\lambda }^{-},$$

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

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

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 → ∞.

• (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 .

• (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$\left\{u_n\right\}\subset {\mathcal{N}}_{\lambda }$.

• (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, $\mu <\bar{\mu }$, 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

• (i) $J_{\lambda }\left(u_{\lambda }\right)={\alpha }_{\lambda }={\alpha }_{\lambda }^{+}$ ,

• (ii) u _λ is a positive solution of (1.1),

• (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

$$J_{\lambda }\left(u_n\right)={\alpha }_{\lambda }+o\left(1\right)and{J}_{\lambda }^{\prime }\left(u_n\right)=o\left(1\right)in{\left({\mathcal{D}}_0^{1,p}\left(\Omega \right)\right)}^{-1}.$$

(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 → ∞

$$\left\{\begin{array}{c}\hfill u_n\rightharpoonup u_{\lambda }weaklyin{\mathcal{D}}_0^{1,p}\left(\Omega \right),\hfill \\ \hfill u_n\rightharpoonup u_{\lambda }weaklyin{L}^{p^{*}}\left(\Omega \right),\hfill \\ \hfill u_n\to u_{\lambda }stronglyin{L}_{loc}^r\left(\Omega \right)forall1\le r<p^{*},\hfill \\ \hfill u_n\to u_{\lambda }a.e.in\Omega .\hfill \end{array}\right.$$

(4.2)

By (f₁) and Lemma 2.3, we obtain

$$\begin{array}{c}\hfill \lambda {\int }_{\Omega }f\mid u_n{\mid }^q=\lambda {\int }_{\Omega }f\mid u_{\lambda }{\mid }^q+o\left(1\right)asn\to \infty .\hfill \end{array}$$

(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

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

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

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

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

$$J_{\lambda }\left(t_{\lambda }^{+}{u}_{\lambda }\right)<J_{\lambda }\left(\bar{t}{u}_{\lambda }\right)\le J_{\lambda }\left(t_{\lambda }^{-}{u}_{\lambda }\right)=J_{\lambda }\left(u_{\lambda }\right),$$

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 $\mu <\bar{\mu }$, 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 → ∞

$$\left\{\begin{array}{c}\hfill u_n\rightharpoonup u_{0}weaklyin{\mathcal{D}}_0^{1,p}\left(\Omega \right),\hfill \\ \hfill u_n\rightharpoonup u_{0}weaklyin{L}^{p^{*}}\left(\Omega \right),\hfill \\ \hfill u_n\to u_{0}stronglyin{L}_{loc}^r\left(\Omega \right)for1\le r<p^{*},\hfill \\ \hfill u_n\to u_{0}a.e.in\Omega .\hfill \end{array}\right.$$

(5.1)

By (f₁), (g₁), (5.1) and Lemma 2.3, we have that $J_{\lambda }^{\prime }\left(u_0\right)=0$ and

$$\lambda {\int }_{\Omega }f\mid u_n{\mid }^q=\lambda {\int }_{\Omega }f\mid u_0{\mid }^q+o\left(1\right)asn\to \infty .$$

(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 = lim_n→∞J _λ (u _n ) = 0, which is a contradiction. Thus, we conclude that l > 0. Furthermore, the Sobolev embedding theorem implies that

$$\begin{array}{ll}\hfill \parallel u_n{\parallel }_{\mu }^p& \ge S_{\mu }{\left({\int }_{\Omega }g\mid u_n{\mid }^{p^{*}}\right)}^{\frac{p}{p^{*}}}\\ \ge S_{\mu }{\left({\int }_{\Omega }\frac{g}{\mid g^{+}{\mid }_{\infty }}\mid u_n{\mid }^{p^{*}}\right)}^{\frac{p}{p^{*}}}\\ =S_{\mu }\mid g^{+}{\mid }_{\infty }^{-\frac{N-p}{N}}{\left({\int }_{\Omega }g\mid u_n{\mid }^{p^{*}}\right)}^{\frac{p}{p^{*}}}.\end{array}$$

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

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

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$0<\mu <\bar{\mu }$, x₀ = 0 and β ≥ pγ, 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

$$\underset{t\ge 0}{sup}{J}_{\lambda }\left(t{u}_{\epsilon }\right)=J_{\lambda }\left(t_{\epsilon }{u}_{\epsilon }\right)\text{and}0<C_1\le t_{\epsilon }\le C_2<\infty .$$

(5.6)

By (g₂), we conclude that

$$\begin{array}{ll}\hfill \left|{\int }_{\Omega }g\left(x\right)\mid u_{\epsilon }{\mid }^{p^{*}}-{{\int }_{\Omega }g\left(0\right)\mid u_{\epsilon }\mid }^{p^{*}}\right|& \le {\int }_{\Omega }\mid g\left(x\right)-g\left(0\right)\mid \mid u_{\epsilon }{\mid }^{p^{*}}\\ =O\left({\int }_{B\left(0;\rho \right)}\mid x{\mid }^{\beta }\mid u_{\epsilon }{\mid }^{p^{*}}\right)\\ =O\left({\epsilon }^{\beta }\right),\end{array}$$

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 ≥ pγ and

$$\underset{t\ge 0}{max}\left(\frac{t^p}{p}{B}_1-\frac{t^{p^{*}}}{p^{*}}{B}_2\right)=\frac{1}{N}{B}_1^{\frac{N}{p}}{B}_2^{-\frac{N-p}{p}},B_1>0,B_2>0,$$

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

$$\begin{array}{ll}\hfill J_{\lambda }\left(t_{\epsilon }{u}_{\epsilon }\right)& =\frac{t_{\epsilon }^p}{p}\parallel u_{\epsilon }{\parallel }_{\mu }^p-\frac{t_{\epsilon }^{p^{*}}}{p^{*}}{\int }_{\Omega }g\mid u_{\epsilon }{\mid }^{p^{*}}-\lambda \frac{t_{\epsilon }^q}{q}{\int }_{\Omega }f\mid u_{\epsilon }{\mid }^q\\ \le \frac{1}{N}\parallel u_{\epsilon }{\parallel }_{\mu }^{\frac{N}{p}}{\left({\int }_{\Omega }g\mid u_{\epsilon }{\mid }^{p^{*}}\right)}^{-\frac{N-p}{p}}-\lambda \frac{C_1^q}{q}{\beta }_0{\int }_{\Omega }\mid u_{\epsilon }{\mid }^q\\ =\frac{1}{N}{\left(S_{\mu }^{\frac{N}{p}}+O\left({\epsilon }^{p\gamma }\right)\right)}^{\frac{N}{p}}{\left(g\left(0\right)S_{\mu }^{\frac{N}{p}}+O\left({\epsilon }^{p^{*}\gamma }\right)+O\left({\epsilon }^{\beta }\right)\right)}^{-\frac{N-p}{p}}\\ -\lambda \frac{C_1^q}{q}{\beta }_0{\int }_{\Omega }\mid u_{\epsilon }{\mid }^q\\ =\frac{1}{N}g{\left(0\right)}^{-\frac{N-p}{p}}{S}_{\mu }^{\frac{N}{p}}+O\left({\epsilon }^{p\gamma }\right)+O\left({\epsilon }^{\beta }\right)-\lambda \frac{C_1^q}{q}{\beta }_0{\int }_{\Omega }\mid u_{\epsilon }{\mid }^q.\end{array}$$

(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)

• (i) If $1<q<\frac{N}{b\left(\mu \right)}$, 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

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

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

  $${\int }_{\Omega }\mid u_{\epsilon }{\mid }^q=\left\{\begin{array}{cc}\hfill O_1\left({\epsilon }^{\theta }\right),\hfill & \hfill q>\frac{N}{b\left(\mu \right)},\hfill \\ \hfill O_1\left({\epsilon }^{\theta }\mid ln\epsilon \mid \right),\hfill & \hfill q=\frac{N}{b\left(\mu \right)},\hfill \end{array}\right.$$

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

$${\int }_{\Omega }f\mid u_{{\epsilon }_{\lambda }}{\mid }^q>0and{\int }_{\Omega }g\mid u_{{\epsilon }_{\lambda }}{\mid }^{p^{*}}>0.$$

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$0<\mu <\bar{\mu }$, x₀ = 0, β ≥ pγ and$\lambda \in \left(0,\frac{q}{p}{\Lambda }_1\right)$, then there exists$U_{\lambda }\in {\mathcal{N}}_{\lambda }^{-}$such that

• (i) $J_{\lambda }\left(U_{\lambda }\right)={\alpha }_{\lambda }^{-}$,

• (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 inf_n→∞||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 N ≤ p², μ < 0, x₀ ≠ 0 and$\frac{\beta }{p}\ge \tilde{\gamma }:=\frac{N-p}{p\left(p-1\right)}$, 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]:

$$V_{\epsilon }\left(x\right)=\bar{C}{\epsilon }^{-\frac{N-p}{p}}{\left(1+{\left(\frac{\mid x-x_0\mid }{\epsilon }\right)}^{\frac{p}{p-1}}\right)}^{-\frac{N-p}{p}},\forall \epsilon >0,x_0\in {ℝ}^N,$$

where $\bar{C}>0$ is a particular constant. Take ρ > 0 small enough such that B(x₀; ρ) ⊂ Ω\{0} and set ${\tilde{u}}_{\epsilon }(x)=\phi (x)V_{\epsilon }(x)$, where $\phi (x)\in C_0^{\infty }(B(x_0;\rho )$ is a cutoff function such that φ(x) ≡ 1 in B(x₀; ρ/2). Arguing as in Lemma 2.2, we have

$${\int }_{\Omega }\mid \nabla {ũ}_{\epsilon }{\mid }^p=S_0^{\frac{N}{p}}+O\left({\epsilon }^{p\tilde{\gamma }}\right),$$

(6.2)

$${\int }_{\Omega }\mid {ũ}_{\epsilon }{\mid }^{p^{*}}=S_0^{\frac{N}{P}}++O\left({\epsilon }^{p^{*}\tilde{\gamma }}\right),$$

(6.3)

$${\int }_{\Omega }\mid {ũ}_{\epsilon }{\mid }^q=\left\{\begin{array}{c}\hfill O_1\left({\epsilon }^{\theta }\right),\hfill \\ \hfill O_1\left({\epsilon }^{\theta }\mid ln\epsilon \mid \right),\hfill \\ \hfill O_1\left({\epsilon }^{q\tilde{\gamma }}\right),\hfill \end{array}\begin{array}{c}\hfill \frac{N\left(p-1\right)}{N-p}<q<p^{*},\hfill \\ \hfill q=\frac{N\left(p-1\right)}{N-p},\hfill \\ \hfill 1\le q<\frac{N\left(p-1\right)}{N-p},\hfill \end{array}\right.$$

(6.4)

where $\theta =N-\frac{N-p}{p}q$. Note that $\beta \ge p\tilde{\gamma }$, $p^{*}\tilde{\gamma }>p\tilde{\gamma }$. Arguing as in Lemma 5.2, we deduce that there exists ${\tilde{t}}_{\epsilon }$ satisfying $0<C_1\le {\tilde{t}}_{\epsilon }\le C_2$, such that

$$\begin{array}{ll}\hfill J_{\lambda }\left(t{ũ}_{\epsilon }\right)& \le \underset{t\ge 0}{sup}{J}_{\lambda }\left(t{ũ}_{\epsilon }\right)=J_{\lambda }\left({\tilde{t}}_{\epsilon }{ũ}_{\epsilon }\right)\\ =\frac{{\tilde{t}}_{\epsilon }^p}{p}{\int }_{\Omega }\mid \nabla {ũ}_{\epsilon }{\mid }^p-\frac{{\tilde{t}}_{\epsilon }^{p^{*}}}{p^{*}}{\int }_{\Omega }g\mid {ũ}_{\epsilon }{\mid }^{p^{*}}-\lambda \frac{{\tilde{t}}_{\epsilon }^q}{q}{\int }_{\Omega }f\mid {ũ}_{\epsilon }{\mid }^q-\mu \frac{{\tilde{t}}_{\epsilon }^p}{p}{\int }_{\Omega }\frac{\mid {ũ}_{\epsilon }{\mid }^p}{\mid x{\mid }^p}\\ \le \frac{1}{N}g{\left(x_0\right)}^{-\frac{N-p}{p}}{S}_{\mu }^{\frac{N}{p}}+O\left({\epsilon }^{p\tilde{\gamma }}\right)-\lambda \frac{C_1^q}{q}{\beta }_0{\int }_{\Omega }\mid {ũ}_{\epsilon }{\mid }^q\\ -\mu \parallel x_0\mid -\rho {\mid }^{-p}\frac{C_2^p}{p}{\int }_{\Omega }\mid {ũ}_{\epsilon }{\mid }^p.\end{array}$$

(6.5)

From $\left(\mathcal{H}\right)$, N ≤ p² and (6.4), we can deduce that

$$1<q\tilde{\gamma }<p\tilde{\gamma }=\frac{N-p}{p-1}\le p\le \frac{N\left(p-1\right)}{N-p}$$

and

$${\int }_{\Omega }\mid {ũ}_{\epsilon }{\mid }^q=O_1\left({\epsilon }^{q\tilde{\gamma }}\right)and{\int }_{\Omega }\mid {ũ}_{\epsilon }{\mid }^p=\left\{\begin{array}{c}\hfill O_1\left({\epsilon }^p\mid ln\epsilon \mid \right),\hfill \\ \hfill O_1\left({\epsilon }^{p\tilde{\gamma }}\right),\hfill \end{array}\begin{array}{c}\hfill p=\frac{N\left(p-1\right)}{N-p},\hfill \\ \hfill 1<p<\frac{N\left(p-1\right)}{N-p}.\hfill \end{array}\right.$$

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

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

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. □