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

- Δ p u - μ ∣ u ∣ p - 2 u ∣ x ∣ p = λ f ( x ) ∣ u ∣ q - 2 u + g ( x ) ∣ u ∣ p * - 2 u , u = 0 , i n Ω , o n ∂ Ω ,

where Δ _pu = div(|∇u| ^p-2∇u), 1 < p < N, μ < μ̄ = (N - p p )p , μ ̄ is the best Hardy constant, λ > 0, 1 < q < p, p * = N p N - p is the critical Sobolev exponent and the weight functions f , g : Ω ̄ → ℝ are continuous, which change sign on Ω.

Let D 0 1 , p ( Ω ) be the completion of C 0 ∞ ( Ω ) with respect to the norm ( ∫ Ω ∣ ∇ u ∣ p d x ) 1 ∕ p . The energy functional of (1.1) is defined on D 0 1 , p ( Ω ) by

J λ ( u ) = 1 p ∫ Ω ∣ ∇ u ∣ p - μ ∣ u ∣ p ∣ x ∣ p d x - λ q ∫ Ω f ∣ u ∣ q d x - 1 p * ∫ Ω g ∣ u ∣ p * d x .

Then J λ ∈ C 1 ( D 0 1 , p ( Ω ) , ℝ ) . u ∈ D 0 1 , p ( Ω ) \ { 0 } is said to be a solution of (1.1) if 〈 J λ ′ ( u ) , v 〉 = 0 for all v ∈ D 0 1 , p ( Ω ) 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 :

∫ Ω ∣ u ∣ p ∣ x ∣ p d x ≤ 1 μ ̄ ∫ Ω ∣ ∇ u ∣ p d x , ∀ u ∈ C 0 ∞ ( Ω ) .

By the Hardy inequality, D 0 1 , p ( Ω ) has the equivalent norm ||u|| _μ , where

∥ u ∥ μ p = ∫ Ω ∣ ∇ u ∣ p - μ ∣ u ∣ p ∣ x ∣ p d x , μ ∈ ( - ∞ , μ ̄ ) .

Therefore, for 1 < p < N, and μ < μ ̄ , we can define the best Sobolev constant:

S μ ( Ω ) = inf u ∈ D 0 1 , p ( Ω ) \ { 0 } ∫ Ω ∣ ∇ u ∣ p - μ ∣ u ∣ p ∣ x ∣ p d x ( ∫ Ω ∣ u ∣ p * d x ) p p * .

It is well known that S_μ (Ω) = S_μ (ℝ ^N ) = S_μ . Note that S_μ = S ₀ 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 5 6 and for p > 1 to 7 8 9 10 11 , while in ℝ ^N and for p = 2 to 12 13 , and for p > 1 to 3 14 15 16 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, μ ∈ [ 0 , μ ̄ ) , 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 > p ², - ∞ < μ < μ ̄ 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, - ∞ < μ < μ ̄ , 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:

( H ) μ < μ ̄ , λ > 0, 1 < q < p < N, N ≥ 3.

(f ₁) f ∈ C ( Ω ̄ ) ∩ L q * ( Ω ) ( q * = p * p * - q ) f ⁺ = max{f, 0} ≢ 0 in Ω.

(f ₂) There exist β ₀ and ρ ₀ > 0 such that B(x ₀; 2ρ ₀) ⊂ Ω and f (x) ≥ β ₀ for all x ∈ B(x ₀; 2ρ ₀)

(g ₁) g ∈ C ( Ω ̄ ) ∩ L ∞ ( Ω ) and g ⁺ = max{g, 0} ≢ 0 in Ω.

(g ₂) There exist x ₀ ∈ Ω and β > 0 such that

∣ g ∣ ∞ = g ( x 0 ) = max x ∈ Ω ̄ g ( x ) , g ( x ) > 0 , ∀ x ∈ Ω , g ( x ) = g ( x 0 ) + o ( ∣ x - x 0 ∣ β ) as  x → 0

where | · |_∞ denotes the L ^∞(Ω) norm.

Set

Λ 1 = Λ 1 ( μ ) = p - q ( p * - q ) ∣ g + ∣ ∞ p - q p * - p p * - p ( p * - q ) ∣ f + ∣ q * S μ N p 2 ( p - q ) + q p .

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 ( H ) , (f ₁) and (g ₁) hold. Then, (1.1) has at least one positive solution for all λ ∈ (0, Λ₁).

Theorem 1.2 Suppose ( H ) , (f ₁) - (g ₂) hold, and γ is the constant defined as in Lemma 2.2. If 0 ≤ μ < μ ̄ , x ₀ = 0 and β ≥ pγ, then (1.1) has at least two positive solutions for all λ ∈ ( 0 , q p Λ 1 ) .

Theorem 1.3 Suppose ( H ) , (f ₁) - (g ₂) hold. If μ < 0, x ₀ ≠ 0, β ≥ N - p p - 1 and N ≤ p ², then (1.1) has at least two positive solutions for all λ ∈ ( 0 , q p Λ 1 ( 0 ) ) .

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 μ < μ ̄ , λ > 0 and 1 < q < p < p ² < N. As 0 ≤ μ < μ ̄ 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 ≤ p ², 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(x ₀; R) is the ball centered at x ₀ ∈ ℝ ^N with the radius R > 0, ( D 0 1 , p ( Ω ) ) - 1 denotes the dual space of D 0 1 , p ( Ω ) , 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 O ₁(ε^t ) means that there exist C ₁, C ₂ > 0 such that C ₁ ε^t ≤ O ₁(ε^t ) ≤ C ₂ ε^t as ε is small enough.

Throughout this paper, (f ₁) and (g ₁) 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 ≤ μ < μ ̄ . Then, the limiting problem

- Δ p u - μ u p - 1 ∣ x ∣ p = u p * - 1 , i n ℝ N \ { 0 } , u ∈ D 1 , p ( ℝ N ) , u > 0 , i n ℝ N \ { 0 } ,

has positive radial ground states

V p , μ , ε ( x ) = ε - N - p p U p , μ x ε = ε - N - p p U p , μ ∣ x ∣ ε , f o r a l l ε > 0 ,

that satisfy

∫ ℝ N ( | ∇ V p , μ , ε ( x ) | p − μ | V p , μ , ε ( x ) | p | x | p ) = ∫ ℝ N | V p , μ , ε ( x ) | p * = S μ N p .

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

U p , μ ( 1 ) = ( N ( μ ¯ − μ ) N − p ) 1 p * − p , lim r → 0 + r a ( μ ) U p , μ ( r ) = c 1 > 0,   lim r → 0 + r a ( μ ) + 1 | U ′ p , μ ( r ) | = c 1 a ( μ ) ≥ 0, lim r → + ∞ r b ( μ ) U p , μ ( r ) = c 2 > 0,   lim r → + ∞ r b ( μ ) + 1 | U ′ p , μ ( r ) | = c 2 b ( μ ) > 0, c 3 ≤ U p , μ ( r ) ( r a ( μ ) δ + r b ( μ ) δ ) δ ≤ c 4 ,   δ : = N − p p ,

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 ≤ a ( μ ) < N - p p < b ( μ ) ≤ N - p p - 1 .

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

u ε ( x ) = η ( x ) V p , μ , ε ( x ) = ε - N - p p η ( x ) U p , μ ∣ x ∣ ε ,

where η ∈ C 0 ∞ ( B ( 0 ; ρ ) is a cutoff function such that η(x) ≡ 1 in B ( 0 , ρ 2 ) .

Lemma 2.2 9 20 Suppose 1 < p < N and 0 ≤ μ < μ ̄ . Then, the following estimates hold when ε → 0.

∥ u ε ∥ μ p = S μ N p + O ( ε p γ ) , ∫ Ω ∣ u ε ∣ p * = S μ N p + O ( ε p * γ ) , ∫ Ω ∣ u ε ∣ q = O 1 ( ε θ ) , O 1 ( ε θ ∣ ) l n ε ∣ , O 1 ( ε q γ ) , N b ( μ ) < q < p * , q = N b ( μ ) , 1 ≤ q < N b ( μ ) ,

where δ = N - p p , θ = N - 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 ≤ q < p * = p N N - p and k ( x ) ∈ L p * p * - q ( Ω ) Then, the functional

D 0 1 , p ( Ω ) → ℝ : u ↦ ∫ ℝ N k ( x ) ∣ u ∣ q d x

is well-defined and weakly continuous.

As J_λ is not bounded below on D 0 1 , p ( Ω ) , we need to study J_λ on the Nehari manifold

N λ = { u ∈ D 0 1 , p ( Ω ) \ { 0 } : 〈 J λ ′ ( u ) , u 〉 = 0 } .

Note that N λ contains all solutions of (1.1) and u ∈ N λ if and only if

∥ u ∥ μ p - λ ∫ Ω f ∣ u ∣ q - ∫ Ω g ∣ u ∣ p * = 0 .

Lemma 3.1 J_λ is coercive and bounded below on N λ .

Proof Suppose u ∈ N λ . From (f ₁), (3.1), the Hölder inequality and Sobolev embedding theorem, we can deduce that

J λ ( u ) = p * - p p p * ∥ u ∥ μ p - λ p * - q p * q ∫ Ω f ∣ u ∣ q ≥ 1 N ∥ u ∥ μ p - λ p * - q p * q ∣ f + ∣ q * ∣ u ∣ p * q ≥ 1 N ∥ u ∥ μ p - λ p * - q p * q ∣ f + ∣ q * S μ - q p ∥ u ∥ μ q .

Thus, J_λ is coercive and bounded below on N λ . □

Define ψ λ ( u ) = 〈 J λ ′ ( u ) , u 〉 . Then, for u ∈ N λ ,

〈 ψ ′ λ ( u ) , u 〉 = p ∥ u ∥ μ p - q λ ∫ Ω f ∣ u ∣ q - p * ∫ Ω g ∣ u ∣ p * = ( p - q ) ∥ u ∥ μ p - ( p * - q ) ∫ Ω g ∣ u ∣ p * = λ ( p * - q ) ∫ Ω f ∣ u ∣ q - ( p * - p ) ∥ u ∥ μ p .

Arguing as in 22 , we split N λ into three parts:

N λ + = { u ∈ N λ : 〈 ψ ′ λ ( u ) , u 〉 > 0 } , N λ 0 = { u ∈ N λ : 〈 ψ ′ λ ( u ) , u 〉 = 0 } , N λ - = { u ∈ N λ : 〈 ψ ′ λ ( u ) , u 〉 < 0 } .

Lemma 3.2 Suppose u_λ is a local minimizer of J_λ on N λ and u λ ∉ N λ 0 .

Then, J λ ′ ( u λ ) = 0 in ( D 0 1 , p ( Ω ) ) - 1 .

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

Lemma 3.3 N λ 0 = ∅ for all λ ∈ (0, Λ₁).

Proof We argue by contradiction. Suppose that there exists λ ∈ (0, Λ₁) such that N λ 0 ≠ ∅ . Then, the fact u ∈ N λ 0 and (3.3) imply that

∥ u ∥ μ p = p * - q p - q ∫ Ω g ∣ u ∣ p * ,

and

∥ u ∥ μ p = λ p * - q p * - p ∫ Ω f ∣ u ∣ q .

By (f ₁), (g ₁), the Hölder inequality and Sobolev embedding theorem, we have that

∥ u ∣ ∣ μ ≥ p - q ( p * - q ) ∣ g + ∣ ∞ 1 p * - p S μ N p 2 ,

and

∥ u ∣ ∣ μ ≤ λ p * - q p * - p ∣ f + ∣ q * S μ - q p 1 p - q .

Consequently,

λ ≥ p - q ( p * - q ) ∣ g + ∣ ∞ p - q p * - p p * - p ( p * - q ) ∣ f + ∣ q * S μ N p 2 ( p - q ) + q p = Λ 1 ,

which is a contradiction. □

For each u ∈ D 0 1 , p ( Ω ) with ∫ Ω g ∣ u ∣ p * > 0 , we set

t m a x = ( p - q ) ∥ u ∥ μ p ( p * - q ) ∫ Ω g ∣ u ∣ p * 1 p * - p > 0 .

Lemma 3.4 Suppose that λ ∈ (0, Λ₁) and u ∈ D 0 1 , p ( Ω ) is a function satisfying with ∫ Ω g ∣ u ∣ p * > 0 .

(i) If ∫ Ω f ∣ u ∣ q ≤ 0 , then there exists a unique t ^- > t _max such that t - u ∈ N λ - and

J λ ( t - u ) = sup t ≥ 0 J λ ( t u ) .

(ii) If ∫ Ω f ∣ u ∣ q ≤ 0 , then there exists a unique t ^± such that 0 < t ⁺ < t _max < t ^-, t + u ∈ N λ + and t - u ∈ N λ - . Moreover,

J λ ( t + u ) = inf 0 ≤ t ≤ t max J λ ( t u ) , J λ ( t - u ) = sup t ≥ t + J λ ( t u ) .

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

We remark that it follows Lemma 3.3, N λ = N λ + ∪ N λ - for all λ ∈ (0, Λ₁). Furthermore, by Lemma 3.4, it follows that N λ + and N λ - are nonempty, and by Lemma 3.1, we may define

α λ = inf u ∈ N λ J λ ( u ) , α λ + = inf u ∈ N λ + J λ ( u ) , α λ - = inf u ∈ N λ - J λ ( u ) .

Lemma 3.5 (i) If λ ∈ (0, Λ₁), then we have α λ ≤ α λ + < 0 .

(ii) If λ ∈ ( 0 , q p Λ 1 ) , then α λ - > d 0 for some positive constant d ₀.

In particular, for each λ ∈ ( 0 , q p Λ 1 ) , we have α λ = α λ + < 0 < α λ - .

Proof (i) Suppose that u ∈ N λ + . From (3.3), it follows that

p - q p * - q ∥ u ∥ μ p > ∫ Ω g ∣ u ∣ p * .

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

J λ ( u ) = 1 p - 1 q ∥ u ∥ μ p + 1 q - 1 p * ∫ Ω g ∣ u ∣ p * < 1 p - 1 q + 1 q - 1 p * p - q p * - q ∥ u ∥ μ p = - p - q q N ∥ u ∥ μ p < 0 .

By the definitions of α_λ and α λ + , we get that α λ ≤ α λ + < 0 .

(ii) Suppose λ ∈ ( 0 , q p Λ 1 ) and u ∈ N λ - . Then, (3.3) implies that

p - q p * - q ∥ u ∥ μ p < ∫ Ω ∣ u ∣ p * .

Moreover, by (g ₁) and the Sobolev embedding theorem, we have

∫ Ω g ∣ u ∣ p * ≤ ∣ g + ∣ ∞ S μ - p * p ∥ u ∥ μ p * .

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

∥ u ∣ ∣ μ > p - q ( p * - q ) ∣ g + ∣ ∞ 1 p * - p S μ N p 2 f o r a l l u ∈ N λ - .

By (3.2) and (3.7), we get

J λ ( u ) ≥ ∥ u ∥ μ q 1 N ∥ u ∥ μ p - q - λ p * - q p * q ∣ f + ∣ q * S μ - q p > p - q ( p * - q ) ∣ g + ∣ ∞ q p * - p S μ q N p 2 1 N p - q ( p * - q ) ∣ g + ∣ ∞ p - q p * - p S μ N ( p - q ) p 2 - λ p * - q p * q ∣ f + ∣ q * S μ - q p

which implies that

J λ ( u ) > d 0 f o r a l l u ∈ N λ - ,

for some positive constant d ₀. □

Remark 3.6 If λ ∈ ( 0 , q p Λ 0 ) , then by Lemmas 3.4 and 3.5, for each u ∈ D 0 1 , p ( Ω ) with ∫ Ω g ∣ u ∣ p * > 0 , we can easily deduce that

t - u ∈ N λ - a n d J λ ( t - u ) = sup t ≥ 0 J λ ( t u ) ≥ α λ - > 0 .

First, we define the Palais-Smale (simply by (PS)) sequences, (PS)-values and (PS)-conditions in D 0 1 , p ( Ω ) for J_λ as follows:

Definition 4.1 (i) For c ∈ ℝ, a sequence {u_n } is a (PS) _c-sequence in D 0 1 , p ( Ω ) for J_λ if J_λ (u_n ) = c + o(1) and (J_λ )'(u_n ) = o(1) strongly in ( D 0 1 , p ( Ω ) ) - 1 as n → ∞.

(ii) c ∈ ℝ is a (PS)-value in D 0 1 , p ( Ω ) for J_λ if there exists a (PS) _c-sequence in D 0 1 , p ( Ω ) for J_λ .

(iii) J_λ satisfies the (PS) _c-condition in D 0 1 , p ( Ω ) if any (PS) _c-sequence {u_n } in D 0 1 , p ( Ω ) for J_λ contains a convergent subsequence.

Lemma 4.2 (i) If λ ∈ (0, Λ₁), then J_λ has a ( P S ) α λ -sequence { u n } ⊂ N λ .

(ii) If λ ∈ ( 0 , q p Λ 1 ) , then J_λ has a ( P S ) α λ -sequence { u n } ⊂ N λ - .

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 N λ .

Theorem 4.3 Suppose that N ≥ 3, μ < μ ̄ , 1 < q < p < N and the conditions (f ₁), (g ₁) hold. If λ ∈ (0, Λ₁), then there exists u λ ∈ N λ + such that

(i) J λ ( u λ ) = α λ = α λ + ,

(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 { u n } ⊂ N λ such that

J λ ( u n ) = α λ + o ( 1 ) a n d J λ ′ ( u n ) = o ( 1 ) i n ( D 0 1 , p ( Ω ) ) - 1 .

Since J_λ is coercive on N λ (see Lemma 2.1), we get that (u_n ) is bounded in D 0 1 , p ( Ω ) . Passing to a subsequence, there exists u λ ∈ D 0 1 , p ( Ω ) such that as n → ∞

u n ⇀ u λ w e a k l y i n D 0 1 , p ( Ω ) , u n ⇀ u λ w e a k l y i n L p * ( Ω ) , u n → u λ s t r o n g l y i n L l o c r ( Ω ) f o r a l l 1 ≤ r < p * , u n → u λ a . e . i n Ω .

By (f ₁) and Lemma 2.3, we obtain

λ ∫ Ω f ∣ u n ∣ q = λ ∫ Ω f ∣ u λ ∣ q + o ( 1 ) a s n → ∞ .

From (4.1)-(4.3), a standard argument shows that u_λ is a critical point of J_λ . Furthermore, the fact { u n } ⊂ N λ implies that

λ ∫ Ω f ∣ u n ∣ q = q ( p * - p ) p ( p * - q ) ∥ u n ∥ μ p - p * q p * - q J λ ( u n ) .

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

λ ∫ Ω f ∣ u λ ∣ q ≥ - p * q p * - q α λ > 0 .

Thus, u λ ∈ N λ is a nontrivial solution of (1.1).

Next, we prove that u_n → u_λ strongly in D 0 1 , p ( Ω ) and J_λ (u_λ ) = α_λ . From (4.3), the fact u n , u λ ∈ N λ and the Fatou's lemma it follows that

α λ ≤ J λ ( u λ ) = 1 N ∥ u λ ∥ μ p - λ p * - q p * q ∫ Ω f ∣ u λ ∣ q ≤ liminf n → ∞ 1 N ∥ u n ∥ μ p - λ p * - q p * q ∫ Ω f ∣ u n ∣ q = liminf n → ∞ J λ ( u n ) = α λ ,

which implies that J_λ (u_λ ) = α_λ and lim n → ∞ ∥ u n ∥ μ p = ∥ u λ ∥ μ p . Standard argument shows that u_n → u_λ strongly in D 0 1 , p ( Ω ) . Moreover, u λ ∈ N λ + . Otherwise, if u λ ∈ N λ - , by Lemma 3.4, there exist unique t λ + and t λ - such that t λ + u λ ∈ N λ + , t λ - u λ ∈ N λ - and t λ + < t λ - = 1 . Since

d d t J λ ( t λ + u λ ) = 0 a n d d 2 d t 2 J λ ( t λ + u λ ) > 0 ,

there exists t ̄ ∈ ( t λ + , t λ - ) such that J λ ( t λ + u λ ) < J λ ( t ̄ u λ ) . By Lemma 3.4, we get that

J λ ( t λ + u λ ) < J λ ( t ̄ u λ ) ≤ J λ ( t λ - u λ ) = J λ ( u λ ) ,

which is a contradiction. If u ∈ N λ + , then ∣ u ∣ ∈ N λ + , and by J_λ (u_λ ) = J_λ (|u_λ |) = α_λ , we get ∣ u λ ∣ ∈ N λ + is a local minimum of J_λ on N λ . 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

∥ u λ ∥ μ p - q < λ p * - q p * - p ∣ f + ∣ q * S μ - 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 λ ∈ N λ + for all λ ∈ (0, Λ₀). □

For 1 < p < N and μ < μ ̄ , let

c * = 1 N ∣ g + ∣ ∞ - N - p p S μ N p .

Lemma 5.1 Suppose {u_n } is a bounded sequence in D 0 1 , p ( Ω ) . 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 { u n } ⊂ D 0 1 , p ( Ω ) be a (PS) _c -sequence for J_λ with c ∈ (0, c ^*). Since {u_n } is bounded in D 0 1 , p ( Ω ) , passing to a subsequence if necessary, we may assume that as n → ∞

u n ⇀ u 0 w e a k l y i n D 0 1 , p ( Ω ) , u n ⇀ u 0 w e a k l y i n L p * ( Ω ) , u n → u 0 s t r o n g l y i n L l o c r ( Ω ) f o r 1 ≤ r < p * , u n → u 0 a . e . i n Ω .

By (f ₁), (g ₁), (5.1) and Lemma 2.3, we have that J λ ′ ( u 0 ) = 0 and

λ ∫ Ω f ∣ u n ∣ q = λ ∫ Ω f ∣ u 0 ∣ q + o ( 1 ) a s n → ∞ .

Next, we verify that u ₀ ≢ 0. Arguing by contradiction, we assume u ₀ ≡ 0. Since J λ ′ ( u n ) = o ( 1 ) as n → ∞ and {u_n } is bounded in D 0 1 , p ( Ω ) , then by (5.2), we can deduce that

0 = 〈 lim n → ∞ J λ ′ ( u n ) , u n 〉 = lim n → ∞ ∥ u n ∥ μ p - ∫ Ω g ∣ u n ∣ p * .