Open Access Research

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

Tsing-San Hsu

Author Affiliations

Center for General Education, Chang Gung University, Kwei-San, Tao-Yuan 333, Taiwan ROC

Boundary Value Problems 2011, 2011:37  doi:10.1186/1687-2770-2011-37


The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2011/1/37


Received:13 April 2011
Accepted:19 October 2011
Published:19 October 2011

© 2011 Hsu; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we are concerned with 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 in  Ω , u = 0 on  Ω ,

where Ω ⊂ ℝN is a smooth domain with smooth boundary ∂Ω such that 0 ∈ Ω, Δpu = div(|∇u|p-2u), 1 < p < N, μ < μ ̄ = ( N - p p ) p , λ >0, 1 < q < p, sign-changing weight functions f and g are continuous functions on Ω ̄ , μ ̄ = ( N - p p ) p is the best Hardy constant and p * = N p 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.

Keywords:
Multiple positive solutions; critical Sobolev exponent; concave-convex; Hardy terms; sign-changing weights

1 Introduction and main results

Let Ω be a smooth domain (not necessarily bounded) in ℝN (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 , (1.1)

where Δpu = div(|∇u|p-2u), 1 < p < N, μ < μ ̄ = ( 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 ̄ 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 ) , ) . 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 Jλ.

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 * . (1.2)

It is well known that Sμ(Ω) = Sμ(ℝN) = Sμ. Note that Sμ = 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 ℝN 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, 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 > p2, - < 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 ℝN (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.

(f1) f C ( Ω ̄ ) L q * ( Ω ) ( q * = p * p * - q ) 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 C ( Ω ̄ ) L ( Ω ) and g+ = max{g, 0} ≢ 0 in Ω.

(g2) There exist x0 ∈ 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 . (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 ( H ) , (f1) and (g1) hold. Then, (1.1) has at least one positive solution for all λ ∈ (0, Λ1).

Theorem 1.2 Suppose ( H ) , (f1) - (g2) hold, and γ is the constant defined as in Lemma 2.2. If 0 μ < μ ̄ , x0 = 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 ) , (f1) - (g2) hold. If μ < 0, x0 ≠ 0, β N - p p - 1 and N p2, 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 < p2 < 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 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 Ci to denote various positive constants and omit dx in integral for convenience. B(x0; R) is the ball centered at x0 ∈ ℝN with the radius R > 0, ( D 0 1 , p ( Ω ) ) - 1 denotes the dual space of D 0 1 , p ( Ω ) , 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 C1εt O1(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 μ < μ ̄ . Then, the limiting problem

- Δ p u - μ u p - 1 x p = u p * - 1 , i n N \ { 0 } , u D 1 , p ( N ) , u > 0 , i n N \ { 0 } , (2.1)

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

N ( | V p , μ , ε ( x ) | p μ | V p , μ , ε ( x ) | p | x | p ) = N | V p , μ , ε ( x ) | p * = S μ N p .

Furthermore, Up,μ(|x|) = Up,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 ci (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 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 , (2.2)

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 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 ) : u N k ( x ) u q d x

is well-defined and weakly continuous.

3 Nehari manifold

As Jλ is not bounded below on D 0 1 , p ( Ω ) , we need to study Jλ 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 . (3.1)

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

Proof Suppose u N λ . From (f1), (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 . (3.2)

Thus, Jλ 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 . (3.3)

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 uλ is a local minimizer of Jλ 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 (f1), (g1), 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- > tmax 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+ < tmax < 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 d0.

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 * . (3.4)

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 αλ 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 * . (3.5)

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

Ω g u p * g + S μ - p * p u μ p * . (3.6)

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 - . (3.7)

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 d0. □

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 .

4 Proof of Theorem 1.1

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

Definition 4.1 (i) For c ∈ ℝ, a sequence {un} is a (PS)c-sequence in D 0 1 , p ( Ω ) for Jλ if Jλ(un) = c + o(1) and (Jλ)'(un) = 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 Jλ if there exists a (PS)c-sequence in D 0 1 , p ( Ω ) for Jλ.

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

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

(ii) If λ ( 0 , q p Λ 1 ) , then Jλ 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 Jλ on N λ .

Theorem 4.3 Suppose that N ≥ 3, μ < μ ̄ , 1 < q < p < N and the conditions (f1), (g1) hold. If λ ∈ (0, Λ1), then there exists u λ N λ + such that

(i) J λ ( u λ ) = α λ = α λ + ,

(ii) uλ is a positive solution of (1.1),

(iii) ||uλ||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 . (4.1)

Since Jλ is coercive on N λ (see Lemma 2.1), we get that (un) 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 . (4.2)

By (f1) and Lemma 2.3, we obtain

λ Ω f u n q = λ Ω f u λ q + o ( 1 ) a s n . (3)

From (4.1)-(4.3), a standard argument shows that uλ is a critical point of Jλ. 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 ) . (4.4)

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

λ Ω f u λ q - p * q p * - q α λ > 0 . (4.5)

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

Next, we prove that un uλ strongly in D 0 1 , p ( Ω ) and Jλ(uλ) = αλ. 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 Jλ(uλ) = αλ and lim n u n μ p = u λ μ p . Standard argument shows that un uλ 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 Jλ(uλ) = Jλ(|uλ|) = αλ, we get u λ N λ + is a local minimum of Jλ on N λ . Then, by Lemma 3.2, we may assume that uλ is a nontrivial nonnegative solution of (1.1). By Harnack inequality due to Trudinger [26], we obtain that uλ > 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 ||uλ||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). □

5 Proof of Theorem 1.2

For 1 < p < N and μ < μ ̄ , let

c * = 1 N g + - N - p p S μ N p .

Lemma 5.1 Suppose {un} is a bounded sequence in D 0 1 , p ( Ω ) . If {un} is a (PS)c-sequence for Jλ with c ∈ (0, c*), then there exists a subsequence of {un} converging weakly to a nonzero solution of (1.1).

Proof Let { u n } D 0 1 , p ( Ω ) be a (PS)c-sequence for Jλ with c ∈ (0, c*). Since {un} 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 . (5.1)

By (f1), (g1), (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 . (5.2)

Next, we verify that u0 ≢ 0. Arguing by contradiction, we assume u0 ≡ 0. Since J λ ( u n ) = o ( 1 ) as n → ∞ and {un} 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 * .

Then, we can set

lim n u n μ p = lim n Ω g u n p * = l . (5.3)

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

u n μ p S μ Ω g u n p * p p * S μ Ω g g + u n p * p p * = S μ g + - N - p N Ω g u n p * p p * .

Then, as n → ∞ we have l = lim n u n μ p S μ g + - N - p N l p p * , which implies that

l g + - N - p p S μ N p . (5.4)

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

c = lim n J λ ( u n ) = 1 p lim n u n μ p - λ q lim n Ω f u n q - 1 p * lim n Ω g u n p * = 1 p - 1 p * l 1 N g + - N - p p S μ N p .

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

Lemma 5.2 Suppose ( H ) and (f1) - (g2) hold. If 0 < μ < μ ̄ , x0 = 0 and β pγ, then for any λ > 0, there exists v λ D 0 1 , p ( Ω ) such that

sup t 0 J λ ( t v λ ) < c * . (5.5)

In particular, α λ - < c * for all λ ∈ (0, Λ1).

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

sup t 0 J λ ( t u ε ) = J λ ( t ε u ε ) and 0 < C 1 t ε C 2 < . (5.6)

By (g2), we conclude that

Ω g ( x ) u ε p * - Ω g ( 0 ) u ε p * Ω g ( x ) - g ( 0 ) u ε p * = O B ( 0 ; ρ ) x β u ε p * = O ( ε β ) ,

which together with Lemma 2.2 implies that

Ω g ( x ) u ε p * = g ( 0 ) S μ N p + O ( ε p * γ ) + O ( ε β ) . (5.7)

From the fact λ > 0, 1 < q < p, β and

max t 0 t p p B 1 - t p * p * B 2 = 1 N B 1 N p B 2 - N - p p , B 1 > 0 , B 2 > 0 ,

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

J λ ( t ε u ε ) = t ε p p u ε μ p - t ε p * p * Ω g u ε p * - λ t ε q q Ω f u ε q 1 N u ε μ N p Ω g u ε p * - N - p p - λ C 1 q q β 0 Ω u ε q = 1 N S μ N p + O ( ε p γ ) N p g ( 0 ) S μ N p + O ( ε p * γ ) + O ( ε β ) - N - p p - λ C 1 q q β 0 Ω u ε q = 1 N g ( 0 ) - N - p p S μ N p + O ( ε p γ ) + O ( ε β ) - λ C 1 q q β 0 Ω u ε q . (5.8)

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

sup t 0 J λ ( t u ε ) c * + O ( ε p γ ) + O ( ε β ) - λ C 1 q q β 0 Ω u ε q . (5.9)

(i) If 1 < q < N b ( μ ) , then by Lemma 2.2 and γ = b ( μ ) - δ = b ( μ ) - N - p p > 0 we have that

Ω u ε q = O 1 ( ε q γ ) .

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

sup t 0 J λ ( t u ε λ ) < c * .

(ii) If N b ( μ ) q < p , then by Lemma 2.2 and γ > 0 we have that

Ω u ε q = O 1 ( ε θ ) , q > N b ( μ ) , O 1 ( ε θ l n ε ) , q = N b ( μ ) ,

and

p γ = b ( μ ) p + p - N > N + ( 1 - N p ) q = θ .

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

sup t 0 J λ ( t u ε λ ) < c * .

From (i) and (ii), (5.5) holds by taking v λ = u ε λ .

In fact, by (f2), (g2) and the definition of u ε λ , we have that

Ω f u ε λ q > 0 a n d Ω g u ε λ p * > 0 .

From Lemma 3.4, the definition of α λ - and (5.5), for any λ ∈ (0, Λ0), there exists t ε λ > 0 such that t ε λ u ε λ N λ - and

α λ - J λ ( t ε λ u ε λ ) sup t 0 J λ ( t t ε λ u ε λ ) < c * .

The proof is thus complete. □

Now, we establish the existence of a local minimum of Jλ on N λ - .

Theorem 5.3 Suppose ( H ) and (f1) - (g2) hold. If 0 < μ < μ ̄ , x0 = 0, β pγ and λ ( 0 , q p Λ 1 ) , then there exists U λ N λ - such that

(i) J λ ( U λ ) = α λ - ,

(ii) Uλ is a positive solution of (1.1).

Proof If λ ( 0 , q p Λ 1 ) , then by Lemmas 3.5 (ii), 4.2 (ii) and 5.2, there exists a ( P S ) α λ -sequence { u n } N λ - in D 0 1 , p ( Ω ) for Jλ with α λ - ( 0 , c * ) . Since Jλ is coercive on N λ - (see Lemma 3.1), we get that {un} is bounded in D 0 1 , p ( Ω ) . From Lemma 5.1, there exists a subsequence still denoted by {un} and a nontrivial solution U λ D 0 1 , p ( Ω ) of (1.1) such that un Uλ weakly in D 0 1 , p ( Ω ) .

First, we prove that U λ N λ - . On the contrary, if U λ N λ + , then by N λ - { 0 } is closed in D 0 1 , p ( Ω ) , we have ||Uλ||μ < lim infn→∞ ||un||μ. From (g2) and Uλ ≢ 0 in Ω, we have Ω g U λ p * > 0 . Thus, by Lemma 3.4, there exists a unique tλ such that t λ U λ N λ - . If u N λ , then it is easy to see that

J λ ( u ) = 1 N u μ p - λ p * - q p * q Ω f u q . (5.10)

From Remark 3.6, u n N λ - and (5.10), we can deduce that

α λ - J λ ( t λ U λ ) < lim n J λ ( t λ u n ) lim n J λ ( u n ) = α λ - .

This is a contradiction. Thus, U λ N λ - .

Next, by the same argument as that in Theorem 4.3, we get that un Uλ strongly in D 0 1 , p ( Ω ) and J λ ( U λ ) = α λ - > 0 for all λ ( 0 , q p Λ 1 ) . Since Jλ(Uλ) = Jλ(|Uλ|) and U λ N λ - , by Lemma 3.2, we may assume that Uλ is a nontrivial nonnegative solution of (1.1). Finally, by Harnack inequality due to Trudinger [26], we obtain that Uλ is a positive solution of (1.1). □

Proof of Theorem 1.2 From Theorem 4.3, we get the first positive solution u λ N λ + for all λ ∈ (0, Λ0). From Theorem 5.3, we get the second positive solution U λ N λ + for all λ ( 0 , q p Λ 0 ) . Since N λ - N λ - = , this implies that uλ and Uλ are distinct. □

6 Proof of Theorem 1.3

In this section, we consider the case μ ≤ 0. In this case, it is well-known Sμ = S0 where Sμ is defined as in (1.2). Thus, we have c * = 1 N g + - N - p p S 0 N p when μ ≤ 0.

Lemma 6.1 Suppose ( H ) and (f1) - (g2) hold. If N p2, μ < 0, x0 ≠ 0 and β p γ ̃ : = N - p p ( p - 1 ) , then for any λ > 0 and μ < 0, there exists v λ , μ D 0 1 , p ( Ω ) such that

sup t 0 J λ ( t v λ , μ ) < c * . (6.1)

In particular, α λ - < c * for all λ ∈ (0, Λ1).

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

V ε ( x ) = C ̄ ε - N - p p 1 + x - x 0 ε p p - 1 - N - p p , ε > 0 , x 0 N ,

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

Ω ũ ε p = S 0 N p + O ( ε p γ ̃ ) , (6.2)

Ω ũ ε p * = S 0 N P + + O ( ε p * γ ̃ ) , (6.3)

Ω ũ ε q = O 1 ( ε θ ) , O 1 ( ε θ l n ε ) , O 1 ( ε q γ ̃ ) , N ( p - 1 ) N - p < q < p * , q = N ( p - 1 ) N - p , 1 q < N ( p - 1 ) N - p , (6.4)

where θ = N - N - p p q . Note that β p γ ̃ , p * γ ̃ > p γ ̃ . Arguing as in Lemma 5.2, we deduce that there exists t ̃ ε satisfying 0 < C 1 t ̃ ε C 2 , such that

J λ ( t ũ ε ) sup t 0 J λ ( t ũ ε ) = J λ ( t ̃ ε ũ ε ) = t ̃ ε p p Ω ũ ε p - t ̃ ε p * p * Ω g ũ ε p * - λ t ̃ ε q q Ω f ũ ε q - μ t ̃ ε p p Ω ũ ε p x p 1 N g ( x 0 ) - N - p p S μ N p + O ( ε p γ ̃ ) - λ C 1 q q β 0 Ω ũ ε q - μ x 0 - ρ - p C 2 p p Ω ũ ε p . (6.5)

From ( H ) , N p2 and (6.4), we can deduce that

1 < q γ ̃ < p γ ̃ = N - p p - 1 p N ( p - 1 ) N - p

and

Ω ũ ε q = O 1 ( ε q γ ̃ ) a n d Ω ũ ε p = O 1 ( ε p l n ε ) , O 1 ( ε p γ ̃ ) , p = N ( p - 1 ) N - p , 1 < p < N ( p - 1 ) N - p .

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

sup t 0 J λ ( t ũ ε λ , μ ) < 1 N g ( x 0 ) - N - p p S 0 N p = c * .

Therefore, (6.1) holds by taking v λ , μ = ũ ε λ , μ .

In fact, by (f2), (g2) and the definition of ũ ε λ , μ , we have that

Ω f ũ ε λ , μ q > 0 a n d Ω g ũ ε λ , μ p * > 0 .

From Lemma 3.4, the definition of α λ - and (6.1), for any λ ∈ (0, Λ0) and μ < 0, there exists t ε λ , μ > 0 such that t ε λ , μ ũ ε λ , μ N λ - and

α λ - J λ ( t ε λ , μ ũ ε λ , μ ) sup t 0 J λ ( t t ε λ , μ ũ ε λ , μ ) < 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 λ N λ + for all λ ∈ (0, Λ1(0)) and the second positive solution Ũ λ N λ - for all λ ( 0 , q p Λ 1 ( 0 ) ) . Since N λ + N λ - = , this implies that ũ λ and Ũ λ are distinct. □

Acknowledgements

The author is grateful for the referee's valuable suggestions.

References

  1. Hardy, G, Littlewood, J, Polya, G: Inequalities, Reprint of the 1952 edition, Cambridge Math. Lib. Cambridge University Press, Cambridge (1988)

  2. Caffarelli, L, Kohn, R, Nirenberg, L: First order interpolation inequality with weights. Compos Math. 53, 259–275 (1984)

  3. Abdellaoui, B, Felli, V, Peral, I: Existence and non-existence results for quasilinear elliptic equations involving the p-Laplacian. Boll Unione Mat Ital Scz B. 9, 445–484 (2006)

  4. Cao, D, Han, P: Solutions for semilinear elliptic equations with critical exponents and Hardy potential. J Differ Equ. 205, 521–537 (2004). Publisher Full Text OpenURL

  5. Ghoussoub, N, Robert, F: The effect of curvature on the best constant in the Hardy-Sobolev inequalities. Geom Funct Anal. 16, 897–908 (2006)

  6. Kang, D, Peng, S: Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy potential. Appl Math Lett. 18, 1094–1100 (2005). Publisher Full Text OpenURL

  7. Cao, D, Kang, D: Solutions of quasilinear elliptic problems involving a Sobolev exponent and multiple Hardy-type terms. J Math Anal Appl. 333, 889–903 (2007). Publisher Full Text OpenURL

  8. Ghoussoub, N, Yuan, C: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans Am Math Soc. 352, 5703–5743 (2000). Publisher Full Text OpenURL

  9. Han, P: Quasilinear elliptic problems with critical exponents and Hardy terms. Nonlinear Anal. 61, 735–758 (2005). Publisher Full Text OpenURL

  10. Kang, D: On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms. Nonlinear Anal. 68, 1973–1985 (2008). Publisher Full Text OpenURL

  11. Wang, L, Wei, Q, Kang, D: Multiple positive solutions for p-Laplace elliptic equations involving concave-convex nonlinearities and a Hardytype term. Nonlinear Anal. 74, 626–638 (2011). Publisher Full Text OpenURL

  12. Felli, V, Terracini, S: Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity. Commun Part Differ Equ. 31, 469–495 (2006). Publisher Full Text OpenURL

  13. Li, J: Existence of solution for a singular elliptic equation with critical Sobolev-Hardy exponents. Intern J Math Math Sci. 20, 3213–3223 (2005)

  14. Filippucci, R, Pucci, P, Robert, F: On a p-Laplace equation with multiple critical nonlinearities. J Math Pures Appl. 91, 156–177 (2009). Publisher Full Text OpenURL

  15. Gazzini, M, Musina, R: On a Sobolev-type inequality related to the weighted p-Laplace operator. J Math Anal Appl. 352, 99–111 (2009). Publisher Full Text OpenURL

  16. Kang, D: Solution of the quasilinear elliptic problem with a critical Sobolev-Hardy exponent and a Hardy term. J Math Anal Appl. 341, 764–782 (2008). Publisher Full Text OpenURL

  17. Pucci, P, Servadei, R: Existence, non-existence and regularity of radial ground states for p-Laplace weights. Ann Inst H Poincaré Anal Non Linéaire. 25, 505–537 (2008). Publisher Full Text OpenURL

  18. Hsu, TS, Lin, HL: Multiple positive solutions for singular elliptic equations with concave-convex nonlinearities and sign-changing weights. Boundary Value Problems. 2009, 17 Article ID 584203) (2009)

  19. Hsu, TS: Multiplicity results for p-Laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions. Abstr Appl Anal. 2009, 24 Article ID 652109) (2009)

  20. Kang, D, Huang, Y, Liu, S: Asymptotic estimates on the extremal functions of a quasilinear elliptic problem. J S Cent Univ Natl. 27, 91–95 (2008)

  21. Ben-Naoum, AK, Troestler, C, Willem, M: Extrema problems with critical Sobolev exponents on unbounded domains. Nonlinear Anal. 26, 823–833 (1996). Publisher Full Text OpenURL

  22. Tarantello, G: On nonhomogeneous elliptic involving critical Sobolev exponent. Ann Inst H Poincaré Anal Non Linéaire. 9, 281–304 (1992)

  23. Brown, KJ, Zhang, Y: The Nehari manifold for a semilinear elliptic equation with a sign-changing weigh function. J Differ Equ. 193, 481–499 (2003). Publisher Full Text OpenURL

  24. Brown, KJ, Wu, TF: A semilinear elliptic system involving nonlinear boundary condition and sign-changing weigh function. J Math Anal Appl. 337, 1326–1336 (2008). Publisher Full Text OpenURL

  25. Wu, TF: On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J Math Anal Appl. 318, 253–270 (2006). Publisher Full Text OpenURL

  26. Trudinger, NS: On Harnack type inequalities and their application to quasilinear elliptic equations. Commun Pure Appl Math. 20, 721–747 (1967). Publisher Full Text OpenURL

  27. Talenti, G: Best constant in Sobolev inequality. Ann Mat Pura Appl. 110, 353–372 (1976). Publisher Full Text OpenURL