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

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

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

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

(1.1)

where Δpu = div(|∇u|p-2u), 1 < p < N, , is the best Hardy constant, λ > 0, 1 < q < p, is the critical Sobolev exponent and the weight functions are continuous, which change sign on Ω.

Let be the completion of with respect to the norm . The energy functional of (1.1) is defined on by

Then . is said to be a solution of (1.1) if for all 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]:

By the Hardy inequality, has the equivalent norm ||u||μ, where

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

(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, , 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, 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, , 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:

, λ > 0, 1 < q < p < N, N ≥ 3.

(f1) 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) and g+ = max{g, 0} ≢ 0 in Ω.

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

where | · |denotes the L(Ω) norm.

Set

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

Theorem 1.2 Suppose , (f1) - (g2) hold, and γ is the constant defined as in Lemma 2.2. If , x0 = 0 and β pγ, then (1.1) has at least two positive solutions for all .

Theorem 1.3 Suppose , (f1) - (g2) hold. If μ < 0, x0 ≠ 0, and N p2, then (1.1) has at least two positive solutions for all .

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 , λ > 0 and 1 < q < p < p2 < N. As 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, denotes the dual space of , 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 . Then, the limiting problem

(2.1)

that satisfy

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

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 .

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

(2.2)

where is a cutoff function such that η(x) ≡ 1 in .

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

where , 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, and Then, the functional

is well-defined and weakly continuous.

3 Nehari manifold

As Jλ is not bounded below on , we need to study Jλ on the Nehari manifold

Note that contains all solutions of (1.1) and if and only if

(3.1)

Lemma 3.1 Jλ is coercive and bounded below on .

Proof Suppose . From (f1), (3.1), the Hölder inequality and Sobolev embedding theorem, we can deduce that

(3.2)

Thus, Jλ is coercive and bounded below on . □

Define . Then, for ,

(3.3)

Arguing as in [22], we split into three parts:

Lemma 3.2 Suppose uλ is a local minimizer of Jλ on and .

Then, in .

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

Lemma 3.3 for all λ ∈ (0, Λ1).

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

and

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

and

Consequently,

For each with , we set

Lemma 3.4 Suppose that λ ∈ (0, Λ1) and is a function satisfying with .

(i) If , then there exists a unique t- > tmax such that and

(ii) If , then there exists a unique t± such that 0 < t+ < tmax < t-, and . Moreover,

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

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

Lemma 3.5 (i) If λ ∈ (0, Λ1), then we have .

(ii) If , then for some positive constant d0.

In particular, for each , we have .

Proof (i) Suppose that . From (3.3), it follows that

(3.4)

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

By the definitions of αλ and , we get that .

(ii) Suppose and . Then, (3.3) implies that

(3.5)

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

(3.6)

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

(3.7)

By (3.2) and (3.7), we get

which implies that

for some positive constant d0. □

Remark 3.6 If , then by Lemmas 3.4 and 3.5, for each with , we can easily deduce that

4 Proof of Theorem 1.1

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

Definition 4.1 (i) For c ∈ ℝ, a sequence {un} is a (PS)c-sequence in for Jλ if Jλ(un) = c + o(1) and (Jλ)'(un) = o(1) strongly in as n → ∞.

(ii) c ∈ ℝ is a (PS)-value in for Jλ if there exists a (PS)c-sequence in for Jλ.

(iii) Jλ satisfies the (PS)c-condition in if any (PS)c-sequence {un} in for Jλ contains a convergent subsequence.

Lemma 4.2 (i) If λ ∈ (0, Λ1), then Jλ has a -sequence .

(ii) If , then Jλ has a -sequence .

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 .

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

(i) ,

(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 such that

(4.1)

Since Jλ is coercive on (see Lemma 2.1), we get that (un) is bounded in . Passing to a subsequence, there exists such that as n → ∞

(4.2)

By (f1) and Lemma 2.3, we obtain

(3)

From (4.1)-(4.3), a standard argument shows that uλ is a critical point of Jλ. Furthermore, the fact implies that

(4.4)

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

(4.5)

Thus, is a nontrivial solution of (1.1).

Next, we prove that un uλ strongly in and Jλ(uλ) = αλ. From (4.3), the fact and the Fatou's lemma it follows that

which implies that Jλ(uλ) = αλ and . Standard argument shows that un uλ strongly in . Moreover, . Otherwise, if , by Lemma 3.4, there exist unique and such that , and . Since

there exists such that . By Lemma 3.4, we get that

which is a contradiction. If , then , and by Jλ(uλ) = Jλ(|uλ|) = αλ, we get is a local minimum of Jλ on . 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

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 for all λ ∈ (0, Λ0). □

5 Proof of Theorem 1.2

For 1 < p < N and , let

Lemma 5.1 Suppose {un} is a bounded sequence in . 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 be a (PS)c-sequence for Jλ with c ∈ (0, c*). Since {un} is bounded in , passing to a subsequence if necessary, we may assume that as n → ∞

(5.1)

By (f1), (g1), (5.1) and Lemma 2.3, we have that and

(5.2)

Next, we verify that u0 ≢ 0. Arguing by contradiction, we assume u0 ≡ 0. Since as n → ∞ and {un} is bounded in , then by (5.2), we can deduce that

Then, we can set

(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

Then, as n → ∞ we have , which implies that

(5.4)

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

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

Lemma 5.2 Suppose and (f1) - (g2) hold. If , x0 = 0 and β pγ, then for any λ > 0, there exists such that

(5.5)

In particular, 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

(5.6)

By (g2), we conclude that

which together with Lemma 2.2 implies that

(5.7)

From the fact λ > 0, 1 < q < p, β and

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

(5.8)

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

(5.9)

(i) If , then by Lemma 2.2 and we have that

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

(ii) If , then by Lemma 2.2 and γ > 0 we have that

and

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

From (i) and (ii), (5.5) holds by taking .

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

From Lemma 3.4, the definition of and (5.5), for any λ ∈ (0, Λ0), there exists such that and

The proof is thus complete. □

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

Theorem 5.3 Suppose and (f1) - (g2) hold. If , x0 = 0, β pγ and , then there exists such that

(i) ,

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

Proof If , then by Lemmas 3.5 (ii), 4.2 (ii) and 5.2, there exists a -sequence in for Jλ with . Since Jλ is coercive on (see Lemma 3.1), we get that {un} is bounded in . From Lemma 5.1, there exists a subsequence still denoted by {un} and a nontrivial solution of (1.1) such that un Uλ weakly in .

First, we prove that . On the contrary, if , then by is closed in , we have ||Uλ||μ < lim infn→∞ ||un||μ. From (g2) and Uλ ≢ 0 in Ω, we have . Thus, by Lemma 3.4, there exists a unique tλ such that . If , then it is easy to see that

(5.10)

From Remark 3.6, and (5.10), we can deduce that

This is a contradiction. Thus, .

Next, by the same argument as that in Theorem 4.3, we get that un Uλ strongly in and for all . Since Jλ(Uλ) = Jλ(|Uλ|) and , 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 for all λ ∈ (0, Λ0). From Theorem 5.3, we get the second positive solution for all . Since , 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 when μ ≤ 0.

Lemma 6.1 Suppose and (f1) - (g2) hold. If N p2, μ < 0, x0 ≠ 0 and , then for any λ > 0 and μ < 0, there exists such that

(6.1)

In particular, for all λ ∈ (0, Λ1).

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

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

(6.2)

(6.3)

(6.4)

where . Note that , . Arguing as in Lemma 5.2, we deduce that there exists satisfying , such that

(6.5)

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

and

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

Therefore, (6.1) holds by taking .

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

From Lemma 3.4, the definition of and (6.1), for any λ ∈ (0, Λ0) and μ < 0, there exists such that and

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 for all λ ∈ (0, Λ1(0)) and the second positive solution for all . Since , this implies that and are distinct. □

Acknowledgements

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

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