## Boundary Value Problems

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

Boundary Value Problems20112011:37

DOI: 10.1186/1687-2770-2011-37

Accepted: 19 October 2011

Published: 19 October 2011

## Abstract

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

where Ω N is a smooth domain with smooth boundary ∂Ω such that 0 Ω, Δ p u = 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 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
(1.1)

where Δ p u = 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 J λ .

Problem (1.1) is related to the well-known Hardy inequality [1, 2]:
By the Hardy inequality, has the equivalent norm ||u||μs, where
Therefore, for 1 < p < N, and , we can define the best Sobolev constant:
(1.2)

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

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

In the present paper, our research is mainly related to (1.1) with 1 < q < p < N, the critical exponent and weight functions f, g that change sign on Ω. When p = 2, 1 < q < 2, , 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 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:

, 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 βpγ, then (1.1) has at least two positive solutions for all.

Theorem 1.3 Suppose, (f1) - (g2) hold. If μ < 0, x0 ≠ 0, and Np2, 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. Asand 1 w< q < p < N, the results of Theorems 1.1, 1.2 are improvements of the main results of[11]. As μ < 0 and 1 < q < p < Np2, Theorem 1.3 is the complement to the results in [[11], Theorem 1.3].

This paper is organized as follows. Some preliminaries and properties of the Nehari manifold are established in Sections 2 and 3, and Theorems 1.1-1.3 are proved in Sections 4-6, respectively. Before ending this section, we explain some notations employed in this paper. In the following argument, we always employ C and C i to denote various positive constants and omit dx in integral for convenience. B(x0; R) is the ball centered at x0 N with the radius R > 0, denotes the dual space of , 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 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 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)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, andThen, the functional

is well-defined and weakly continuous.

## 3 Nehari manifold

As J λ is not bounded below on , we need to study J λ on the Nehari manifold
Note that contains all solutions of (1.1) and if and only if
(3.1)

Lemma 3.1 J λ 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, J λ is coercive and bounded below on . □

Define . Then, for ,
(3.3)
Arguing as in [22], we split into three parts:

Lemma 3.2 Suppose u λ is a local minimizer of J λ onand.

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) andis a function satisfying with.
1. (i)
If , then there exists a unique t - > t max such that and

2. (ii)
If , then there exists a unique t ± such that 0 < t + < t max < 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.
1. (ii)

If , then for some positive constant d 0.

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 α λ 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 eachwith, 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 J λ as follows:

Definition 4.1 (i) For c , a sequence {u n } is a (PS) c -sequence infor J λ if J λ (u n ) = c + o(1) and (J λ )'(u n ) = o(1) strongly inas n → ∞.
1. (ii)

c is a (PS)-value in for J λ if there exists a (PS) c -sequence in for J λ .

2. (iii)

J λ satisfies the (PS) c -condition in if any (PS) c -sequence {u n } in for J λ contains a convergent subsequence.

Lemma 4.2 (i) If λ (0, Λ1), then J λ has a-sequence.
1. (ii)

If , then J λ 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 J λ 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
1. (i)

,

2. (ii)

u λ is a positive solution of (1.1),

3. (iii)

||u λ || μ → 0 as λ → 0+.

Proof By Lemma 4.2 (i), there exists a minimizing sequence such that
(4.1)
Since J λ is coercive on (see Lemma 2.1), we get that (u n ) 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 u λ is a critical point of J λ . Furthermore, the fact implies that
(4.4)
Taking n → ∞ in (4.4), by (4.1), (4.3) and the fact α λ < 0, we get
(4.5)

Thus, is a nontrivial solution of (1.1).

Next, we prove that u n u λ strongly in and J λ (u λ ) = α λ . From (4.3), the fact and the Fatou's lemma it follows that
which implies that J λ (u λ ) = α λ and . Standard argument shows that u n u λ 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 J λ (u λ ) = J λ (|u λ |) = α λ , we get is a local minimum of J λ on . 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

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 for all λ (0, Λ0). □

## 5 Proof of Theorem 1.2

For 1 < p < N and , let

Lemma 5.1 Suppose {u n } is a bounded sequence in. 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 be a (PS) c -sequence for J λ with c (0, c*). Since {u n } 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 {u n } 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→∞J λ (u n ) = 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 Supposeand (f1) - (g2) hold. If, x0 = 0 and βpγ, then for any λ > 0, there existssuch 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 t ε > 0 and the positive constants C i (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)
1. (i)
If , then by Lemma 2.2 and we have that

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

Theorem 5.3 Supposeand (f1) - (g2) hold. If, x0 = 0, βpγ and, then there existssuch that
1. (i)

,

2. (ii)

U λ 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 J λ with . Since J λ is coercive on (see Lemma 3.1), we get that {u n } is bounded in . From Lemma 5.1, there exists a subsequence still denoted by {u n } and a nontrivial solution of (1.1) such that u n U λ weakly in .

First, we prove that . On the contrary, if , then by is closed in , we have ||U λ || μ < lim infn→∞||u n || μ . From (g2) and U λ 0 in Ω, we have . Thus, by Lemma 3.4, there exists a unique t λ 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 u n U λ strongly in and for all . Since J λ (U λ ) = J λ (|U λ |) and , 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 for all λ (0, Λ0). From Theorem 5.3, we get the second positive solution for all . Since , 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 when μ ≤ 0.

Lemma 6.1 Supposeand (f1) - (g2) hold. If Np2, μ < 0, x0 ≠ 0 and, then for any λ > 0 and μ < 0, there existssuch 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 , Np2 and (6.4), we can deduce that
and
Combining this with (6.5), for any λ > 0 and μ < 0, we can choose ε λ,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. □

## Declarations

### Acknowledgements

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

## Authors’ Affiliations

(1)
Center for General Education, Chang Gung University, Kwei-San

## 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 1984, 53: 259-275.
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 2006, 9: 445-484.
4. Cao D, Han P: Solutions for semilinear elliptic equations with critical exponents and Hardy potential. J Differ Equ 2004, 205: 521-537. 10.1016/j.jde.2004.03.005
5. Ghoussoub N, Robert F: The effect of curvature on the best constant in the Hardy-Sobolev inequalities. Geom Funct Anal 2006, 16: 897-908.
6. Kang D, Peng S: Solutions for semilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy potential. Appl Math Lett 2005, 18: 1094-1100. 10.1016/j.aml.2004.09.016
7. Cao D, Kang D: Solutions of quasilinear elliptic problems involving a Sobolev exponent and multiple Hardy-type terms. J Math Anal Appl 2007, 333: 889-903. 10.1016/j.jmaa.2006.12.005
8. Ghoussoub N, Yuan C: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents. Trans Am Math Soc 2000, 352: 5703-5743. 10.1090/S0002-9947-00-02560-5
9. Han P: Quasilinear elliptic problems with critical exponents and Hardy terms. Nonlinear Anal 2005, 61: 735-758. 10.1016/j.na.2005.01.030
10. Kang D: On the quasilinear elliptic problems with critical Sobolev-Hardy exponents and Hardy terms. Nonlinear Anal 2008, 68: 1973-1985. 10.1016/j.na.2007.01.024
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 2011, 74: 626-638. 10.1016/j.na.2010.09.017
12. Felli V, Terracini S: Elliptic equations with multi-singular inverse-square potentials and critical nonlinearity. Commun Part Differ Equ 2006, 31: 469-495. 10.1080/03605300500394439
13. Li J: Existence of solution for a singular elliptic equation with critical Sobolev-Hardy exponents. Intern J Math Math Sci 2005, 20: 3213-3223.View Article
14. Filippucci R, Pucci P, Robert F: On a p-Laplace equation with multiple critical nonlinearities. J Math Pures Appl 2009, 91: 156-177. 10.1016/j.matpur.2008.09.008
15. Gazzini M, Musina R: On a Sobolev-type inequality related to the weighted p-Laplace operator. J Math Anal Appl 2009, 352: 99-111. 10.1016/j.jmaa.2008.06.021
16. Kang D: Solution of the quasilinear elliptic problem with a critical Sobolev-Hardy exponent and a Hardy term. J Math Anal Appl 2008, 341: 764-782. 10.1016/j.jmaa.2007.10.058
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 2008, 25: 505-537. 10.1016/j.anihpc.2007.02.004
18. Hsu TS, Lin HL: Multiple positive solutions for singular elliptic equations with concave-convex nonlinearities and sign-changing weights. Boundary Value Problems 2009, 2009: 17. Article ID 584203)MathSciNet
19. Hsu TS: Multiplicity results for p -Laplacian with critical nonlinearity of concave-convex type and sign-changing weight functions. Abstr Appl Anal 2009, 2009: 24. Article ID 652109)View Article
20. Kang D, Huang Y, Liu S: Asymptotic estimates on the extremal functions of a quasilinear elliptic problem. J S Cent Univ Natl 2008, 27: 91-95.
21. Ben-Naoum AK, Troestler C, Willem M: Extrema problems with critical Sobolev exponents on unbounded domains. Nonlinear Anal 1996, 26: 823-833. 10.1016/0362-546X(94)00324-B
22. Tarantello G: On nonhomogeneous elliptic involving critical Sobolev exponent. Ann Inst H Poincaré Anal Non Linéaire 1992, 9: 281-304.
23. Brown KJ, Zhang Y: The Nehari manifold for a semilinear elliptic equation with a sign-changing weigh function. J Differ Equ 2003, 193: 481-499. 10.1016/S0022-0396(03)00121-9
24. Brown KJ, Wu TF: A semilinear elliptic system involving nonlinear boundary condition and sign-changing weigh function. J Math Anal Appl 2008, 337: 1326-1336. 10.1016/j.jmaa.2007.04.064
25. Wu TF: On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function. J Math Anal Appl 2006, 318: 253-270. 10.1016/j.jmaa.2005.05.057
26. Trudinger NS: On Harnack type inequalities and their application to quasilinear elliptic equations. Commun Pure Appl Math 1967, 20: 721-747. 10.1002/cpa.3160200406
27. Talenti G: Best constant in Sobolev inequality. Ann Mat Pura Appl 1976, 110: 353-372. 10.1007/BF02418013