Abstract
In this paper, we are concerned with the following quasilinear elliptic equation
where Ω ⊂ ℝ^{N }is a smooth domain with smooth boundary ∂Ω such that 0 ∈ Ω, Δ_{p}u = div(∇u^{p2}∇u), 1 < p < N,
Keywords:
Multiple positive solutions; critical Sobolev exponent; concaveconvex; Hardy terms; signchanging weights1 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
where Δ_{p}u = div(∇u^{p2}∇u), 1 < p < N,
Let
Then
Problem (1.1) is related to the wellknown Hardy inequality [1,2]:
By the Hardy inequality,
Therefore, for 1 < p < N, and
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 [46] and for p > 1 to [711], while in ℝ^{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,
The following assumptions are used in this paper:
(f_{1})
(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_{2}) There exist x_{0 }∈ Ω and β > 0 such that
where  · _{∞ }denotes the L^{∞}(Ω) norm.
Set
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
Theorem 1.2 Suppose
Theorem 1.3 Suppose
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
This paper is organized as follows. Some preliminaries and properties of the Nehari
manifold are established in Sections 2 and 3, and Theorems 1.11.3 are proved in Sections
46, 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,
2 Preliminaries
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
has positive radial ground states
that satisfy
Furthermore, U_{p,μ}(x) = U_{p,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)t^{p1 }+ μ, t ≥ 0, satisfying
Take ρ > 0 small enough such that B(0; ρ) ⊂ Ω, and define the function
where
Lemma 2.2 [9,20]Suppose 1 < p < N and
where
We also recall the following known result by BenNaoum, 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,
is welldefined and weakly continuous.
3 Nehari manifold
As J_{λ }is not bounded below on
Note that
Lemma 3.1 J_{λ }is coercive and bounded below on
Proof Suppose
Thus, J_{λ }is coercive and bounded below on
Define
Arguing as in [22], we split
Lemma 3.2 Suppose u_{λ }is a local minimizer of J_{λ }on
Then,
Proof The proof is similar to [[23], Theorem 2.3] and is omitted. □
Lemma 3.3
Proof We argue by contradiction. Suppose that there exists λ ∈ (0, Λ_{1}) such that
and
By (f_{1}), (g_{1}), the Hölder inequality and Sobolev embedding theorem, we have that
and
Consequently,
which is a contradiction. □
For each
Lemma 3.4 Suppose that λ ∈ (0, Λ_{1}) and
(i) If
(ii) If
Proof See BrownWu [[24], Lemma 2.6]. □
We remark that it follows Lemma 3.3,
Lemma 3.5 (i) If λ ∈ (0, Λ_{1}), then we have
(ii) If
In particular, for each
Proof (i) Suppose that
According to (3.1) and (3.4), we have
By the definitions of α_{λ }and
(ii) Suppose
Moreover, by (g_{1}) and the Sobolev embedding theorem, we have
From (3.5) and (3.6), it follows that
By (3.2) and (3.7), we get
which implies that
for some positive constant d_{0}. □
Remark 3.6 If
4 Proof of Theorem 1.1
First, we define the PalaisSmale (simply by (PS)) sequences, (PS)values and (PS)conditions in
Definition 4.1 (i) For c ∈ ℝ, a sequence {u_{n}} is a (PS)_{c}sequence in
(ii) c ∈ ℝ is a (PS)value in
(iii) J_{λ }satisfies the (PS)_{c}condition in
Lemma 4.2 (i) If λ ∈ (0, Λ_{1}), then J_{λ }has a
(ii) If
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,
(i)
(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
Since J_{λ }is coercive on
By (f_{1}) and Lemma 2.3, we obtain
From (4.1)(4.3), a standard argument shows that u_{λ }is a critical point of J_{λ}. Furthermore, the fact
Taking n → ∞ in (4.4), by (4.1), (4.3) and the fact α_{λ }< 0, we get
Thus,
Next, we prove that u_{n }→ u_{λ }strongly in
which implies that J_{λ}(u_{λ}) = α_{λ }and
there exists
which is a contradiction. If
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
5 Proof of Theorem 1.2
For 1 < p < N and
Lemma 5.1 Suppose {u_{n}} is a bounded sequence in
Proof Let
By (f_{1}), (g_{1}), (5.1) and Lemma 2.3, we have that
Next, we verify that u_{0 }≢ 0. Arguing by contradiction, we assume u_{0 }≡ 0. Since
Then, we can set
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
Then, as n → ∞ we have
Hence, from (5.2)(5.4), we get
This is contrary to c < c^{*}. Therefore, u_{0 }is a nontrivial solution of (1.1). □
Lemma 5.2 Suppose
In particular,
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
By (g_{2}), we conclude that
which together with Lemma 2.2 implies that
From the fact λ > 0, 1 < q < p, β ≥ pγ and
and by Lemma 2.2, (5.7) and (f_{2}), we get
By (5.6) and (5.8), we have that
(i) If
Combining this with (5.9), for any λ > 0, we can choose ε_{λ }small enough such that
(ii) If
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 (f_{2}), (g_{2}) and the definition of
From Lemma 3.4, the definition of
The proof is thus complete. □
Now, we establish the existence of a local minimum of J_{λ }on
Theorem 5.3 Suppose
(i)
(ii) U_{λ }is a positive solution of (1.1).
Proof If
First, we prove that
From Remark 3.6,
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
Proof of Theorem 1.2 From Theorem 4.3, we get the first positive solution
6 Proof of Theorem 1.3
In this section, we consider the case μ ≤ 0. In this case, it is wellknown S_{μ }= S_{0 }where S_{μ }is defined as in (1.2). Thus, we have
Lemma 6.1 Suppose
In particular,
Proof Note that S_{0}_{} has the following explicit extremals [27]:
where
where
From
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 (f_{2}), (g_{2}) and the definition of
From Lemma 3.4, the definition of
The proof is thus complete. □
Proof of Theorem 1.3 Let Λ_{1}(0) be defined as in (1.3). Arguing as in Theorems 4.3 and 5.3, we can get the first
positive solution
Acknowledgements
The author is grateful for the referee's valuable suggestions.
References

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

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

Abdellaoui, B, Felli, V, Peral, I: Existence and nonexistence results for quasilinear elliptic equations involving the pLaplacian. Boll Unione Mat Ital Scz B. 9, 445–484 (2006)

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

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

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

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

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

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

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

Wang, L, Wei, Q, Kang, D: Multiple positive solutions for pLaplace elliptic equations involving concaveconvex nonlinearities and a Hardytype term. Nonlinear Anal. 74, 626–638 (2011). Publisher Full Text

Felli, V, Terracini, S: Elliptic equations with multisingular inversesquare potentials and critical nonlinearity. Commun Part Differ Equ. 31, 469–495 (2006). Publisher Full Text

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

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

Gazzini, M, Musina, R: On a Sobolevtype inequality related to the weighted pLaplace operator. J Math Anal Appl. 352, 99–111 (2009). Publisher Full Text

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

Pucci, P, Servadei, R: Existence, nonexistence and regularity of radial ground states for pLaplace weights. Ann Inst H Poincaré Anal Non Linéaire. 25, 505–537 (2008). Publisher Full Text

Hsu, TS, Lin, HL: Multiple positive solutions for singular elliptic equations with concaveconvex nonlinearities and signchanging weights. Boundary Value Problems. 2009, 17 Article ID 584203) (2009)

Hsu, TS: Multiplicity results for pLaplacian with critical nonlinearity of concaveconvex type and signchanging weight functions. Abstr Appl Anal. 2009, 24 Article ID 652109) (2009)

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)

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

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

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

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

Wu, TF: On semilinear elliptic equations involving concaveconvex nonlinearities and signchanging weight function. J Math Anal Appl. 318, 253–270 (2006). Publisher Full Text

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

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