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.
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