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, , λ >0, 1 < q < p, signchanging weight functions f and g are continuous functions on , is the best Hardy constant and is the critical Sobolev exponent. By extracting the PalaisSmale sequence in the Nehari manifold, the multiplicity of positive solutions to this equation is verified.
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, , 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 wellknown 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:
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, , 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 > p^{2}, and f, g are nonnegative. They also proved that there existence of Λ_{0 }> 0 such that for λ ∈ (0, Λ_{0}), (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 ℝ^{N }(N ≥ 3). By extracting the PalaisSmale 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.
(f_{1}) f^{+ }= max{f, 0} ≢ 0 in Ω.
(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}) and g^{+ }= max{g, 0} ≢ 0 in Ω.
(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 , (f_{1}) and (g_{1}) hold. Then, (1.1) has at least one positive solution for all λ ∈ (0, Λ_{1}).
Theorem 1.2 Suppose , (f_{1})  (g_{2}) hold, and γ is the constant defined as in Lemma 2.2. If , x_{0 }= 0 and β ≥ pγ, then (1.1) has at least two positive solutions for all .
Theorem 1.3 Suppose , (f_{1})  (g_{2}) hold. If μ < 0, x_{0 }≠ 0, and N ≤ p^{2}, 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 < p^{2 }< 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 ≤ p^{2}, 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 1.11.3 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, 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 O_{1}(ε^{t}) means that there exist C_{1}, C_{2 }> 0 such that C_{1}ε^{t }≤ O_{1}(ε^{t}) ≤ C_{2}ε^{t }as ε is small enough.
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 . Then, the limiting problem
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 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.
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, and Then, the functional
is welldefined 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
Lemma 3.1 J_{λ }is coercive and bounded below on .
Proof Suppose . From (f_{1}), (3.1), the Hölder inequality and Sobolev embedding theorem, we can deduce that
Thus, J_{λ }is coercive and bounded below on . □
Arguing as in [22], we split into three parts:
Lemma 3.2 Suppose u_{λ }is a local minimizer of J_{λ }on and .
Proof The proof is similar to [[23], Theorem 2.3] and is omitted. □
Lemma 3.3 for all λ ∈ (0, Λ_{1}).
Proof We argue by contradiction. Suppose that there exists λ ∈ (0, Λ_{1}) such that . Then, the fact and (3.3) imply that
and
By (f_{1}), (g_{1}), the Hölder inequality and Sobolev embedding theorem, we have that
and
Consequently,
which is a contradiction. □
Lemma 3.4 Suppose that λ ∈ (0, Λ_{1}) and is a function satisfying with .
(i) If , then there exists a unique t^{ }> t_{max }such that and
(ii) If , then there exists a unique t^{± }such that 0 < t^{+ }< t_{max }< t^{}, and . Moreover,
Proof See BrownWu [[24], Lemma 2.6]. □
We remark that it follows Lemma 3.3, for all λ ∈ (0, Λ_{1}). 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, Λ_{1}), then we have .
(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
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
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 , 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 PalaisSmale (simply by (PS)) sequences, (PS)values and (PS)conditions in for J_{λ }as follows:
Definition 4.1 (i) For c ∈ ℝ, a sequence {u_{n}} is a (PS)_{c}sequence in for J_{λ }if J_{λ}(u_{n}) = c + o(1) and (J_{λ})'(u_{n}) = o(1) strongly in as n → ∞.
(ii) c ∈ ℝ is a (PS)value in for J_{λ }if there exists a (PS)_{c}sequence in for J_{λ}.
(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, Λ_{1}), then J_{λ }has a sequence .
(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 (f_{1}), (g_{1}) hold. If λ ∈ (0, Λ_{1}), then there exists such that
(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 such that
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 → ∞
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 implies that
Taking n → ∞ in (4.4), by (4.1), (4.3) and the fact α_{λ }< 0, we get
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, Λ_{0}). □
5 Proof of Theorem 1.2
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 → ∞
By (f_{1}), (g_{1}), (5.1) and Lemma 2.3, we have that and
Next, we verify that u_{0 }≢ 0. Arguing by contradiction, we assume u_{0 }≡ 0. Since as n → ∞ and {u_{n}} is bounded in , then by (5.2), we can deduce that
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 , which implies that
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 and (f_{1})  (g_{2}) hold. If , x_{0 }= 0 and β ≥ pγ, then for any λ > 0, there exists such that
In particular, for all λ ∈ (0, Λ_{1}).
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 , 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
(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 (f_{2}), (g_{2}) and the definition of , we have that
From Lemma 3.4, the definition of and (5.5), for any λ ∈ (0, Λ_{0}), 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 Suppose and (f_{1})  (g_{2}) hold. If , x_{0 }= 0, β ≥ pγ and , then there exists such that
(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 inf_{n→∞ }u_{n}_{μ}. From (g_{2}) 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
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, Λ_{0}). 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 wellknown S_{μ }= S_{0 }where S_{μ }is defined as in (1.2). Thus, we have when μ ≤ 0.
Lemma 6.1 Suppose and (f_{1})  (g_{2}) hold. If N ≤ p^{2}, μ < 0, x_{0 }≠ 0 and , then for any λ > 0 and μ < 0, there exists such that
In particular, for all λ ∈ (0, Λ_{1}).
Proof Note that S_{0}_{} has the following explicit extremals [27]:
where is a particular constant. Take ρ > 0 small enough such that B(x_{0}; ρ) ⊂ Ω\{0} and set , where is a cutoff function such that φ(x) ≡ 1 in B(x_{0}; ρ/2). Arguing as in Lemma 2.2, we have
where . Note that , . Arguing as in Lemma 5.2, we deduce that there exists satisfying , such that
From , N ≤ p^{2 }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 (f_{2}), (g_{2}) and the definition of , we have that
From Lemma 3.4, the definition of and (6.1), for any λ ∈ (0, Λ_{0}) and μ < 0, there exists such that and
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 for all λ ∈ (0, Λ_{1}(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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