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Multiple positive solutions for a class of quasi-linear elliptic equations involving concave-convex nonlinearities and Hardy terms
Boundary Value Problems volume 2011, Article number: 37 (2011)
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|p-2∇u), 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.
1 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|p-2∇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 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:
It is well known that S μ (Ω) = S μ (ℝ 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 [4–6] and for p > 1 to [7–11], while in ℝ N 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 ℝ 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
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. 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 < 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 C i to denote various positive constants and omit dx in integral for convenience. B(x0; R) is the ball centered at x0 ∈ ℝ N 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
has positive radial ground states
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)tp - (N - p)tp-1+ 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.
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 ℝ 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
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
Thus, J λ is coercive and bounded below on . □
Define . Then, for ,
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,
which is a contradiction. □
For each with , we set
Lemma 3.4 Suppose that λ ∈ (0, Λ1) andis 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 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 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 (g1) 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 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 ∈ ℝ, 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 → ∞.
-
(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, Λ1), 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 (f1), (g1) hold. If λ ∈ (0, Λ1), then there exists such that
-
(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 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 (f1) 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, 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 → ∞
By (f1), (g1), (5.1) and Lemma 2.3, we have that and
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
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
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 β ≥ pγ, then for any λ > 0, there existssuch that
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
By (g2), 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 (f2), 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 (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, β ≥ pγ and, then there existssuch that
-
(i)
,
-
(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
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 N ≤ p2, μ < 0, x0 ≠ 0 and, then for any λ > 0 and μ < 0, there existssuch that
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
where . Note that , . Arguing as in Lemma 5.2, we deduce that there exists satisfying , such that
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 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. □
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Hsu, TS. Multiple positive solutions for a class of quasi-linear elliptic equations involving concave-convex nonlinearities and Hardy terms. Bound Value Probl 2011, 37 (2011). https://doi.org/10.1186/1687-2770-2011-37
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DOI: https://doi.org/10.1186/1687-2770-2011-37
Keywords
- Multiple positive solutions
- critical Sobolev exponent
- concave-convex
- Hardy terms
- sign-changing weights