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

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_pu = 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.

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_pu = 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||_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 = 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. 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 ≤ 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 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.

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 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^p-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, and Then, the functional

is well-defined and weakly continuous.

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 on and .

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) 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 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 d₀.

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 each with , we can easily deduce that

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 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, 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). □

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 = 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, u₀ is a nontrivial solution of (1.1). □

Lemma 5.2 Suppose and (f₁) - (g₂) hold. If , x₀ = 0 and β ≥ pγ, then for any λ > 0, there exists such 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 Suppose and (f₁) - (g₂) hold. If , x₀ = 0, β ≥ pγ and , then there exists such 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 inf_n→∞ ||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. □

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 Suppose and (f₁) - (g₂) hold. If N ≤ p², μ < 0, x₀ ≠ 0 and , then for any λ > 0 and μ < 0, there exists such 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. □

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

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