Multiple positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in ℝ^N

In this article, we investigate the effect of the coefficient f(z) of the sub-critical nonlinearity. For sufficiently large λ > 0, there are at least k + 1 positive solutions of the semilinear elliptic equations

where 1 ≤ q < 2 < p < 2* = 2N/(N - 2) for N ≥ 3.

AMS (MOS) subject classification: 35J20; 35J25; 35J65.

For N ≥ 3, 1 ≤ q < 2 < p < 2* = 2N/(N - 2), we consider the semilinear elliptic equations

where λ > 0.

Let f and h satisfy the following conditions:

(f 1) f is a positive continuous function in ℝ^N and lim_{|z| → ∞} f(z) = f_∞ > 0.

(f2) there exist k points a¹, a²,..., a^k in ℝ^N such that

and f_∞ < f_max.

(h 1) and .

Semilinear elliptic problems involving concave-convex nonlinearities in a bounded domain

have been studied by Ambrosetti et al. [1] (h ≡ 1, and 1 < q < 2 < p ≤ 2* = 2N/(N- 2)) and Wu [2] and changes sign, 1 < q < 2 < p < 2*). They proved that this equation has at least two positive solutions for sufficiently small c > 0. More general results of Equation (E_c) were done by Ambrosetti et al. [3], Brown and Zhang [4], and de Figueiredo et al. [5].

In this article, we consider the existence and multiplicity of positive solutions of Equation (E_λ) in ℝ^N. For the case q = λ = 1 and f(z) ≡ 1 for all z ∈ ℝ^N, suppose that h is nonnegative, small, and exponential decay, Zhu [6] showed that Equation (E_λ) admits at least two positive solutions in ℝ^N. Without the condition of exponential decay, Cao and Zhou [7] and Hirano [8] proved that Equation (E_λ) admits at least two positive solutions in ℝ^N. For the case q = λ = 1, by using the idea of category and Bahri-Li's minimax argument, Adachi and Tanaka [9] asserted that Equation (E_λ) admits at least four positive solutions in ℝ^N, where f(z) ≢ 1, f(z) ≥ 1 - C exp((-(2 + δ) |z|) for some C, δ > 0, and sufficiently small . Similarly, in Hsu and Lin [10], they have studied that there are at least four positive solutions of the general case -Δu + u = f(z)v^p-1 + λh(z) v^q-1 in ℝ^N for sufficiently small λ > 0.

By the change of variables

Equation (E_λ) is transformed to

Associated with Equation (E_ε), we consider the C¹-functional J_ε, for u ∈ H¹ (ℝ^N),

where is the norm in H¹ (ℝ^N) and u₊ = max{u, 0} ≥ 0. We know that the nonnegative weak solutions of Equation (E_ε) are equivalent to the critical points of J_ε. This article is organized as follows. First of all, we use the argument of Tarantello [11] to divide the Nehari manifold M_ε into the two parts and . Next, we prove that the existence of a positive ground state solution of Equation (E_ε). Finally, in Section 4, we show that the condition (f2) affects the number of positive solutions of Equation (E_ε), that is, there are at least k critical points of J_ε such that for 1 ≤ i ≤ k.

Let

then

For the semilinear elliptic equations

we define the energy functional , and

where N_ε = {u ∈ H¹ (ℝ^N) \ {0} | u₊ ≢ 0 and }. Note that

(i) if f ≡ f_∞, we define and

where N_∞ = {u ∈ H¹ (ℝ^N) \ {0} | u₊ ≢ 0 and };

(ii) if f ≡ f_max, we define and

where N_max = {u ∈ H¹ (ℝ^N) \ {0} | u₊ ≢ 0 and }.

Lemma 1.1

Proof. It is similar to Theorems 4.12 and 4.13 in Wang [[12], p. 31].

Our main results are as follows.

(I) Let Λ = ε^2(p-q)/(p-2). Under assumptions (f 1) and (h1), if

where ∥h∥_# is the norm in , then Equation (E_ε) admits at least a positive ground state solution. (See Theorem 3.4)

(II) Under assumptions (f1) - (f2) and (h1), if λ is sufficiently large, then Equation (E_λ) admits at least k + 1 positive solutions. (See Theorem 4.8)

First of all, we define the Palais-Smale (denoted by (PS)) sequences and (PS)-conditions in H¹(ℝ^N) for some functional J.

Definition 2.1 (i) For β ∈ ℝ, a sequence {u_n} is a (PS)_β-sequence in H¹(ℝ^N) for J if J(u_n) = β + o_n(1) and J'(u_n) = o_n(1) strongly in H^-1 (ℝ^N) as n → ∞, where H^-1 (ℝ^N) is the dual space of H¹(ℝ^N);

(ii) J satisfies the (PS)_β-condition in H¹(ℝ^N) if every (PS)_β-sequence in H¹(ℝ^N) for J contains a convergent subsequence.

Next, since J_ε is not bounded from below in H¹ (ℝ^N), we consider the Nehari manifold

where

Note that M_ε contains all nonnegative solutions of Equation (E_ε). From the lemma below, we have that J_ε is bounded from below on M_ε.

Lemma 2.2 The energy functional J_ε is coercive and bounded from below on M_ε.

Proof. For u ∈ M_ε, by (2.1), the Hölder inequality and the Sobolev embedding theorem (1.1), we get

Hence, we have that J_ε is coercive and bounded from below on M_ε.

Define

Then for u ∈ M_ε, we get

We apply the method in Tarantello [11], let

Lemma 2.3 Under assumptions (f1) and (h1), if 0 < Λ (= ε^2(p-q)/(p-2)) < Λ₀, then .

Proof. See Hsu and Lin [[10], Lemma 5].

Lemma 2.4 Suppose that u is a local minimizer for J_ε on M_ε and . Then in H^-1 (ℝ^N).

Proof. See Brown and Zhang [[4], Theorem 2.3].

Lemma 2.5 We have the following inequalities.

(i) for each ;

(ii) for each ;

(iii) for each ;

(iv) If , then J_ε(u) > 0 for each .

Proof. (i) It can be proved by using (2.2).

(ii) For any , by (2.2), we apply the Hölder inequality to obtain that

(iii) For any , by (2.3), we have that

(iv) For any , by (iii), we get that

Thus, if , we get that J_ε(u) ≥ d₀ > 0 for some constant d₀ = d₀(ε, p, q, S, ∥h∥_# , f_max).

For u ∈ H¹ (ℝ^N) \ {0} and u₊ ≢ 0, let

Lemma 2.6 For each u ∈ H¹ (ℝ^N)\ {0} and u₊ ≢ 0, we have that

(i) if , then there exists a unique positive number such that and J_ε(t^-u) = sup_{t ≥ 0} J_ε(tu);

(ii) if 0 < Λ ( = ε^2(p-q)/(p-2)) < Λ₀ and , then there exist unique positive numbers such that and

Proof. See Hsu and Lin [[10], Lemma 7].

Applying Lemma 2.3 , we write , where

Define

Lemma 2.7 (i) If 0 < Λ ( = ε^2(p-q)/(p-2)) < Λ₀, then ;

(ii) If 0 < Λ < qΛ₀/2, then for some constant d₀ = d₀ (ε, p, q, S, ∥h∥_#, f_max).

Proof. (i) Let , by (2.2), we get

Then

By the definitions of α_ε and , we deduce that .

(ii) See the proof of Lemma 2.5 (iv).

Applying Ekeland's variational principle and using the same argument in Cao and Zhou [7] or Tarantello [11], we have the following lemma.

Lemma 2.8 (i) There exists a -sequence {u_n} in M_ε for J_ε;

(ii) There exists a -sequence {u_n} in for J_ε;

(iii) There exists a -sequence {u_n} in for J_ε.

In order to prove the existence of positive solutions, we claim that J_ε satisfies the (PS)_β-condition in H¹(ℝ^N) for , where Λ = ε^2(p-q)/(p-2) and C₀ is defined in the following lemma.

Lemma 3.1 Assume that h satisfies (h1) and 0 < Λ ( = ε^2(p-q)/(p-2)) < Λ₀. If {u_n} is a (PS)_β-sequence in H¹(ℝ^N) for J_ε with u_n ⇀ u weakly in H¹ (ℝ^N), then in H^-1 (ℝ^N) and , where

and

Proof. Since {u_n} is a (PS)_β-sequence in H¹(ℝ^N) for J_ε with u_n ⇀ u weakly in H¹ (ℝ^N), it is easy to check that in H^-1(ℝ^N) and u ≥ 0. Then we have , that is, . Hence, by the Young inequality

Lemma 3.2 Assume that f and h satisfy (f1) and (h1). If 0 < Λ ( = ε^2(p-q)/(p-2)) < Λ₀, then J_ε satisfies the (PS)_β-condition in H¹(ℝ^N) for .

Proof. Let {u_n} be a (PS)_β-sequence in H¹(ℝ^N) for J_ε such that J_ε(u_n) = β + o_n(1) and (1) in H^-1(ℝ^N). Then

where c_n = o_n(1), d_n = o_n(1) as n → ∞. It follows that {u_n} is bounded in H¹(ℝ^N). Hence, there exist a subsequence {u_n} and a nonnegative u ∈ H¹ (ℝ^N) such that in H^-1 (ℝ^N), u_n ⇀ u weakly in H¹ (ℝ^N), u_n ⇀ u a.e. in ℝ^N, u_n ⇀ u strongly in for any 1 ≤ s < 2*. Using the Brézis-Lieb lemma to get (3.1) and (3.2) below.

Next, claim that

For any σ > 0, there exists r > 0 such that . By the Hölder inequality and the Sobolev embedding theorem, we get

Applying (f1) and u_n → u in , we get that

Let p_n = u_n - u. Suppose p_n ↛ 0 strongly in H¹ (ℝ^N). By (3.1)-(3.4), we deduce that

Then

By Theorem 4.3 in Wang [12], there exists a sequence {s_n} ⊂ ℝ⁺ such that s_n = 1 + o_n(1), {s_n p_n} ⊂ N_∞ and I_∞(s_n p_n) = I_∞(p_n) + o_n(1). It follows that

which is a contradiction. Hence, u_n → u strongly in H¹(ℝ^N).

Remark 3.3 By Lemma 1.1, we obtain

and for 0 < Λ < Λ₀.

By Lemma 2.8 (i), there is a -sequence {u_n} in M_ε for J_ε. Then we prove that Equation (E_ε) admits a positive ground state solution u₀ in ℝ^N.

Theorem 3.4 Under assumptions (f1), (h1), if 0 < Λ ( = ε^2(p-q)/(p-2)) < Λ₀, then there exists at least one positive ground state solution u₀ of Equation (E_ε) in ℝ^N. Moreover, we have that and

Proof. By Lemma 2.8 (i), there is a minimizing sequence {u_n} ⊂ M_ε for J_ε such that J_ε(u_n) = α_ε + o_n(1) and in H^-1 (ℝ^N). Since , by Lemma 3.2, there exist a subsequence {u_n} and u₀ ∈ H¹ (ℝ^N) such that u_n → u₀ strongly in H¹ (ℝ^N). It is easy to see that is a solution of Equation (E_ε) in ℝ^N and J_ε(u₀) = α_ε. Next, we claim that . On the contrary, assume that .

We get that

Otherwise,

It follows that

which contradicts to α_ε < 0. By Lemma 2.6 (ii), there exist positive numbers such that and

which is a contradiction. Hence, and

By Lemma 2.4 and the maximum principle, then u₀ is a positive solution of Equation (E_ε) in ℝ^N.

From now, we assume that f and h satisfy (f1)-(f2) and (h1). Let w ∈ H¹ (ℝ^N) be the unique, radially symmetric, and positive ground state solution of Equation (E0) in ℝ^N for f = f_max. Recall the facts (or see Bahri and Li [13], Bahri and Lions [14], Gidas et al. [15], and Kwong [16]).

(i) for some 0 < θ < 1 and ;

(ii) for any ε > 0, there exist positive numbers C₁, , and such that for all z ∈ ℝ^N

and

For 1 ≤ i ≤ k, we define

Clearly, . By Lemma 2.6 (ii), there is a unique number such that , where 1 ≤ i ≤ k.

We need to prove that

Lemma 4.1 (i) There exists a number t₀ > 0 such that for 0 ≤ t ≤t₀ and any ε > 0, we have that

(ii) There exist positive numbers t₁ and ε₁ such that for any t > t₁ and ε < ε₁, we have that

Proof. (i) Since J_ε is continuous in is uniformly bounded in H¹ (ℝ^N) for any ε > 0, and γ_max > 0, there is t₀ > 0 such that for 0 ≤ t ≤ t₀ and any ε > 0

(ii) There is an r₀ > 0 such that f (z) ≥ f_max/2 for z ∈ B^N (aⁱ; r₀) uniformly in i. Then there exists ε₁ > 0 such that for ε < ε₁

Thus, there is t₁ >0 such that for any t > t₁ and ε < ε₁

Lemma 4.2 Under assumptions (f1), (f2), and (h1). If 0 < Λ ( = ε^2(p-q)/(p-2)) < q Λ₀/2, then

Proof. By Lemma 4.1, we only need to show that

We know that sup_{t ≥0} I_max (tw) = γ_max. For t₀ ≤ t ≤ t₁, we get

Since

and

then , that is, uniformly in i.

Applying the results of Lemmas 2.6, 2.7(ii), and 4.2, we can deduce that

Since γ_max < γ_∞, there exists ε₀ > 0 such that

Choosing 0 < ρ₀ < 1 such that

where and f(aⁱ) = f_max. Define K = {aⁱ | 1 ≤ i ≤ k} and . Suppose for some r₀ > 0.

Let Q_ε : H¹ (ℝ^N) \ {0} → ℝ^N be given by

where χ : ℝ^N → ℝ^N, χ (z) = z for |z| ≤ r₀ and χ (z) = r₀z/|z| for |z| > r₀.

Lemma 4.3 There exists 0 < ε⁰ ≤ ε₀ such that if ε < ε⁰, then for each 1 ≤ i ≤ k.

Proof. Since

there exists ε⁰ > 0 such that

Lemma 4.4 There exists a number such that if u ∈ N_ε and , then for any 0 < ε < ε⁰.

Proof. On the contrary, there exist the sequences {ε_n} ⊂ ℝ⁺ and such that (1) as n → ∞ and for all n ∈ ℕ. It is easy to check that {u_n} is bounded in H¹ (ℝ^N). Suppose u_n → 0 strongly in L^p (ℝ^N). Since

and

then

which is a contradiction. Thus, u_n ↛ 0 strongly in L^p (ℝ^N). Applying the concentration-compactness principle (see Lions [17] or Wang [[12], Lemma 2.16]), then there exist a constant d₀ > 0 and a sequence such that

Let , there are a subsequence {v_n} and v ∈ H¹ (ℝ^N) such that v_n ⇀ v weakly in H¹ (ℝ^N). Using the similar computation in Lemma 2.6, there is a sequence such that and

We deduce that a convergent subsequence satisfies . Then there are subsequences and such that weakly in H¹ (ℝ^N). By (4.2), then . Moreover, we can obtain that strongly in H¹ (ℝ^N) and . Now, we want to show that there exists a subsequence such that z_n → z₀ ∈ K.

(i) Claim that the sequence {z_n} is bounded in ℝ^N. On the contrary, assume that |z_n| → ∞, then

which is a contradiction.

(ii) Claim that z₀ ∈ K. On the contrary, assume that z₀ ∉ K, that is, f(z₀) < f_max. Then using the above argument to obtain that

which is a contradiction. Since v_n → v ≠ 0 in H¹ (ℝ^N), we have that

which is a contradiction.

Hence, there exists a number such that if u ∈ N_ε and , then for any 0 < ε < ε⁰.

From (4.1), choosing such that

For each 1 ≤ i ≤ k, define

and .

Lemma 4.5 If and J_ε (u) ≤ γ_max + δ₀/2, then there exists a number such that for any .

Proof. We use the similar computation in Lemma 2.6 to get that there is a unique positive number

such that . We want to show that for some constant c > 0 (independent of u). First, since ,

and J_ε is coercive on M_ε, then for some constants c₁ and c₂ (independent of u). Next, we claim that for some constant c₃ > 0 (independent of u). On the contrary, there exists a sequence such that

By (2.3),

which is a contradiction. Thus, for some constant c > 0 (independent of u). Now, we get that

From the above inequality, we deduce that

Hence, there exists such that for

By Lemma 4.4, we obtain

or for any .

Applying the above lemma, we get that

By Lemmas 4.2, 4.3, and Equation (4.3), there exists such that

Lemma 4.6 Given , then there exist an η > 0 and a differentiable functional l : B(0; η) ⊂ H¹(ℝ^N) → ℝ⁺ such that for any v ∈ B(0;η) and

where .

Proof. See Cao and Zhou [7].

Lemma 4.7 For each 1 ≤ i ≤ k, there is a -sequence in H¹(ℝ^N) for J_ε.

Proof. For each 1 ≤ i ≤ k, by (4.4) and (4.5),

Then

Let be a minimizing sequence for . Applying Ekeland's variational principle, there exists a subsequence such that and

Using (4.7), we may assume that for sufficiently large n. By Lemma 4.6, then there exist an and a differentiable functional such that , and for . Let v_σ = σv with ║v║_H = 1 and . Then and . From (4.8) and by the mean value theorem, we get that as σ → 0

Hence,

Since we can deduce that for all n and i from (4.6), then strongly in H^-1 (ℝ^N) as n → ∞.

Theorem 4.8 Under assumptions (f1), (f₂), and (h1), there exists a positive number λ*(λ* = (ε*)^-2) such that for λ > λ*, Equation (E_λ) has k + 1 positive solutions in ℝ^N.

Proof. We know that there is a -sequence in H¹(ℝ^N) for J_ε for each 1 ≤ i ≤ k, and (4.5). Since J_ε satisfies the (PS)_β-condition for , then J_ε has at least k critical points in for 0 < ε < ε*. It follows that Equation (E_λ) has k nonnegative solutions in ℝ^N. Applying the maximum principle and Theorem 3.4, Equation (E_λ) has k + 1 positive solutions in ℝ^N.

The author declares that he has no competing interests.

The author was grateful for the referee's helpful suggestions and comments.

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