Abstract
In this article, we investigate the effect of the coefficient f(z) of the subcritical 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.
Keywords:
semilinear elliptic equations; concave and convex; positive solutions1 Introduction
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 a^{1}, a^{2},..., a^{k }in ℝ^{N }such that
and f_{∞ }< f_{max}.
Semilinear elliptic problems involving concaveconvex 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 BahriLi'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^{p1 }+ λh(z) v^{q1 }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 C^{1}functional J_{ε}, for u ∈ H^{1 }(ℝ^{N}),
where is the norm in H^{1 }(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 ∈ H^{1 }(ℝ^{N}) \ {0}  u_{+ }≢ 0 and }. Note that
(i) if f ≡ f_{∞}, we define and
where N_{∞ }= {u ∈ H^{1 }(ℝ^{N}) \ {0}  u_{+ }≢ 0 and };
(ii) if f ≡ f_{max}, we define and
where N_{max }= {u ∈ H^{1 }(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(pq)/(p2)}. 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)
2 The Nehari manifold
First of all, we define the PalaisSmale (denoted by (PS)) sequences and (PS)conditions in H^{1}(ℝ^{N}) for some functional J.
Definition 2.1 (i) For β ∈ ℝ, a sequence {u_{n}} is a (PS)_{β}sequence in H^{1}(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 H^{1}(ℝ^{N});
(ii) J satisfies the (PS)_{β}condition in H^{1}(ℝ^{N}) if every (PS)_{β}sequence in H^{1}(ℝ^{N}) for J contains a convergent subsequence.
Next, since J_{ε }is not bounded from below in H^{1 }(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(pq)/(p2)}) < Λ_{0}, 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.
(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) ≥ d_{0 }> 0 for some constant d_{0 }= d_{0}(ε, p, q, S, ∥h∥_{# }, f_{max}).
For u ∈ H^{1 }(ℝ^{N}) \ {0} and u_{+ }≢ 0, let
Lemma 2.6 For each u ∈ H^{1 }(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(pq)/(p2)}) < Λ_{0 }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(pq)/(p2)}) < Λ_{0}, then ;
(ii) If 0 < Λ < qΛ_{0}/2, then for some constant d_{0 }= d_{0 }(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};
3 Existence of a ground state solution
In order to prove the existence of positive solutions, we claim that J_{ε }satisfies the (PS)_{β}condition in H^{1}(ℝ^{N}) for , where Λ = ε^{2(pq)/(p2) }and C_{0 }is defined in the following lemma.
Lemma 3.1 Assume that h satisfies (h1) and 0 < Λ ( = ε^{2(pq)/(p2)}) < Λ_{0}. If {u_{n}} is a (PS)_{β}sequence in H^{1}(ℝ^{N}) for J_{ε }with u_{n }⇀ u weakly in H^{1 }(ℝ^{N}), then in H^{1 }(ℝ^{N}) and , where
and
Proof. Since {u_{n}} is a (PS)_{β}sequence in H^{1}(ℝ^{N}) for J_{ε }with u_{n }⇀ u weakly in H^{1 }(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(pq)/(p2)}) < Λ_{0}, then J_{ε }satisfies the (PS)_{β}condition in H^{1}(ℝ^{N}) for .
Proof. Let {u_{n}} be a (PS)_{β}sequence in H^{1}(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 H^{1}(ℝ^{N}). Hence, there exist a subsequence {u_{n}} and a nonnegative u ∈ H^{1 }(ℝ^{N}) such that in H^{1 }(ℝ^{N}), u_{n }⇀ u weakly in H^{1 }(ℝ^{N}), u_{n }⇀ u a.e. in ℝ^{N}, u_{n }⇀ u strongly in for any 1 ≤ s < 2*. Using the BrézisLieb 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 H^{1 }(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 H^{1}(ℝ^{N}).
Remark 3.3 By Lemma 1.1, we obtain
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 u_{0 }in ℝ^{N}.
Theorem 3.4 Under assumptions (f1), (h1), if 0 < Λ ( = ε^{2(pq)/(p2)}) < Λ_{0}, then there exists at least one positive ground state solution u_{0 }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 u_{0 }∈ H^{1 }(ℝ^{N}) such that u_{n }→ u_{0 }strongly in H^{1 }(ℝ^{N}). It is easy to see that is a solution of Equation (E_{ε}) in ℝ^{N }and J_{ε}(u_{0}) = 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 u_{0 }is a positive solution of Equation (E_{ε}) in ℝ^{N}.
4 Existence of k + 1 solutions
From now, we assume that f and h satisfy (f1)(f2) and (h1). Let w ∈ H^{1 }(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]).
(ii) for any ε > 0, there exist positive numbers C_{1}, , 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 t_{0 }> 0 such that for 0 ≤ t ≤t_{0 }and any ε > 0, we have that
(ii) There exist positive numbers t_{1 }and ε_{1 }such that for any t > t_{1 }and ε < ε_{1}, we have that
Proof. (i) Since J_{ε }is continuous in is uniformly bounded in H^{1 }(ℝ^{N}) for any ε > 0, and γ_{max }> 0, there is t_{0 }> 0 such that for 0 ≤ t ≤ t_{0 }and any ε > 0
(ii) There is an r_{0 }> 0 such that f (z) ≥ f_{max}/2 for z ∈ B^{N }(a^{i}; r_{0}) uniformly in i. Then there exists ε_{1 }> 0 such that for ε < ε_{1}
Thus, there is t_{1 }>0 such that for any t > t_{1 }and ε < ε_{1}
Lemma 4.2 Under assumptions (f1), (f2), and (h1). If 0 < Λ ( = ε^{2(pq)/(p2)}) < q Λ_{0}/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 t_{0 }≤ t ≤ t_{1}, 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 ε_{0 }> 0 such that
Choosing 0 < ρ_{0 }< 1 such that
where and f(a^{i}) = f_{max}. Define K = {a^{i } 1 ≤ i ≤ k} and . Suppose for some r_{0 }> 0.
Let Q_{ε }: H^{1 }(ℝ^{N}) \ {0} → ℝ^{N }be given by
where χ : ℝ^{N }→ ℝ^{N}, χ (z) = z for z ≤ r_{0 }and χ (z) = r_{0}z/z for z > r_{0}.
Lemma 4.3 There exists 0 < ε^{0 }≤ ε_{0 }such that if ε < ε^{0}, then for each 1 ≤ i ≤ k.
Proof. Since
there exists ε^{0 }> 0 such that
Lemma 4.4 There exists a number such that if u ∈ N_{ε }and , then for any 0 < ε < ε^{0}.
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 H^{1 }(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 concentrationcompactness principle (see Lions [17] or Wang [[12], Lemma 2.16]), then there exist a constant d_{0 }> 0 and a sequence such that
Let , there are a subsequence {v_{n}} and v ∈ H^{1 }(ℝ^{N}) such that v_{n }⇀ v weakly in H^{1 }(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 H^{1 }(ℝ^{N}). By (4.2), then . Moreover, we can obtain that strongly in H^{1 }(ℝ^{N}) and . Now, we want to show that there exists a subsequence such that z_{n }→ z_{0 }∈ 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 z_{0 }∈ K. On the contrary, assume that z_{0 }∉ K, that is, f(z_{0}) < f_{max}. Then using the above argument to obtain that
which is a contradiction. Since v_{n }→ v ≠ 0 in H^{1 }(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 < ε^{0}.
From (4.1), choosing such that
For each 1 ≤ i ≤ k, define
Lemma 4.5 If and J_{ε }(u) ≤ γ_{max }+ δ_{0}/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 c_{1 }and c_{2 }(independent of u). Next, we claim that for some constant c_{3 }> 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
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) ⊂ H^{1}(ℝ^{N}) → ℝ^{+ }such that for any v ∈ B(0;η) and
Proof. See Cao and Zhou [7].
Lemma 4.7 For each 1 ≤ i ≤ k, there is a sequence in H^{1}(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), (f_{2}), 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 H^{1}(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}.
Competing interests
The author declares that he has no competing interests.
Acknowledgements
The author was grateful for the referee's helpful suggestions and comments.
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