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Multiple positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in ℝN
Boundary Value Problems volume 2012, Article number: 24 (2012)
Abstract
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.
1 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 ℝNand lim|z| → ∞f(z) = f∞ > 0.
(f 2) there exist k points a1, a2,..., akin ℝNsuch that
and f∞ < fmax.
(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 ℝN. For the case q = λ = 1 and f(z) ≡ 1 for all z ∈ ℝN, suppose that h is nonnegative, small, and exponential decay, Zhu [6] showed that Equation (E λ ) admits at least two positive solutions in ℝN. 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 ℝN. 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 ℝN, 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)vp-1+ λh(z) vq-1in ℝNfor 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 (ℝN),
where is the norm in H1 (ℝN) 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 (f 2) 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 (ℝN) \ {0} | u+ ≢ 0 and }. Note that
(i) if f ≡ f∞, we define and
where N∞ = {u ∈ H1 (ℝN) \ {0} | u+ ≢ 0 and };
(ii) if f ≡ fmax, we define and
where Nmax = {u ∈ H1 (ℝN) \ {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 = ε2(p-q)/(p-2). Under assumptions (f 1) and (h 1), 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 (f 1) - (f 2) and (h 1), 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 Palais-Smale (denoted by (PS)) sequences and (PS)-conditions in H1(ℝN) for some functional J.
Definition 2.1 (i) For β ∈ ℝ, a sequence {u n } is a (PS) β -sequence in H1(ℝN) for J if J(u n ) = β + o n (1) and J'(u n ) = o n (1) strongly in H-1 (ℝN) as n → ∞, where H-1 (ℝN) is the dual space of H1(ℝN);
(ii) J satisfies the (PS) β -condition in H1(ℝN) if every (PS) β -sequence in H1(ℝN) for J contains a convergent subsequence.
Next, since J ε is not bounded from below in H1 (ℝN), 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 (f 1) and (h 1), if 0 < Λ (= ε2(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 (ℝN).
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∥ # , fmax).
For u ∈ H1 (ℝN) \ {0} and u + ≢ 0, let
Lemma 2.6 For each u ∈ H1 (ℝN)\ {0} and u + ≢ 0, we have that
(i) if , then there exists a unique positive number such that and J ε (t-u) = supt ≥ 0J ε (tu);
(ii) if 0 < Λ ( = ε2(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 ( = ε2(p-q)/(p- 2)) < Λ0, then ;
(ii) If 0 < Λ < q Λ0/2, then for some constant d0 = d0 (ε, p, q, S, ∥h∥ # , fmax).
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 .
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 H1(ℝN) for , where Λ = ε2(p-q)/(p- 2)and C0 is defined in the following lemma.
Lemma 3.1 Assume that h satisfies (h 1) and 0 < Λ ( = ε2(p-q)/(p- 2)) < Λ0. If {u n } is a (PS) β -sequence in H1(ℝN) for J ε with u n ⇀ u weakly in H1 (ℝN), then in H-1 (ℝN) and , where
and
Proof. Since {u n } is a (PS) β -sequence in H1(ℝN) for J ε with u n ⇀ u weakly in H1 (ℝN), it is easy to check that in H-1(ℝN) and u ≥ 0. Then we have , that is, . Hence, by the Young inequality
Lemma 3.2 Assume that f and h satisfy (f 1) and (h 1). If 0 < Λ ( = ε2(p-q)/(p- 2)) < Λ0, then J ε satisfies the (PS) β -condition in H1(ℝN) for .
Proof. Let {u n } be a (PS) β -sequence in H1(ℝN) for J ε such that J ε (u n ) = β + o n (1) and (1) in H-1(ℝN). Then
where c n = o n (1), d n = o n (1) as n → ∞. It follows that {u n } is bounded in H1(ℝN). Hence, there exist a subsequence {u n } and a nonnegative u ∈ H1 (ℝN) such that in H-1 (ℝN), u n ⇀ u weakly in H1 (ℝN), u n ⇀ u a.e. in ℝN, 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 (f 1) and u n → u in , we get that
Let p n = u n - u. Suppose p n ↛ 0 strongly in H1 (ℝN). 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(ℝN).
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 ℝN.
Theorem 3.4 Under assumptions (f 1), (h 1), if 0 < Λ ( = ε2(p-q)/(p- 2)) < Λ0, then there exists at least one positive ground state solution u0 of Equation (E ε ) in ℝN. 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 (ℝN). Since , by Lemma 3.2, there exist a subsequence {u n } and u0 ∈ H1 (ℝN) such that u n → u0 strongly in H1 (ℝN). It is easy to see that is a solution of Equation (E ε ) in ℝNand 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 ℝN.
4 Existence of k+ 1 solutions
From now, we assume that f and h satisfy (f 1)-(f 2) and (h 1). Let w ∈ H1 (ℝN) be the unique, radially symmetric, and positive ground state solution of Equation (E 0) in ℝNfor f = fmax. 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 ∈ ℝN
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 (ℝN) for any ε > 0, and γmax > 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) ≥ fmax/2 for z ∈ BN(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 (f 1), (f 2), and (h 1). If 0 < Λ ( = ε2(p-q)/(p- 2)) < q Λ0/ 2, then
Proof. By Lemma 4.1, we only need to show that
We know that supt ≥0Imax (tw) = γmax. 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 γmax < γ∞, there exists ε0 > 0 such that
Choosing 0 < ρ0 < 1 such that
where and f(ai) = fmax. Define K = {ai| 1 ≤ i ≤ k} and . Suppose for some r0 > 0.
Let Q ε : H1 (ℝN) \ {0} → ℝNbe given by
where χ : ℝN→ ℝN, 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 (ℝN). Suppose u n → 0 strongly in Lp(ℝN). Since
and
then
which is a contradiction. Thus, u n ↛ 0 strongly in Lp(ℝN). 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 (ℝN) such that v n ⇀ v weakly in H1 (ℝN). 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 (ℝN). By (4.2), then . Moreover, we can obtain that strongly in H1 (ℝN) 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 ℝN. 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) < fmax. Then using the above argument to obtain that
which is a contradiction. Since v n → v ≠ 0 in H1 (ℝN), 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) ≤ γmax + 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(ℝN) → 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(ℝN) 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 (ℝN) as n → ∞.
Theorem 4.8 Under assumptions (f 1), (f2), and (h 1), there exists a positive number λ*(λ* = (ε*)-2) such that for λ > λ*, Equation (E λ ) has k + 1 positive solutions in ℝN.
Proof. We know that there is a -sequence in H1(ℝN) 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 ℝN. Applying the maximum principle and Theorem 3.4, Equation (E λ ) has k + 1 positive solutions in ℝN.
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Lin, Hl. Multiple positive solutions of semilinear elliptic equations involving concave and convex nonlinearities in ℝN. Bound Value Probl 2012, 24 (2012). https://doi.org/10.1186/1687-2770-2012-24
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DOI: https://doi.org/10.1186/1687-2770-2012-24