Multiple positive solutions of saturable nonlinear Schrödinger equations with intensity functions

In this paper, we study saturable nonlinear Schrodinger equations with nonzero intensity function which makes the nonlinearity become not superlinear near zero. Using the Nehari manifold and the Lusternik-Schnirelman category, we prove the existence of multiple positive solutions for saturable nonlinear Schrodinger equations with nonzero intensity function which satisfies suitable conditions. The ideas contained here might be useful to obtain multiple positive solutions of the other non-homogeneous nonlinear elliptic equations.

To get positive solutions of equation (E), we may consider the following energy minimization problem. where . This approach was taken in [20], not in this one. Note that the Euler-Lagrange equation of problem (2) is equation (E) and λ > 0 is an eigenvalue which comes from the Lagrange multiplier of the L 2 -normalization condition R N u 2 = 1. When function I becomes zero, spatial dimension N = 2 and Γ > T 1 > 0, the existence of minimizer of problem (2) can be proved by the energy estimate method (cf. [20]), where T 1 is a positive constant. Moreover, Lin et al. [21] get the estimate of the eigenvalue λ and the minimum (ground state) energy e Γ by developing a virial theorem. Very recently, as function I becomes nonzero, Lin et al. [22] use a convexity argument to prove the existence of minimizer of problem (2) with orbital stability of equation (1) On the other hand, under the assumption I ≡ 0, equation (E) can be regarded as a perturbation problem of the nonlinear Schrödinger equation as follows: It is well known that the assumptions on the nonlinearity like g (s) = s 3 / 1 + s 2 , asymptotically linear at infinity, can be found in Berestycki and Lions [4]. The existence of the radial ground state positive solution for equation (4) was given by Stuart and Zhou [30] and the uniqueness is guaranteed by Serrini and Tang [29]. Furthermore, a more general situation has been studied by various authors who considered the following problem: where the nonlinear term f (x, s) s is asymptotically linear in s at infinity, that is, f (x, s) ∼ q(x) ∈ L ∞ R N as |s| → +∞. We refer readers to [6,7,8,14,15,16,17,19,25,26,30,31] and the references therein. In most of these papers, f (x, s)s is generally assumed to be superlinear in s near zero, that is, f (x, s) ∼ 0 as s → 0.
In addition, either q(x) is a constant or lim |x|→+∞ q(x) exists, or f (x, s) ≡ f (s) is required. For example, by using Mountain Pass Theorem suggested by Ambrosetti-Rabinowitz [3], the existence of positive solution for equation (5) was obtained in [6]. Applying the sub-supersolution method, the existence of positive solution for equation (5) was also discussed in [8] when f (x, u) ∼ p(x) ∈ L ∞ R N and q(x) ∈ L ∞ R N , respectively, as u approaches the origin and infinity, where both p(x) and q(x) are required to vanish at infinity. In fact, there are a few papers concerned with the existence of infinitely many solutions for the asymptotically linear case when f is periodic in x; e.g. see [7]. However, it is remarkable that only the existence of positive solution is considered on this problem while the multiplicity of positive solutions is not concerned so far, because of limitation of research methods. In view of this, the main purpose of this paper is to present a new approach to search for multiple positive solutions of asymptotically linear Schrödinger equations. Here we require that the nonlinear term in the equation is non-homogeneous and non-autonomous, leading to complicate the problem. By showing that all functions on an open subset of H 1 R N \{0} can be projected on the Nehari manifold N and minimizing the energy functional on an appropriate subset of N , we prove the existence and multiplicity of positive solutions for equation (E) , with the aid of the shape of the graph of I (x) and Lusternik Schnirelman category.
Here we consider the following nonlinear Schrödinger equation: where the parameter µ ≥ 0. We assume that I µ (x) = µI + (x) − I − (x) , where the functions I ± = max {±I, 0} satisfy the following conditions: (D1) I ∈ C R N and there is positive number r < 2 such that 0 ≤ I − (x) ≤ c 0 for some c 0 < 1 and for all x ∈ R N and I + (x) ≥ d 0 exp (−r |x|) for some d 0 > 0 and for all x ∈ R N ; (D2) suppI − is a bounded set with positive measure; (D3) I + (x) → 0 as |x| → ∞. Then we summarize our main result as follows.
Remark 1. Suppose that function I ∈ C R N , R satisfies lim |x|→∞ I (x) = 0 and I(x) > −1 for x ∈ R N . Then this is easy to see that equation (E) does not admit any nontrivial solution, for all λ > Γ. Indeed, if λ > Γ and u 0 is a nontrivial solution, then which is a contradiction.
For the proof of Theorem 1.1, we have to face two challenges because of condition lim s→0 f (·, s) = Γ I 1+I ≡ 0. The first one is how to prove that the Nehari manifold is a natural constraint. The second one is to show that all functions on an open subset of H 1 R N \{0} can be projected on the Nehari manifold, since the uniqueness of critical point of fibering map associated with the energy functional is not sure. In order to overcome these difficulties, we establish some accurate inequalities using two power-law nonlinearities to estimate the saturable nonlinearity with nonzero intensity function (see Lemma 2.2, 2.3), which can help us to verify that the Nehari manifold is a natural constraint with nice properties (see Lemma 2.4-2.7). Furthermore, we provide a new estimation method to prove that the center mass function is non-zero on an appropriate sublevel set of the Nehari manifold. It is worth emphasizing that the ideas contained here might be useful to obtain multiple positive solutions of the other non-homogeneous nonlinear elliptic equations.
The rest of this paper is organized as follows. After giving some notations and preliminaries in Section 2, we establish some energy estimates in Section 3. In Section 4, we prove the existence of least energy positive solutions of equation (E µ ). In Section 5, the existence of two positive solutions is obtained for µ sufficiently small.

Notations and preliminaries. Let us define the energy functional
is the standard norm in H 1 R N . It is well known that J µ ∈ C 1 H 1 R N , R and the solutions of equation (E µ ) are the critical points of the energy functional J µ in H 1 R N . Next, we define the Palais-Smale (or simply (PS)-) sequences, (PS)-values, and (PS)-conditions in H 1 R N for J µ as follows.
We establish the following estimates on the nonlinearity.
such that Proof. Let f (s) = c (x) s p − b (x) s 2 + ln 1 + s 2 a(x) for 2 < p ≤ min {4, 2 * } . Clearly, f (0) = 0 and a direct calculation shows that We take a (x) = 1 where g (s) = for all s ≥ 0. This implies that f (s) ≥ 0 and This completes the proof.
Proof. The proof is almost the same as Lemma 2.2, and we omit it here.

Remark 2.
From the proof of Proposition 2.2 in Costa-Tehrani [6], one can see that the uniqueness of critical points of fibering map h u (t) = J µ (tu) is not obvious. However, by the inequalities obtained in Lemma 2.3, we can prove that the uniqueness of critical points of h u (t) satisfying same condition as in Proposition 2.2 [6].
As the energy functional J µ is not bounded from below on H 1 R N , it is useful to consider the functional on the Nehari manifold Then we have the following results.
Proof. (i) We divide the proof into the following two cases: (i − a) For N = 1, 2, 3 and 0 < µ < µ * (N ) . By Lemma 2.2 and the Sobolev inequality, we take p = 4, where A 2 * is as in (6) . Then by the Sobolev inequality, This shows that N µ is a C 1 manifold.
(iv) The proofs of all cases N ≥ 1 are similar. So, we only prove the cases N ≥ 4. For u ∈ N µ , it follows that then by Lemma 2.3 and the Sobolev inequality, for 0 < µ < µ * (N ) This implies that Thus, This completed the proof.
Proof. Let u be a critical point of the functional J µ restricted to the manifold N µ . By the theorem of Lagrange multipliers, there exists a ζ ∈ R such that From (9), we have ζ = 0, gives J µ (u) = 0 and so u is a critical point of J µ .
Lemma 2.6. Suppose that 0 < λ < Γ and 0 < µ < µ * (N ) . Then for each u ∈ Proof. The proofs of all cases N ≥ 1 are similar. So, we only prove the cases N ≥ 4. Then dx and the arguments of (10) show that h u (t) > 0 and h u (t) > 0 for t > 0 small.
The proofs of these claims are similar and the second one is simpler. So we only prove Claim (I) (see also [30]). We have so that Lebesgue's dominated convergence theorem yields Since Iµ+s 2 1+Iµ+s 2 as a increasing function for s ≥ 0, (13) shows that h u (t) /t is a decreasing function of t and Claim (I) implies Similarly, we obtain from (12) and Claim (II) that Next, we show that t (u) ≤ t µ (u) . First, we consider the function m u : Clearly, m u (0) = m u t (u) = 0 and lim t→∞ m u (t) = −∞, where Moreover, which implies that m u is strictly increasing on (0, t (u)) and is strictly decreasing Then by Lemma 2.3, Thus, by the uniqueness of t µ (u) and m u t (u) = 0, we can conclude that t (u) ≤ t µ (u) .
(iii) By the uniqueness of t µ (u) and the extrema property of t µ (u) , we have t µ (u) is a continuous function for Then by part (iii) , Thus, This completes the proof.
Furthermore, we have the following results.
(i) The energy functional J µ is bounded below on N µ . Furthermore, there exists Proof. (i) The proofs of all cases N ≥ 1 are similar. So, we only prove the cases N ≥ 4. By Lemma 2.6 (v) , if u ∈ N µ , then Moreover, by Lemma 2.2 and the Sobolev inequality, . This implies that (ii) Similar argument to the proof of proposition 1.5 in [6]. Now we consider the following elliptic problem: We consider the energy functional J ∞ in H 1 R N associated to equation (E ∞ ) , Consider the minimizing problem: It is known that equation (E ∞ ) has a unique positive radial solution w (x) such that J ∞ (w) = α ∞ and w (0) = max x∈R N w (x) (see [13,29]). Then we have the following results.
In addition, if u n ≥ 0, then v µ ≥ 0 and w i ≥ 0 for each 1 ≤ i ≤ m.
For µ ≥ 0, we define Then, by Proposition 1, we have the following compactness result.
Then there exists a subsequence {u n } and a non-zero u µ in H 1 R N such that u n → u µ strongly in H 1 R N and J µ (u µ ) = β. Furthermore, u µ is a non-zero solution of equation (E µ ) .
3. Existence of positive solutions. Let w (x) be a positive radial solution of equation (E ∞ ) such that J ∞ (w) = α ∞ . Then by Gidas, Ni and Nirenberg [13], for any ε > 0, there exist positive numbers A ε and B 0 such that Let e ∈ S N −1 = x ∈ R N | |x| = 1 . Define w e,l (x) = w (x − le) for l ≥ 0 and e ∈ S N −1 .
Clearly, w e,l is also least energy positive solutions of equation (E ∞ ) for all l ≥ 0.
Furthermore, there is a unique t µ (w e,l ) > 0 such that t µ (w e,l ) w e,l ∈ N µ .
Proof. We have this implies that J µ (tw e,l ) → −∞ as t → ∞ for all e ∈ S N −1 . Thus, there exists t 1 > 0 such that for every l ≥ 0, J µ (tw e,l ) < α ∞ for all t ≥ t 1 and for all e ∈ S N −1 .
Moreover, by Lemma 2.6, there is a unique t µ (w e,l ) > 0 such that t µ (w e,l ) w e,l ∈ N µ . This completes the proof.
First, we establish the existence of least energy positive solutions of equation (E µ ).
Theorem 3.1. For each 0 < µ < µ * (N ) , equation (E µ ) has a least energy positive solution u µ such that Proof. By analogy with the proof of Ni and Takagi [28], one can show that by the Ekeland variational principle (see [11]), there exists a minimizing sequence {u n } ⊂ N µ such that Since inf u∈Nµ J µ (u) < α ∞ from Proposition 2 (ii) , by Lemma 2.7 and Corollary 1 there exists a subsequence {u n } and u µ ∈ N µ , a nonzero solution of equation (E µ ) , such that u n → u µ strongly in H 1 (R N ) and J µ (u µ ) = inf u∈Nµ J µ (u) .
Since J µ (u µ ) = J µ (|u µ |) and |u µ | ∈ N µ , by Lemma 2.5 and the maximum principle, we obtain u µ > 0 in R N . This completes the proof. Proof. Let w e,l be as in (18) . Then, by w e,l 2 H 1 − R N w 2 e,l dx < 0 and Lemma 2.6 (ii), there is a unique t µ (w e,l ) > 0 such that t 0 (w e,l ) w e,l ∈ N 0 for all e ∈ S N −1 and for µ = 0 that is Then and so α 0 ≥ α ∞ . Therefore, Next, we will show that for µ = 0, equation (E µ ) does not admit any positive solution u 0 such that J 0 (u 0 ) = α 0 . Suppose the contrary. Then we can assume that there exists u 0 ∈ N 0 such that J 0 (u 0 ) = α 0 . Then, by Lemma 2.6 (i) , J 0 (u 0 ) = sup t≥0 J 0 (tu 0 ) . Moreover, there is a unique t ∞ (u 0 ) > 0 such that t ∞ (u 0 ) u 0 ∈ N ∞ . Thus, which is a contradiction. This completes the proof. 4. Existence of two positive solutions. By Theorem 3.2, for µ = 0, equation (E µ ) does not admit any solution u 0 such that J 0 (u 0 ) = inf u∈N0 J 0 (u) and  Furthermore, there exists a subsequence {u n } which is (PS) α ∞ -sequence for J ∞ in H 1 R N .

TAI-CHIA LIN AND TSUNG-FANG WU
Proof. For each n, by u n 2 H 1 − Γ R N u 2 n dx < 0 and Lemma 2.6 (ii) , there is a unique t ∞ n > 0 such that t ∞ n u n ∈ N ∞ , that is We will show that there exists C 0 > 0 such that t ∞ n > C 0 for all n. Suppose the contrary. Then we may assume t n → 0 as n → ∞. Since J 0 (u n ) = α ∞ + o (1) , by Lemma 2.7, we have u n is uniformly bounded and so t n u n H 1 → 0 or J ∞ (t n u n ) → 0, and this contradicts the fact that J ∞ (t n u n ) ≥ α ∞ > 0. Thus, this implies that u n → 0 a.e. in Ω.
by the Lebesgue dominated convergence theorem, which implies that and Then by the Ekeland variational principle (see [11]) , there exists a subsequence We need the following result.
Proof. Suppose the contrary. Then there exists a sequence {u n } ⊂ N 0 such that J 0 (u) = α ∞ + o (1) and Moreover, by Lemma 4.1, there exists a subsequence {u n } which is (PS) α ∞ -sequence in H 1 R N for J ∞ . By the concentration-compactness principle (see Lions [23,24]) and the fact that α ∞ > 0, there exist a subsequence {u n } , a sequence {x n } ⊂ R N , and a positive solution w ∈ H 1 R N of equation (E ∞ ) such that Now we will show that |x n | → ∞ as n → ∞. Suppose the contrary. Then we may assume that {x n } is bounded and x n → x 0 for some x 0 ∈ R N . Thus, by (25) , which contradicts the result of Lemma 4.1: Hence we may assume xn |xn| → e 0 as n → ∞, where e 0 ∈ S N −1 . Then, by the Lebesgue dominated convergence theorem, we have , which is a contradiction. This completes the proof.
For µ > 0 and u ∈ N µ , by Lemma 2.6, there is a unique t 0 (u) > 0 such that t 0 (u) u ∈ N 0 where N 0 = N µ with µ = 0. Moreover, by the proof of Proposition 2, there exist positive numbers t µ (w e,l ) and l 1 such that t µ (w e,l ) w e,l ∈ N µ and J µ (t µ (w e,l ) w e,l ) < α ∞ for all l > l 1 .
Then we have the following result.
There exists a positive number µ 0 ≤ µ * (N ) such that for every µ ∈ (0, µ 0 ) , we have Proof. Let u ∈ N µ with J µ (u) < α ∞ . Then, by Lemma 2.6, there exists t 0 (u) > 0 such that t 0 (u) u ∈ N 0 . Moreover, We have that Let ξ 0 > 0 be as in Lemma 4.2. Then there exists a positive number µ 0 ≤ µ * (N ) such that for µ ∈ (0, µ 0 ) , Since t 0 (u) u ∈ N 0 and t 0 (u) > 0, by Lemma 4.2 and (27) R N x |x| |∇ (t 0 (u) u)| 2 + (t 0 (u) u) 2 dx = 0, which implies that there exists a positive number µ 0 such that for every µ ∈ (0, µ 0 ) , In the following, we use an idea of Adachi and Tanaka [1]. For c ∈ R + , we define We then try to show that for a sufficiently small σ > 0, To prove (28) , we need some preliminaries. Recall the definition of the Lusternik-Schnirelman category. When there do not exist finitely many closed subsets Y 1 , ..., Y k ⊂ X such that Y j is contractible to a point in X for all j and k ∪ j=1 Y j = X, we say that cat (X) = ∞.
We need the following two lemmas. Lemma 4.5. Suppose that X is a Hilbert manifold and F ∈ C 1 (X, R) . Assume that there exist c 0 ∈ R and k ∈ N such that (i) F (x) satisfies the Palais-Smale condition for energy levels c ≤ c 0 ; We have the following results.
For l > l 1 , we may define a map Φ µ,l : where t µ (w 0 (x − le)) w 0 (x − le) is as in the proof of Proposition 2. Then we have the following result.
We are now ready to prove Theorem 1.1: Theorem 1.1 can be obtained directly from Theorems 3.1, 4.9.

5.
Conclusion. To obtain multiple positive solutions of saturable nonlinear Schrödinger equations with nonzero intensity function, we use two power-law nonlinearities to estimate the saturable nonlinearity with nonzero intensity function and derive some accurate inequalities (see Lemmas 2.2, 2.3) which can be used to verify that the Nehari manifold is a natural constraint with nice properties (see Lemmas 2.4-2.7). Furthermore, we provide a new estimation method to prove that the center mass function is non-zero on an appropriate sublevel set of the Nehari manifold.