Multiplicity and existence of solutions for generalized quasilinear Schrödinger equations with sign-changing potentials

<jats:p>We consider a class of generalized quasilinear Schrödinger equations <jats:disp-formula><jats:alternatives><jats:tex-math>$$ -\operatorname{div}\bigl(l^{2}(u)\nabla u\bigr)+l(u)l'(u) \vert \nabla u \vert ^{2}+V(x)u= f(u),\quad x\in \mathbb{R}^{N}, $$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>−</mml:mo><mml:mo>div</mml:mo><mml:mrow><mml:mo>(</mml:mo><mml:msup><mml:mi>l</mml:mi><mml:mn>2</mml:mn></mml:msup><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo><mml:mi>∇</mml:mi><mml:mi>u</mml:mi><mml:mo>)</mml:mo></mml:mrow><mml:mo>+</mml:mo><mml:mi>l</mml:mi><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo><mml:msup><mml:mi>l</mml:mi><mml:mo>′</mml:mo></mml:msup><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo><mml:mo>|</mml:mo><mml:mi>∇</mml:mi><mml:mi>u</mml:mi><mml:msup><mml:mo>|</mml:mo><mml:mn>2</mml:mn></mml:msup><mml:mo>+</mml:mo><mml:mi>V</mml:mi><mml:mo>(</mml:mo><mml:mi>x</mml:mi><mml:mo>)</mml:mo><mml:mi>u</mml:mi><mml:mo>=</mml:mo><mml:mi>f</mml:mi><mml:mo>(</mml:mo><mml:mi>u</mml:mi><mml:mo>)</mml:mo><mml:mo>,</mml:mo><mml:mspace /><mml:mi>x</mml:mi><mml:mo>∈</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mi>N</mml:mi></mml:msup><mml:mo>,</mml:mo></mml:math></jats:alternatives></jats:disp-formula> where <jats:inline-formula><jats:alternatives><jats:tex-math>$l(t): \mathbb{R}\to\mathbb{R}^{+}$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mi>l</mml:mi><mml:mo>(</mml:mo><mml:mi>t</mml:mi><mml:mo>)</mml:mo><mml:mo>:</mml:mo><mml:mi>R</mml:mi><mml:mo>→</mml:mo><mml:msup><mml:mi>R</mml:mi><mml:mo>+</mml:mo></mml:msup></mml:math></jats:alternatives></jats:inline-formula> is a nondecreasing function with respect to <jats:inline-formula><jats:alternatives><jats:tex-math>$|t|$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>|</mml:mo><mml:mi>t</mml:mi><mml:mo>|</mml:mo></mml:math></jats:alternatives></jats:inline-formula>, the potential function <jats:italic>V</jats:italic> is allowed to be sign-changing so that the Schrödinger operator <jats:inline-formula><jats:alternatives><jats:tex-math>$-\Delta+V$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mo>−</mml:mo><mml:mi>Δ</mml:mi><mml:mo>+</mml:mo><mml:mi>V</mml:mi></mml:math></jats:alternatives></jats:inline-formula> possesses a finite-dimensional negative space. We obtain existence and multiplicity results for the problem via the Symmetric Mountain Pass Theorem and Morse theory.</jats:p>


Introduction
In our article, we study the generalized quasilinear Schrödinger problem as follows: div l 2 (u)∇u + l(u)l (u)|∇u| 2 + V (x)u = f (u), x ∈ R N , (1.1) where N ≥ 1, f ∈ C(R, R) and the function l satisfies the following assumptions: (l) l ∈ C 2 (R, R + ), l(t) = l(-t), l(0) = 1, l (t) ≥ 0 for all t ≥ 0, tl (t) ≤ l(t) for all t ∈ R and l (t) ≥ 0 is strict on a subset of positive measure in R. Solutions of (1.1) are related to the solitary wave solutions for the following quasilinear Schrödinger equations: i∂ t z =z + W (x)zf (z)g |z| 2 g |z| 2 z, (1.2) where z : R N × R → C, W : R N → R is a given potential, and f , g are real functions. The above problem (1.2) has been studied in several areas of physics corresponding to various types of g. For example, the case g(t) = t was used in [9] for the superfluid film equation in plasma physics. If g(t) = (1 + t) 1/2 , equation (1.2) models the self-channeling of a highpower ultrashort laser in matter (see [2] and [3]). Equation (1.2) also has relations with condensed matter theory (see [12]). Taking z(x, t) = exp(-iEt)u(x) in (1.2), with E > 0, we are led to investigate the following elliptic equation: with V (x) = W (x) -E. If we choose l 2 (u) = 1 + ((g(u 2 )) ) 2 2 , then (1.3) turns into (1.1). In particular, if g(t) = t, we have For equation (1.4), to the best of our knowledge, the first results were proved by Poppenberg, Schmitt, and Wang in [13]. The idea in [13] is a constrained minimization argument. Subsequently, a general existence result for (1.4) was derived by Liu, Wang, and Wang [10]. The main existence results were obtained, through making a change of variable, reducing the quasilinear problem (1.4) to a semilinear one, and an Orlicz space framework was used to prove the existence of a positive solutions via Mountain Pass Theorem. The same method of variable change was also used by Colin and Jeanjean in [4], but the usual Sobolev space H 1 (R N ) framework was chosen as the working space. We refer the readers to [5,6,11,15,17,18,20,22,23] for more results.
In all these papers, it is required that the potential V satisfies the positivity condition With this assumption and suitable conditions on the nonlinearity f , one can show that u = 0 is a local minimizer of the energy functional associated with (1.4), which would then verify the mountain pass geometry and so Mountain Pass Theorem can be applied to produce a solution. However, from V (x) = W (x) -E we can see that, if the frequency E is large, then the potential V (x) in (1.4) could not satisfy (1.5).
In the literature (see [7]), there are some existence results which allow the potential V to be negative somewhere. The strategy is to write is still a norm on the function space. This is the key to verify that 0 is a local minimizer of the corresponding energy functional.
Recently, Shi and Chen [19] studied problem (1.1) with a sign-changing potential V . Compared with [7], (1.6) fails to be a norm any more. To overcome this difficulty, the authors chose a constant V 0 > 0 satisfying and considered the equivalent problem div l 2 (u)∇u + l(u)l (u)|∇u| 2 in their condition (f 3 ), we cannot ensure Here we give an example to indicate why (1.7) fails. Consider where It is easy to find that f (t) and l(t) satisfy conditions (f 1 )-(f 4 ) (see [19]) and (l), respectively. Then we denote Due to this implies that, for some M 1, there exist T 1 , Since L(t) is continuous on [T 1 , T 2 ], there exists K > 0 such that Thus, for V 0 ≥ 4K M , we have Therefore, unlike what the authors declared at the beginning of [19], this new nonlinearity f (t) does not satisfy their condition (f 3 ) any more. For this reason, their result may be valid for the case when the potential V is positive.
To the best of our knowledge, up to now in the literature there is only one research paper devoted to the situation that the quasilinear Schrödinger problems with "strongly" sign-changing potential. Recently, S. Liu et al. [11] proved the multiplicity of solutions of where V is a sign-changing potential.
Our results extend and modify those obtained by S. Liu et al. [11] and H. Shi et al. [19]. Inspired by [11], we now present our hypotheses on the potential V and the nonlinearity f : and there exist C 1 , C 2 > 0 such that for all t ∈ R, p ∈ (2, 2 * ), We now summarize our main results: Remark 1.1 From (V 1 ), the potential V (x) is allowed to be sign-changing.
we can easily obtain for some constants C 3 , C 4 , C 5 > 0. This paper is organized as follows. In Sect. 2, we describe the main preliminaries which we will use in this paper. Theorems 1.1 and 1.2 are proved in Sect. 3 and Sect. 4, respectively.
Notation. In this paper we use the following notations: • |u| s = ( R N |u| s dx) 1/s denotes the usual norm in L s -space.
• We denote the weak and strong convergence in X, as n → ∞, by u n u and u n → u, respectively.

Preliminaries
Since V (x) is bounded from below, there exists V 0 > 0 satisfying We now introduce the working space. Set which is a Hilbert space with the inner product and the corresponding norm Applying the spectral theory of self-adjoint compact operators, let be the sequence of eigenvalues of where each eigenvalue is repeated according to its multiplicity, and let e 1 , e 2 , . . . be the corresponding orthonormal eigenfunctions in L 2 (R N ). Problem (1.1) is the Euler-Lagrange equation of the following energy functional: But I(u) may be ill-behaved in X. To overcome this difficulty, we make a change of variables introduced in Shen and Wang [16], as Firstly, we give some properties for L and L -1 which are defined in Sect. 1.

Lemma 2.1 ([19])
The functions L(t) and L -1 (s) satisfy the following properties: (1) L is odd, from class C 2 and invertible; (2) lim |s|→0 Thus, after the change of variables, we obtain the functional J(v) in the following form: which is well defined in X. By Lemma 2.1, we know that J ∈ C 1 , and the critical points of J are the weak solutions of our problem (1.1) (see [16]). Hence, to prove our main results, we should find critical points of the functional J.
Secondly, we set and rewrite J in the following form: Note that by (f 2 ), the new nonlinearity f satisfies It is easy to see that the nonlinearity f (t) does not satisfy Ambrosetti-Rabinowitz condition any more, hence the boundedness of Palais-Smale sequences seems hard to verify. For this reason, we will show the functional J satisfies the Cerami condition. Thirdly, recall that a (C) c -sequence {v n } in X at the level c is such that Then J is said to satisfy the Cerami condition if any (C) c -sequence has a convergent subsequence in X. for some c ∈ R.
Step 1. We prove that If this conclusion is not true, we can suppose ρ n → +∞. Consider the sequence {h n }, defined by Since l(t) ≥ 1, we obtain Passing to a subsequence, we may assume that Subsequently, by (2.3) and Lemma 2.1(4), we get After multiplying both sides of the above equation by ρ -2 n , for large n, we have Since h n → h in L 2 (R N ) and μ > 2, it implies that h = 0. Thus the set Θ = {x ∈ R N : h(x) = 0} has a positive Lebesgue measure. Due to our assumption (f 2 ), it implies that Noticing that F(t) ≥ 0 and by Fatou's lemma, we obtain Hence, we get which is a contradiction. Thus, we obtain Step 2. We prove that there exists a constant C 7 > 0 such that Indeed, we may assume v n ≡ 0 (otherwise, the conclusion is trivial). If (2.6) is incorrect, passing to a subsequence, we suppose ρ 2 n v n 2 → 0, as n → ∞. (2.7) Setting w n = v n v n and g n = |L -1 (v n )| 2 v n 2 , one has From (2.7), as n → ∞, we have We claim that for each ε > 0, there exists a constant C 8 > 0 independent of n, such that meas(Ω n ) < ε, where Ω n := {x ∈ R N : |v n (x)| ≥ C 8 }. If this claim is not true, there is an ε 0 > 0 and a subsequence {v n k } of {v n } such that for each positive integer k, meas({x ∈ R N : |v n k (x)| ≥ k}) ≥ ε 0 > 0. Set Ω n k := {x ∈ R N : |v n k (x)| ≥ k}. By Lemma 2.1(9), we obtain On the other hand, if |v n (x)| < C 8 , by Lemma 2.1(9)-(10), one has Therefore, there exists a constant C 10 > 0 such that By the absolutely continuity of the Lebesgue integral, there exist ε > 0 and n 0 > 0 for n > n 0 we have meas(Ω n ) < ε and Ω n V (x)w 2 n dx < 1 2 . For this ε, taking n → +∞, we get (2.10) From (2.8), (2.9), and (2.10), we get a contradictory inequality 1 ≤ 1 2 . Thus, summing up the above arguments, we prove that the Cerami sequence {v n } in (2.4) is bounded in X.
Step 3. We prove that v n → v in X. From the boundedness of the sequence {v n } and the compactness of the embedding X → → L s (R N ), up to subsequence, we may assume v n v in X and v n → v in L s R N for s ∈ [2, 2 * ).
By the growth condition (f 1 ), the properties of L -1 described in Lemma 2.1 and Hölder inequality, we have On the other hand, we claim that there exists C 12 > 0 such that There is no harm in supposing v n ≡ v (otherwise, the conclusion is trivial). Denote .
If (2.11) is not true, we can assume that By Lemma 2.1 (8), this implies d ds Therefore, we deduce that d n (x) is positive and By the argument of proving Lemma 3.11 in [8], we can obtain a contradiction.

Consequently,
We deduce that v n → v in X.

Proof of Theorem 1.1
Since 0 is not an eigenvalue of we can assume that there exists an integer d ≥ 0 such that 0 ∈ (λ d , λ d+1 ). For d ≥ 1, we denote Specially, if d = 0, we set X -= {0} and X + = X. Then Xand X + are the negative and positive spaces of the quadratic form respectively. Furthermore, for some η > 0 we get for all v, φ, ψ ∈ X, in particular, since L -1 (0) = 0, l(0) = 1 and |l (t)| ≤ C 5 , we have Applying Taylor's formula, we have To prove Theorem 1.1, we will apply the following Symmetric Mountain Pass Theorem due to Ambrosetti-Rabinowitz [14].
Proof For any {v n } ⊂ W with v n → +∞, consider a n = v n v n .
Then {a n } is a bounded sequence in W . Since dim W < ∞, there exists a ∈ W \ {0} such that a n → a in W , a n → a a.e. on R N .
Proof By condition (f 1 ), there exist positive constants C 1 and C 2 such that Then, similar to Lemma 3.8 in [21], we have the following fact: Let Y = span{e 1 , . . . , e k-1 } and Z k = span{e k , e k+1 , . . . }, where k > d and k will be determined, then Z k ⊂ X + . For v ∈ Z k and v small enough, using (3.1) and Taylor's expansion, we have where we used p > 2. Noticing that β k → 0 as k → ∞, we then take k large enough such that η - U is a neighborhood of u ∈ K, where u is an isolated critical point of J with J(u) = c. Then the qth critical group of J at an isolated critical point u is defined by where H q (·, ·) is a qth singular relative homology group with integer coefficients. If J satisfies the Cerami condition and a < inf u∈K J(u), then the critical groups of J at infinity are defined by In Morse theory, the functional J is always required to satisfy the so-called deformation condition.

Definition 4.1
The functional J satisfies deformation condition if for every ε > 0 small enough, c ∈ R and any neighborhood N of K c , there is a continuous deformation η : We note that if the functional J satisfies the (PS)-condition or the Cerami condition, then J satisfies the deformation condition.
Morse theory tells us that if J satisfies the Cerami condition, v = 0 is a critical point of J and K = {0}, then C q (J, ∞) ∼ = C q (J, 0) for all q ∈ N. It follows that if C q (J, ∞) C q (J, 0) for some q ∈ N then J must have a nontrivial critical point. So one has to compute these groups to get the nontrivial critical point.

Critical groups at zero
In this section, we will use the following proposition to compute the critical groups of J at zero.

Proposition 4.1
Suppose J ∈ C 1 (X, R) has a local linking at zero with respect to the decomposition X = Y ⊕ Z, i.e., for some ε > 0. Proof From (f 1 ) and (f 3 ), for all ε > 0, there exists C ε > 0, such that Hence, we get Using this and (3.2), we obtain From this and (3.1), one obtains that J has a local linking property at zero. Then it follows from Proposition 4.1 that C d (J, 0) 0. Proof Let S ∞ be the unit sphere in X. Firstly, we will establish the following fact:

Critical groups at infinity
Due to |L -1 (sw)| ≤ |sw| and (f 2 ), we deduce Secondly, we will show that the following claim is true.
Claim There exists A > 0 such that if J(v) ≤ -A then d dt t=1 J(tv) < 0.
If this claim is false, there exists a sequence {v n } ⊂ X such that J(v n ) ≤ -n and J (v n ), v n = d dt t=1 J(tv n ) ≥ 0.
By the same argument as in Lemma 2.2, we deduce By the similar arguments of Lemma 2.2, we obtain ρ n → +∞ and {h n } is a bounded sequence in X.
Then h n h in X; h n → h a.e. on R N .
Multiplying both sides of (4.5) by ρ -2 n , we obtain h = 0. By the assumption J (v n ), v n ≥ 0, we have Proof of Theorem 1. 2 We have verified that J satisfies the Cerami condition. By Lemma 4.1, J has a local linking at zero with respect to the decomposition X = X -⊕ X + , hence, by Proposition 4.1, for d = dim X -, we have C d (J, 0) = 0.
On the other hand, Lemma 4.2 says that for all q ∈ N, C q (J, ∞) = 0. Hence, J has a nontrivial critical point v. Now u = L -1 (v) is a nontrivial solution of problem (1.1).