Existence results for quasilinear Schrödinger equations with a general nonlinearity

Consider the quasilinear Schrodinger equation \begin{document}$ \begin{equation*} \label{eq0.1}-\Delta u+V(x)u- \Delta(u^2)u = h(u)\ \ \mbox{in}\ {\mathbb{R}}^N,\tag{A} \end{equation*} $\end{document} where \begin{document}$ N\geq 3 $\end{document} , \begin{document}$ V: {\mathbb{R}}^N\to{\mathbb{R}} $\end{document} and \begin{document}$ h: {\mathbb{R}}\to{\mathbb{R}} $\end{document} are functions. Under some general assumptions on \begin{document}$ V $\end{document} and \begin{document}$ h $\end{document} , we establish two existence results for problem (A) by using variational methods. The main novelty is that, unlike most other papers on this problem, we do not assume the nonlinear term to be 4-superlinear at infinity.


Introduction and main results. Consider the quasilinear elliptic equation
− ∆u + V (x)u − ∆(u 2 )u = |u| p−2 u in R N , (1.1) where N ≥ 3, 2 < p < 2 · 2 * with 2 * = 2N N −2 being the critical Sobolev exponent, and V : R N → R is a continuous function. It is known that, via the ansatz ψ(t, x) = e −iωt u(x), weak solutions of problem (1.1) correspond to stationary waves of the time-dependent quasilinear Schrödinger equation where W (x) = V (x) + ω is a new potential function. Quasilinear Schrödinger equations like (1.2) arise in plasma physics and condensed matter theory, see [19,20,28] and references therein for more details on the physical background of (1.2). The natural energy functional corresponding to problem (1.1) is given by which is not well defined for all u ∈ H 1 (R N ). Due to this fact, the standard variational methods cannot be applied directly to problem (1.1). This difficulty makes problems like (1.1) more interesting and challenging. Indeed, during the last twenty years, a considerable amount of research is devoted to studying (1.1) and related problems. Many existence and multiplicity results were proved by using different approaches, such as minimizations [24,29], change of variables [8,11,22], Nehari method [23] and perturbation method [25,27]. In [22], by a suitable change of variables, problem (1.1) was reduced to a semilinear elliptic equation and existence results were proved under four different types of potentials in an Orlicz space framework. Using a similar change of variables, Colin and Jeanjean [8]  h(s) ds ≤ h(t)t for all t ∈ R + , an existence result was also obtained for problem (1.1) with well potential and the power nonlinearity |u| p−2 u replaced by h. As is well known, (AR) is used to guarantee the boundedness of Palais-Smale sequences for the energy functional.
It is worth pointing out that most of these aforementioned results are based on the condition 4 ≤ p < 2 · 2 * . As observed in [23], the number 2 · 2 * behaves as a critical exponent for problem (1.1). As a matter of fact, nonexistence result of (1.1) can be proved by using a Pohožaev type identity in the case where p ≥ 2 · 2 * and ∇V (x) · x ≥ 0 for all x ∈ R N .
To the best of our knowledge, few results are known about problem (1.1) with 2 < p < 4 and we are only aware of the papers [4,8,12,18,24,29,30]. In [24,29], an unknown Lagrange multiplier appears in the equation. In [8], an existence result was established for problem (1.1) with positive constant potential. In [12], Gloss dealt with a semiclassical problem related to (1.1) and she proved that there exists a positive semiclassical solution which concentrates at a local minimum of the potential. Recently, Ruiz and Siciliano [30] proved the existence of a positive ground state solution of problem (1.1) with 2 < p < 2 · 2 * . The proof relies on a constrained minimization procedure. We remark that a concavity hypothesis was imposed on the potential, which is technique and important in their arguments. In [4,18], the authors investigated the quasilinear elliptic equation where Ω is a bounded domain in R N and λ is a positive parameter. Existence of multiple solutions were established provided that λ is sufficiently large. For more results related to (1.1), we refer the reader to [1,3,9,10,13,14,21,26] and references therein.
Motivated by [4,8,18,30], we are interested in the quasilinear Schrödinger equation where N ≥ 3 and V ∈ C 1 (R N , R) satisfies the following conditions: For the nonlinear term h, we assume that: t = +∞. The first result of this paper is the following theorem.
has at least a positive solution.
The second part of this paper is motivated by the celebrated paper of Berestycki and Lions [6], where the authors proved the existence of a ground state solution for semilinear elliptic equations under some general conditions on the nonlinearity. In particular, a weak subcritical condition is assumed and superlinear condition (h 3 ) is not required. A natural question is that whether or not we can obtain existence result of positive solutions for problem (1.3) under similar assumptions on the nonlinear term as in [6]. In the next theorem, we give an affirmative answer to such an interesting question. For this, we make the following assumptions: The second result of this paper is stated as follows. To analyze the assumptions in the current paper and to compare Theorems 1.1 and 1.2 with results in the literature, some remarks are in order.

Remark 1.
(1) In Theorem 1.1, the potential V is assumed to be well-shaped. In this case, we are allowed to use concentration compactness arguments. However, such an argument does not work for Theorem 1.2 since we do not impose similar geometrical hypothesis on the potential V . Therefore, a symmetric property of the potential V is required in Theorem 1.2.
(2) Condition (V 2 ) shall be used to prove the boundedness of a special Palais-Smale sequence for the energy functional. Similar conditions can be found in [5,17], where semilinear elliptic equations were considered. It should be mentioned that, due to the well properties of the transformation (see Lemma 2.1 and Corollary 1), we only need a weaker condition than that in [17].
(3) As described before, Ambrosetti-Rabinowitz type condition is used to guarantee the boundedness of Palais-Smale sequences for the energy functional. Nevertheless, from the viewpoint of the celebrated paper [6], it seems that Ambrosetti-Rabinowitz type condition is not essential for the existence of a nontrivial solution. In Theorems 1.1 and 1.2, (h 3 ) and (h 5 ) are both weaker than Ambrosetti-Rabinowitz type condition. We also remark that, if V ∞ := lim |x|→∞ V (x) exists, then (h 5 ) can be replaced by a weaker condition (h 5 ) there exists a positive number ζ such that H(ζ) > V∞ 2 ζ 2 .

Remark 2.
(1) In the case V ≡ V ∞ , namely when problem (1.3) is autonomous, Theorems 1.1 and 1.2 are covered by [8,Theorem 1.2]. In this paper, we are interested in non-autonomous problem. Therefore, we assume without loss of generality that V ≡ V ∞ throughout this paper.
(2) Compared to [4,18,30], the current paper investigates problem (1.3) with a more general nonlinearity and conditions on the potential are somewhat different. Especially, there is no parameter in the equation and we do not need a concavity hypothesis on the potential which is essential in [30]. Thus, Theorems 1.1 and 1.2 can be seen as complements of [4,Theorem 4 (3) Since positive solutions are of particular interest, we always assume without restriction that h(t) = 0 for t ≤ 0 in the current paper.
To prove Theorems 1.1 and 1.2, we will face several difficulties. On one hand, due to the presence of quasilinear term ∆(u 2 )u and growth condition on the nonlinearity, the natural energy functional related to problem (1.3) is not well defined for all functions in H 1 (R N ). Therefore, we can not apply standard variational methods directly. To overcome this difficulty, we will employ an argument developed in [8,22] and make a change of variables to reformulate the quasilinear problem to a semilinear one.
On the other hand, it will be shown that the functional I associated with equivalent semilinear problem possesses the mountain pass geometry (see Lemmas 3.2 and 4.1) and so there exists a Palais-Smale sequence for I. However, the boundedness of Palais-Smale sequence seems hard to verify. Our strategy is applying Jeanjean's monotonicity method to find a special Palais-Smale sequence for the functional I. The procedure consists of the following three steps. Firstly, we define a family {I λ } λ∈J of C 1 -functionals such that I 1 = I. By an abstract result of Jeanjean, for almost every λ ∈ J , there is a bounded Palais-Smale sequence for the functional I λ . Secondly, using a version of global compactness lemma due to Adachi and Watanabe when V is a well potential or restricting in the subspace of radially symmetric functions if V is radially symmetric, we obtain a nontrivial critical point v λ of the functional I λ for almost every λ ∈ J . Finally, choosing a sequence {λ n } of numbers satisfying lim n→∞ λ n = 1, we obtain a sequence {v λn } of functions with v λn being a nontrivial critical point of the functional I λn . Then, with the help of Pohožaev type identity and condition (V 2 ), we prove that {v λn } is indeed a bounded Palais-Smale sequence for the functional I.
The paper is organized as follows. In Section 2, following the method in [8,22], we reformulate (1.3) to a semilinear problem. Sections 3 and 4 are devoted to the proofs of Theorems 1.1 and 1.2, respectively.
In the sequel, we will use the following notations. The letters C and C j stand for positive constants which may take different values at different places. The standard norms of L p (R N ) and H 1 (R N ) are denoted by | · | p and · respectively. We also set H 1 r (R N ) = u ∈ H 1 (R N ) | u is radially symmetric .

2.
Preliminaries. The natural energy functional related to problem (1.3) is defined by which is not well defined for all u ∈ H 1 (R N ) as mentioned before. To apply variational methods, we employ an argument developed in [8,22] and make a change of variables.
Let f : R → R be defined by Then the function f is uniquely defined, smooth and invertible. In the next lemma, we summarize some important properties of f which have been proved in [8,10,22].
The function f has the following properties: As a consequence of Lemma 2.1, we have Setting v = f −1 (u), we obtain which is well defined in the Sobolev space H 1 (R N ) and belongs to C 1 under the assumptions of Theorem 1.1 or Theorem 1.2. It is well known that critical points of the functional I are weak solutions of the semilinear elliptic equation 3. Proof of Theorem 1.1. Since we do not assume that the nonlinear term is 4-superlinear at infinity, it seems hard to prove the boundedness of Palais-Smale sequences for the functional I. We will use the following slight modified version of [15, Theorem 1.1 and Lemma 2.3] (see also [16]) to construct a special Palais-Smale sequence for the functional I.
Theorem 3.1. Let X be a Banach space equipped with the norm · X and let J ⊂ R + be an interval. Consider a family {I λ } λ∈J of C 1 -functionals defined on X of the form (1)) < 0} is nonempty and Then, for almost every λ ∈ J , there exists a sequence {v n } ⊂ X such that where λ ∈ [ 1 2 , 1]. To apply Theorem 3.1, we set with B being nonnegative. Next lemma ensures that A is coercive and the functional I λ possesses the mountain pass geometry.
Lemma 3.2. The following statements hold: This inequality implies that A is coercive. In order to prove (2), we consider the functional Then, for t > 0, we have It follows from (h 3 ) that J 1/2 (tū(x/t)) < 0 for t > 0 sufficiently large. Choose u 0 = tū(·/t) with t > 0 sufficiently large and Since λ ≤ 1 and H K (t) ≥ 0 for all t ∈ R, there holds from which we see that c λ > 0. The proof is complete.
In order to proceed, we recall some known results of the "limit" functional Define the minimum energy The following lemma from [8, Theorems 3.2 and 3.4] states the existence of least energy solution for autonomous problem which is crucial to ensure the compactness of bounded Palais-Smale sequences. Lemma 3.3. m ∞ λ > 0 and it is achieved by a positive function w ∞ λ ∈ H 1 (R N ). Furthermore, there exists a path γ λ ∈ C([0, 1], H 1 (R N )) satisfying the following properties: • γ λ (0) = 0 and I ∞ λ (γ λ (1)) < 0; • γ λ (t)(x) > 0 for all x ∈ R N and t ∈ (0, 1]; . Lemma 3.4. For any λ ∈ [ 1 2 , 1], there holds c λ < m ∞ λ . Proof. Let w ∞ λ and γ λ be as in Lemma 3.3. Then, by (V 1 ), we have I λ (γ λ (t)) < I ∞ λ (γ λ (t)), for all t ∈ (0, 1], where we have used the assumption V ≡ V ∞ as stated in Remark 2. Using the definition of c λ leads to The proof is complete. The following lemma was essentially proved in [2, Lemma 4.2], which describes the decomposition of bounded Palais-Smale sequences for the functional I λ . (1) |y k n | → ∞ and |y k n − y k n | → ∞ for k = k , , where we agree that, in the case l = 0, the above conclusion holds without w k λ and {y k n }. Using Lemmas 3.4 and 3.5, we can prove is a bounded Palais-Smale sequence for the functional I λ satisfying lim sup n→∞ I λ (v n ) ≤ c λ and v n 0 as n → ∞, then there is a subsequence of {v n } which converges weakly to a nontrivial critical point v λ of I λ with I λ (v λ ) ≤ c λ .
Proof. By Lemma 3.5, up to a subsequence, there exist a nonnegative integer l and v λ ∈ H 1 (R N ) such that v n v λ in H 1 (R N ), I λ (v λ ) = 0 and where {w k λ } l k=1 are nontrivial critical points of the "limit" functional I ∞ λ . If I λ (v λ ) < 0, then the proof is complete. If I λ (v λ ) ≥ 0, then we claim that l = 0. Otherwise, we have which contradicts the conclusion of Lemma 3.4. Thus v n → v λ in H 1 (R N ) and I λ (v λ ) ≤ c λ . Note that v n 0 as n → ∞. Therefore, v λ is a nontrivial critical point of the functional I λ . The proof is complete. Proof. By (h 1 ), (h 2 ), the definition of h K and Lemma 2.1, we have h K (f (t))f (t)t ≥ 0 for all t ∈ R, and Then there exists C 1 > 0 such that Combining this with I λ (v), v = 0 leads to which implies that Since v = 0 and |f (t)| ≤ |t| for all t ∈ R, we obtain v ≥ for some positive constant δ.
In the proof of Theorem 1.1, we also need the following Pohožaev type identity. Because the proof is standard, we omit it and refer the reader to [6,Section 2].

HAIDONG LIU, LEIGA ZHAO
Using Hölder inequality and Sobolev inequality leads to where we have used assumption (V 2 ) and Corollary 1. Since σ ∈ [1, 2), we see that R N |∇v λn | 2 dx ≤ C 3 for some positive constant C 3 . Secondly, we use the same arguments as in the proof of Lemma 3.7 to obtain Then it follows from the coercivity of A that {v λn } is bounded in H 1 (R N ). Thirdly, we have and, similarly, Palais-Smale sequence for the functional I satisfying lim sup n→∞ I(v λn ) ≤ c 1 and v λn 0 as n → ∞. Using Lemma 3.6 again, we obtain a nontrivial critical point v of I. By a standard argument, we can show that v(x) > 0 for all x ∈ R N . 4. Proof of Theorem 1.2. In this section, we consider problem (1.3) with a more general nonlinearity. In order to apply Theorem 3.1, we define h j = max{(−1) j+1 h,0} for j = 1, 2. Then h = h 1 − h 2 . By (h 5 ), there exists a functionū ∈ H 1 r (R N ) ∩ L ∞ (R N ) (see step 1 in the proof of [6, Theorem 2]) satisfying where H j (t) = t 0 h j (s) ds for j = 1, 2. Thus we can findλ ∈ (0, 1) such that As in Section 3, we introduce a family of C 1 -functionals defined on H 1 r (R N ) where λ ∈ [λ, 1]. Setting we see that I λ (v) = A(v) − λB(v) with B being nonnegative. Similar to Lemma 3.2, we have Lemma 4.1. The following statements hold: Proof. The arguments in the proof of Lemma 3.2 also work for the items (1) and (3) if we observe that H j (t) ≥ 0 for all t ∈ R and j = 1, 2. Next, we prove item (2). Setting we have, for t > 0, where G(t) = t 0 g(s) ds. Proof. We use similar arguments as in the proof of [7,Theorem 1]. First of all, using (4.2) and Fatou's lemma yields that Next we claim that, given ε > 0, there exists C ε > 0 such that for all a, b ∈ R. In fact, using (4.2) and Young inequality, we have Taking a = v n − v and b = v in (4.3) and using (4.2) again leads to Then, defining Letting ε → 0, we finish the proof.
is a bounded Palais-Smale sequence for the functional I λ at level c λ , then, up to a subsequence, {v n } converges to a positive critical point v λ of I λ with I λ (v λ ) = c λ .
Proof. Since {v n } ⊂ H 1 r (R N ) is bounded, we may assume up to a subsequence that v n v λ in H 1 r (R N ) and v n → v λ a.e. in R N . By Lebesgue's dominated convergence theorem and Lemma 4.4 with P (t) = h j (f (t))f (t), Q(t) = |t| 2 * −1 and ϕ ∈ C ∞ 0 (R N , R), we conclude that I λ (v λ ) = 0. Set w n = v n − v λ and define We claim that A 1 (w n ) → 0 as n → ∞. Indeed, it follows from Remark 4.3 that Using the well known Strauss radial lemma (see [31,Lemma 1]) and Lemma 4.4 with P (t) = h j (f (t))f (t)t, Q(t) = t 2 + |t| 2 * and ϕ = 1 yields that Combining (4.4)−(4.6) leads to lim n→∞ I λ (v n ), v n − I λ (v λ ), v λ − I λ (w n ), w n = 0 and then lim n→∞ I λ (w n ), w n = 0. Therefore, using (4.6) again and Lemma 2.1, we have lim sup From the proof of Lemma 3.2, we see that Since lim n→∞ A 1 (w n ) = 0, we have w n → 0 in H 1 r (R N ) and then v n → v λ in H 1 r (R N ). Hence v λ is a nontrivial critical point of I λ with I(v λ ) = c λ . Using standard arguments, we can prove that v λ (x) > 0 for all x ∈ R N . At this point, for almost every λ ∈ [λ, 1], we obtain a positive critical point v λ of the functional I λ . In general, we do not known whether this conclusion holds for λ = 1. However, we have Next we prove that the sequence {v λn } obtained in Lemma 4.6 is bounded in H 1 r (R N ). For this purpose, we need the following Pohožaev type identity, which is similar to Lemma 3.8.
Lemma 4.7. If v ∈ H 1 (R N ) is a critical point of the functional I λ , then Thus it suffices to prove that {A 1 (v λn )} is bounded. Invoking Lemma 4.7 and using the same arguments as in the proof of Theorem 1.1 leads to R N |∇v λn | 2 dx ≤ C 1 for some positive constant C 1 . By (h 1 ), (h 2 ), the definition of h j and Lemma 2.1, we have h j (f (t))f (t)t ≥ 0 for all t ∈ R, and lim t→0 h 1 (f (t))f (t)t f 2 (t) = 0, lim t→∞ h 1 (f (t))f (t)t |t| 2 * = 0.
Using I λn (v λn ), v λn = 0 and Lemma 2.1 yields which implies that The proof is complete. Similarly, there holds I (v λn ) → 0 in H −1 (R N ). Therefore, {v λn } is a bounded Palais-Smale sequence for the functional I at level c 1 . Using Lemma 4.5 again, we obtain a positive critical point v of the functional I.