Solutions of semilinear elliptic equations
are standing waves of the corresponding time-dependent Schrödinger. For the existence of solutions of Equation (1.1), one of the important role is the sign of and . We say Equation (1.1) is linearly indefinite if changes sign, and superlinearly indefinite if changes sign. There are many results of Equation (1.1) for the superlinearly indefinite problem, linearly indefinite or not, we refer to   . In this paper, we consider the following modified Schrödinger equations
This kind of equations arise when we are looking for standing waves for the time-dependent quasilinear Schrödinger equation
which was used for the superfluid film equation in plasma physics by Kurihar . This model also appears in plasma physics and fluid mechanics, dissipative quantum mechanics and condensed matter theory. For more information on the relevance of these models and their deduction, we refer to .
To the best of our knowledge, the first mathematical studies of the Equation (1.2) seem to be Poppenberg et al.  for the one dimensional case and Liu-Wang  for higher dimensional case. The proofs in these papers are based on constrained minimization argument. Formally, Equation (1.2) associates with the Euler functional
Unfortunately, the functional J is not defined for all , unless . Therefore, it is difficult to use the standard variational methods to study the functional J. To overcome this difficulty, Jeanjean  introduced a transformation f so that if v is a critical point of
where f is defined by
Then is a solution of (1.2).
Since the publication of , Problem (1.2) has been studied extensively. For example, the case that the potential V is is studied in Silva-Vieira . By Nehari manifold method, Fang-Szulkin  studied the case that the nonlinearity is 4-superlinear and the potential has a positive lower bound. For problems with critical nonlinearities, see Silva-Vieira .
In all these papers, it is required that the potential V and nonlinearity satisfy the positive condition. With this condition and suitable conditions on the nonlinearity, the mountain pass theorem can be applied to produce a solution of (1.2).
In the literature, there are some existence results which allow the potential V to be negative somewhere. The strategy is to write with . Then if is in some sense small, it can be absorbed and the functional still verifies the mountain pass geometry. We refer the reader to . Recently, by a local linking argument and Morse theory, Liu-Zhou  obtains a nontrivial solution for the problem (1.2) with indefinite potential. For linearly indefinite case, we also refer to .
However, this is a gap in the high dimensional quasilinear Schrödinger equations with indefinite nonlinearity. The one dimensional case has been partially studied in  by critical point theory. The purpose of this paper is to present some results about indefinite quasilinear Schrödinger equations in higher dimensional. More precisely, we present our assumptions on the potential and
(V2) and for each , , where is a constant and denotes the Lebesgue measure of a measurable set ;
(A1) and , where .
(P1) where the critical Sobolev exponent for and for .
Then we have
Theorem 1. Suppose that (V1), (V2), (A1) and (P1) hold. Then Equation (1.2) has at least one nontrivial solutions.
Notation. will denote different positive constants whose exact value is inessential.
Before prove our results, we shall introduce the appropriate space to find critical points of the Euler functional. Let
with the inner product
and the norm
Then X is a Hilbert space. By Bartsch and Wang , we know that the embedding ↪ for is compact for .
Below we summarize the properties of f in (1.3). Proofs may be found in .
Lemma 2.1. The function f has the following properties:
(f1) f is uniquely defined, and invertible.
(f2) and for all . Moreover, .
(f3) For all we have .
(f4) For all we have and .
(f5) There exists a positive constant such that for , for .
By Lemma 2.1, it is easy to see that , moreover
for all .
3. Proof of the Theorem 1
Because the principle part of , denoted by
is not a quadratic form on v, it’s not so obvious to verify that satisfies the mountain pass geometry. Similar to , by taking into account the Taylor expansion of Q at the origin, it is easy to deduce that is a strict local minimizer of .
Lemma 3.1. Under the assumptions of Theorem 1, then
(i) is a strict local minimizer of .
(ii) There is with such that .
Proof. By the properties of the transformation f, it is easy to see that Q is a C2-functional on X. Since , we get . According to the Taylor formula, as , we have
Therefore, combining this with Lemma 2.1 (f2), there exists such that
this implies that the zero function 0 is a strict local minimizer of .
On the other hand, since and is continuous in , we may choose such that and for all . Then for any , using Lemma 2.1 (f2),(f5), we deduce
Since , we know that for s sufficiently large. Thus the conclusion(ii) follows from choosing with large. £
Lemma 3.2. Under the assumptions of Theorem 1. Then the functional satisfies Cerami condition.
Proof. Let be a Cerami sequence of , that is , for some .
First we claim that there exists such that
Let . By direct computation, we get . By (1.3) and (2.1), there exists such that
Therefore, our claim is true.
Next, we claim that there exists such that
Indeed, we may assume that (otherwise the conclusion is trivial). We argue by contradiction and assume that
where and . By direct computation, we have
This implies is strictly increasing. So we get is positive if . Combining this with (3.3), we obtain
We claim that for each , there exists a constant independent of n such that , where . Otherwise, there is an and a subsequence of such that for any positive integer k, , where . By the properties of f described in Lemma 2.1 and (V1), there exists a constant such that
a contradiction. Hence the assertion is true. Then for each , may be chosen so that . Next, keeping in mind. Let . By (3.4), as in the proof of the Lemma 3.10 in , it is easy to see that as , there exists such that
Combining this with (3.3) and the Mean Value Theorem, we have
Since is uniformly bounded, by the integral absolutely continuity there exists such that whenever , . For this , we have
This and (3.6) contradict with (3.5). Therefore, this claim is true.
Lastly, together (3.2) and Lemma 2.1(f4) give us
Combining this with (3.1) implies is bounded in X. Up to a subsequence we may assume in X. Since embedding ↪ is compact for , by a standard argument, we can show that has a convergent subsequence, see  (Theorem 2.1, Step 3). We omit it here. This completes the proof.
To prove Theorem 1, we will apply the following Mountain Pass Theorem .
Theorem 2. Let X be a Banach space and be a functional satisfying the Cerami condition. If and are such that
is a critical value of with , where .
Proof of the Theorem 1
Proof. From Lemma 3.1 and Lemma 3.2, we know satisfies the conditions of Theorem 2. Hence Equation (1.2) has at least one nontrivial solution under assumptions (V1), (V2), (A1) and (P1).
By mountain pass theorem and Taylor expansion, we prove the existence of solutions for the quasilinear Schrödinger equations with indefinite nonlinearity. This indefinite problem had never been considered so far. So our main results can be regarded as complementary work in the literature. On the other hand, our approach seems much simpler than those presented in  .
This project is supported by National Natural Science Foundation of China (Nos.11701114, 11871171).
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