1 Introduction and main result
In the present paper, we consider a class of quasilinear Schrödinger equations of the form
where is a given potential, , κ is a positive constant, and l, ρ are real functions. Corresponding to various types of nonlinear terms ρ, this problem appears naturally in different mathematical physical models; see [28, 23, 21] for an explanation. Here we focus on the case , . As we all know, a standing wave of (1.1) is a solution of the form , , and consequently satisfies the following equation:
where is the new potential and is the new nonlinear term. The main purpose of this paper is to deal with the existence of solutions for equation (1.2). This kind of problem has been studied by many authors; see [28, 23, 21, 3, 6, 22, 11, 10, 20, 26, 31, 24, 32, 27, 9, 7, 25, 13, 8, 30] and the references therein.
In [28, 23], by using a constrained minimization argument, a positive ground state solution has been proved for equation (1.2) with , , where is the Sobolev critical exponent. The case is called subcritical growth; there are many articles that deal with this class of problem (see [28, 23, 21, 3, 6, 22, 11, 10, 20, 26, 31]). In , the existence of both positive and sign-changing ground states of soliton-type solutions were established via the Nehari method. Then by a change of variables, the quasilinear problem is transformed into a semilinear one; see  for an Orlicz space framework and  for a Sobolev space frame. Recently, a perturbation method was developed in  to deal with equation (1.2), which can be applied to more general quasilinear Schrödinger equations (see also ).
For the critical case, we would like to mention [32, 27, 9, 26, 7, 25, 13, 8, 30] and the references therein. It seems that Moameni  first studied the critical case when the potential V is radial and satisfies some geometric conditions. Do Ó, Miyagaki and Soares  obtained a positive classical solution by using the concentration compactness principle of Lions . He and Li  obtained the existence, concentration and multiplicity of weak solutions by employing the minimax theorems and Ljusternik–Schnirelmann theory.
In recent years, the use of the Pohozaev manifold was shown very effective when treading nonlinearities which do not satisfy the Ambrosetti–Rabinowitz condition and the monotonicity condition; see [16, 5, 14, 15, 4]. Lehrer and Maia  employed the minimization methods restricted to the Pohozaev manifold to obtain the existence of positive solutions for the asymptotically linear case. Later in , Carrião, Lehrer and Miyagaki extended the result given in  to more general quasilinear equations. Motivated by [16, 5, 17], we consider equation (1.2) with the nonlinear term being nonhomogeneous and asymptotically linear at infinity. We will use the linking theorem together with the barycenter function restricted to the Pohozaev manifold associated to our problem.
The main obstacle in finding a solution of equation (1.2) is due to the influence of the quasilinear and nonconvex term . The other difficulty is the possible lack of compactness due to the unboundedness of the domain. We will employ an argument developed in  to overcome the first difficult and a splitting lemma to conquer the second one.
We suppose that V satisfies the following assumptions:
, for all ;
for all , and the strict inequality holds on a subset of positive Lebesgue measure of ;
for all ;
for all , where is the Hessian matrix of the function V.
We assume the following conditions on the function g:
and , where .
We employ an argument developed in  to introduce a variational framework associated with equation (1.2). We observe that equation (1.2) is formally the Euler–Lagrange equation associated with the energy functional
We make a change of variables , where f is defined by
After the change of variables from J, we obtain the following functional:
Then and I is well defined on , under the hypotheses (V), (V) and (g)–(g). Moreover, we observe that if v is a critical point of the functional I, then the function is a solution of equation (1.2) (see ).
The critical points of I are weak solutions of the problem
We can demonstrate that
for all . Each solution of equation (1.3) satisfies , where
We define the Pohozaev manifold associated with equation (1.3) by
Let c be the min-max mountain pass level for the functional I given by
Under the previous hypotheses on V and g, we have the following nonexistence result.
Suppose that (V)–(V) and (g)–(g) are satisfied. Then is not a critical level for the functional I. In particular, the infimum p is not achieved.
Consider now also the limiting problem
Its associated energy functional is denoted by .
Each solution of equation (1.4) satisfies the following Pohozaev identity:
We define the Pohozaev manifold associated with equation (1.4) by
We say that a solution of equation (1.4) is a least energy solution if and only if , where
as well as the mountain pass min-max level
Using a method similar to the one in , we can deduce that .
Now we can state our main existence result.
Suppose that (V)–(V) and (g)–(g) are satisfied and the following facts hold:
is sufficiently small;
Then equation (1.2) admits a positive solution whose energy is above .
These results extend the corresponding results in  to the more general quasilinear case. The framework employed and ideas of the proofs for our main results are close to those found in . However, some technical details in this paper are different from those in .
For the case of nonlinearities , uniqueness properties of ground state solutions of equation (1.2) were recently proved in [1, 12, 29, 2], so assumption of Theorem 1.2 (3) is expected to be fulfilled.
Let , where , and is small enough. Then
We can see by a computation that satisfies all conditions in Theorem 1.2.
Conditions (g) and (g) imply that given and , there exists a positive constant such that
We also obtain the estimate
Since we are looking for positive solutions, we set for all . Let v be a critical point of I. Taking , we have
Since , we get
Hence we may conclude that a.e. in and . As , we conclude that u is a nonnegative solution for equation (1.2).
In this paper, we use the following notations:
is the usual Hilbert space endowed with the norm
is the usual Banach space endowed with the norm
denotes the usual norm in .
denotes the Lebesgue measure of the set Ω.
denote various positive constants whose exact value is inessential.
2 Some preliminary results
The function f satisfies the following properties:
f is uniquely defined, and invertible;
for all ;
for all ;
for all ;
for all ;
there exists a positive constant C such that for and for ;
for all .
The functional γ and the Pohozaev manifold satisfy the following properties:
is an isolated point of ;
is a closed set;
is a manifold;
there is a such that for all .
(1) Thanks to Lemma 2.1 (9), we can deduce that there is such that
where S is the best Sobolev constant of the embedding and
by taking sufficiently small. Let such that ; then if , we have
(2) The functional is a functional, thus is a closed subset. Moreover, is an isolated point in and the assertion follows.
(3) It follows from Lemma 2.1 (6) and (g) that
Since , we have
Hence we can deduce that
(4) Since 0 is an isolated point in , there must be a ball which does not intersect and the assertion is proved. ∎
Assume (V), (V) and (g) hold. Then is a nature constraint for the functional I.
Let be a critical point of . By the theorem of Lagrange multipliers, there exists a such that . The proof is complete as soon as we show that . Evaluating the linear functional above at , we obtain
This expression is associated with the equation
which can be rewritten as
Each solution of equation (2.4) satisfies , where
Recalling that , and substituting in the equation above, we get
Since v is a solution of equation (2.4), it satisfies . This yields
From (V) we get that, if , the right-hand side of the above equation is nonnegative, while the left-hand side is negative. If , one gets the same contradiction. Hence . ∎
3 Proof of Theorem 1.1
then we can get the following lemmas. The proof of these lemmas can be found in ; we omit them.
Assume that and . Then there exist unique and such that and .
If , then there exists such that and .
If , then there exists such that and .
Let . Then the function given by such that is continuous.
Let . Then for any , we have . Moreover, there exists such that
There holds and .
There exists a real number such that .
There holds and .
4 Proof of Theorem 1.2
This section is dedicated to proving the existence of a positive solution for equation (1.3). By the previous results, we should search for solutions which have energy levels above . Similarly to what was done in [16, 17], we start by showing that the min-max levels of the mountain pass theorem for the functionals I and are equal.
There holds .
The proof is analogous to the proofs of [16, Lemmas 4.1 and 4.2]; we omit it.
For every , there exists such that intersects .
By the proof of Lemma 2.2 (1), we learn that there exists such that if . Furthermore, we observe that
It follows from (V) that
Therefore, if , we have and . Since , we conclude that there exists such that for which . The function satisfies , which shows that every path intersects . ∎
There exists a sequence where
If is a sequence with , then it has a bounded subsequence.
First of all, we observe that if a sequence satisfies
for some constant , then the sequence is bounded in . For that, we simply need to demonstrate that is bounded. In fact, by Lemma 2.1 (9) and (V), we observe that
Moreover, by the Sobolev inequality and Lemma 2.1 (9), one deduces
Hence there is a constant such that
Therefore, it remains to show that
Let be an arbitrary Cerami sequence for I at level , that is,
and for any ,
from Lemma 2.1 (6), we get and
Thus there exists a constant such that . Recalling that is a Cerami sequence, we get
Thanks to (2.2), we get
By the above inequality and (4.4), one has
Lemma 4.5 (Splitting (see )).
Let be a bounded sequence such that
Then there exists (if necessary, replace by a subsequence) a solution of equation (1.3), a number , k functions and k sequences of points , , satisfying the following properties:
are nontrivial solutions of equation ( 1.4 );
and , ;
If and , then either is relatively compact or the splitting lemma holds with and .
Let us set
Then we have the following lemma.
Assume that is an isolated radial critical level for . Then and I satisfies condition (3) at level . Assume now that the limiting equation (1.4) admits a unique positive radial solution. Then I satisfies condition (3) at level .
The proof is analogous to [18, Lemma 5.9]; we omit it.
If and , then the sequence is bounded.
Since and , we get
where we also used (V). Therefore is bounded. By the Sobolev inequality, the sequence is also bounded.
Since and are bounded, is bounded as well. Hence is bounded in . ∎
Define the barycenter function of a given function by setting
with and μ is a continuous function. Subsequently, take
It follows that . Now we define the barycenter of u by
Then is well defined since has compact support.
The function satisfies the following properties:
β is a continuous function in ;
if u is radial, then ;
if is given and if we define , then .
We shall also need the following lemma.
Assume that are such that and as , where is bounded. Then as .
We simply observe that
thus by Lemma 2.1 (10) we have
Since , we find that there is a constant such that
Since the function is continuous and satisfies (V), we can conclude that
We also find that
Hence as . ∎
It is clear that ; moreover, we have the following lemma.
There holds .
Suppose . By the definition of b, there is a minimizing sequence
such that . By Lemma 4.8, is bounded. Since by Lemma 4.1, is also a minimizing sequence of I on . By Ekeland’s variational principle [33, Theorem 8.5], there is another sequence such that , and as . Now we prove that as . Indeed, if does not go to zero, that means there exist and a subsequence denoted also by such that . Arguing as in the proof of Lemma 4.10, we can deduce that there is a positive constant such that
For sufficiently small, we have , and for all , one has . Now let and . By [33, Lemma 2.3], there is a deformation η on the level p, taking all the points of to the level .
Moreover, for n sufficiently large,
because is a minimizing sequence, , for n sufficiently large, and since , we have
On the other hand, is a path in Γ for M and n large enough, hence
which is a contradiction to , provided by Lemma 4.1; hence as . By Lemma 4.10, we get as . Hereafter, the sequence satisfies the assumptions of Corollary 4.6. Since and p is not attained by Theorem 1.1, the splitting lemma holds with . This yields
where , and is a solution of equation (1.4). Making a translation, we obtain
Calculating the barycenter function on both sides, we have
where the first equality comes from property (3) of the barycenter function β and the second one due to the continuity of β. Since , and , we arrive at a contradiction, yielding . ∎
Let us consider the positive, radially symmetric, ground state solution of equation (1.4). We define the operator by
where is the real number t which projects onto the Pohozaev manifold . Since is unique and is a continuous function of , we have that Π is a continuous function of y.
and as .
Since is translation invariant, the maximum of is attained at and . It follows from (V) and Lemma 2.1 (3) that
which concludes the proof. ∎
Replacing (V) with
Let S be a closed subset of a Banach space X and let Q be a submanifold of X with relative boundary . We say that S and link if the following facts hold:
for any such that there holds .
Moreover, if S and Q are as stated above and B is a subset of , then S and link with respect to B if (1) and (2) hold for any .
Theorem 4.16 (Linking).
Suppose that is a functional satisfying the (C) condition. Consider a closed subset and a submanifold with relative boundary . In addition, suppose the following:
S and link;
If , then the real number defines a critical value of I with .
Now we are ready to prove our main existence result.
Proof of Theorem 1.2.
In order to apply the linking theorem, we take
We will show that the conditions of Theorem 4.16 are satisfied.
(1) Since , from Lemma 4.12 we have that , because if , then , and if , then due to the equality . Now we show that , for any , where
Given , let us define for . The function T is continuous, because it is the composition of continuous functions. Moreover, for any , we have that , thus , because . Hence from Lemma 4.12 we have . By the fixed point theorem of Brouwer, we conclude that there exists such that , which implies . Therefore . From Definition 4.15, we know that S and link, i.e. relation (1) is proved.
(2) From the definitions of b and Q and inequality (4.10) we may write
which implies relation (2).
(3) Let us define
Then we have . Indeed, we have already proved that for all . If h is fixed, then there exists such that w belongs to , which means that for some . Therefore,
From (1), (2) and (3) above we can apply the linking theorem and conclude that d is a critical level for the functional I. Hence there exists a nontrivial solution of equation (1.3). Reasoning as usual, because of the hypotheses on g and f, and using the maximum principle, we conclude that v is positive and Theorem 1.2 is proved. ∎
The authors would like to thank the referee for his/her useful suggestions.
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About the article
Published Online: 2017-05-11
Funding Source: National Natural Science Foundation of China
Award identifier / Grant number: 11471267
This paper was supported by the National Natural Science Foundation of China (No. 11471267).
Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 323–338, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0244.
© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0