1 Introduction and main results
In this paper, we consider the following Kirchhoff-type problem with convolution nonlinearity:
where , , with , is the Riesz potential defined by
, and .
Such a problem is often referred to as being nonlocal due to the appearance of the terms or which implies that (1.1) is no longer a pointwise identity. In particular, if , then (1.1) reduces to the following generalized Choquard equation:
When , and , (1.2) is known as the Choquard–Pekar equation or the stationary nonlinear Hartree equation, which was introduced in 1954, in a work by Pekar  describing the quantum mechanics of a polaron at rest; for more details and applications, we refer to [19, 27]. For the case where and , (1.2) is known to have a solution if and only if (see [24, p. 457], [27, Theorem 1]; see also [11, Lemma 2.7]). As described Moroz and Van Schaftingen in , since the Hardy–Littlewood–Sobolev inequality  implies
where and are the upper and lower critical exponents, which appear as extensions of the exponents 6 and 2 for the corresponding local problem.
, as and as .
If we let in (1.1), then it becomes formally the following Kirchhoff-type problem with local nonlinearity :
which is related to the stationary analogue of the Kirchhoff equation
Equation (1.5) is proposed by Kirchhoff  as an extension of the classical D’Alembert’s wave equation for free vibrations of elastic strings. Kirchhoff’s model takes into account the changes in length of the string produced by transverse vibrations. For more details on the physical aspects, we refer the readers to [2, 3, 7, 8, 26].
After Lions  proposed an abstract functional analysis framework to (1.4), it has received more and more attention from the mathematical community; there have been many works about the existence of nontrivial solutions to (1.4) and its fractional version by using variational methods, for example, see [4, 6, 9, 12, 13, 16, 17, 18, 22, 30, 31, 35, 34, 37, 40, 41, 42] and the references therein. A typical way to deal with (1.4) is to use the mountain-pass theorem. For this purpose, one usually assumes that is superlinear at and super-cubic at . In this case, if g further satisfies the monotonicity condition
is increasing for ,
via the Nehari manifold approach, He and Zou  obtained the first existence result on ground state solutions of (1.4). For the case where is not super-cubic at , Li and Ye  proved that (1.4), with special forms and for , has a ground state positive solution by using a minimizing argument on a new manifold that is defined by a condition which is a combination of the Nehari equation and the Pohoz̆aev equality. This idea comes from Ruiz , in which the nonlinear Schrödinger–Poisson system was studied. Later, by introducing another suitable manifold differing from , Guo  and Tang and Chen  improved the above result to (1.4), where V satisfies
and for all ,
and there exists such that for all ,
and g satisfies
and is increasing on ,
and is nondecreasing on , where .
respectively, and some standard growth assumptions.
where , is a parameter and is a nonnegative steep potential well function. By using the Nehari manifold and the concentration compactness principle, Lü proved the existence of ground state solutions for (1.6) if the parameter μ is large enough. It is worth pointing out that the same result is not available in the case where , even when , since both the mountain pass theorem and the Nehari manifold argument do not work. In fact, in this case, it is more difficult to get a bounded (PS) sequence and to prove that the (PS) sequence converges weakly to a critical point of the corresponding functional in . To the best of our knowledge, there seem to be no results dealt with this case in the literature. As for the related study of problem (1.1) involving the critical exponents, we refer to [25, 32] for more details.
Motivated by the above-mentioned papers, we shall deal with the existence of ground state solutions for (1.1) under (V1) and (F1). It is standard to check, according to (1.3), that under (V1) and (F1), the energy functional defined in by
is continuously differentiable and its critical points correspond to the weak solutions of (1.1). We say a weak solution to (1.1) is a ground state solution if it minimizes the functional Φ among all nontrivial weak solutions.
In addition to (F1), we also need the following assumptions on f:
the function is nondecreasing on .
(F3) is weaker than the following assumption, which is easier to verify:
the function is nondecreasing on .
To overcome the lack of compactness of Sobolev embeddings in unbounded domains, different from  in which a steep potential well was considered, we assume that V satisfies (V1) and the decay condition:
and either of the following cases holds:
for all ,
Now we are in a position to state the first main result.
Next, we further provide a minimax characterization of the ground state energy. To this end, we introduce a new monotonicity condition on V as follows:
and there exists such that
is nonincreasing on for every .
Then every nontrivial solution of (1.1) is contained in . Our second main result is as follows.
where and in the sequel .
where is the Pohoz̆ave type identity related with (1.10). Then we have the following corollary.
In the last part of this paper, we give a non-existence result for the following special form of (1.1):
If or , then problem (1.12) does not admit any nontrivial solution.
There are indeed functions which satisfy (V1)–(V3). An example is given by , where and are two positive constants. Our results extend and improve the previous results on (1.1) in the literature, which are new even when . In particular, Theorems 1.2 and 1.3 fill a gap on (1.1) in the case where , with .
Letting , our results cover the ones in [12, 17, 37], which dealt with (1.4) that can be considered as a limiting problem of (1.1) when . In fact, if , then (V2)’ and (G2) imply case (i) of (V2) and (F3), respectively. Moreover, since (G4) and (F3) imply that and are nondecreasing on , respectively (see Lemma 2.3), one can see that (F3) reduces to (G4) when .
To prove Theorem 1.2, we will use Jeanjean’s monotonicity trick , that is, an approximation procedure to obtain a bounded (PS)-sequence for Φ, instead of starting directly from an arbitrary (PS)-sequence. More precisely, firstly, for , we consider a family of functionals defined by
These functionals have a mountain pass geometry, and we denote the corresponding mountain pass levels by . Moreover, has a bounded (PS)-sequence at level for almost every . Secondly, we use the global compactness lemma to show that the bounded sequence converges weakly to a nontrivial critical point of . To do this, we have to establish the following strict inequality:
where is the associated limited functional defined by
A classical way to obtain (1.14) is to find a positive function such that when nonconstant potential . However, it seems to be impossible to obtain the mentioned above only under (F1)–(F3). So the usual arguments cannot be applied here to prove (1.14). To overcome this difficulty, we follow a strategy introduced in , that is, we first show that there exists such that
and then, by means of the translation invariance for and the crucial inequality
established in Lemma 3.3, we can find such that
(see Lemma 3.5), where
and is the corresponding Pohoz̆ave type identity. In particular, any information on sign of is not required in our arguments. Finally, we choose two sequences and such that and , and by using (1.17) and the global compactness lemma, we get a nontrivial critical point of Φ.
We would like to mention that in the proof of Theorem 1.2, a crucial step is to prove (1.16), which is a corollary of Theorem 1.3. Inspired by [5, 36, 37], we shall prove Theorem 1.3 by following this scheme:
we verify and establish the minimax characterization of ,
we prove that m is achieved,
we show that the minimizer of Φ on is a critical point.
Although we mainly follow the procedure of , we have to face many new difficulties due to the mutual competing effect between and . These difficulties enforce the implementation of new ideas and techniques. More precisely, in step 1, we first establish a key inequality, namely,
in Lemma 2.5, where some more careful analyses on the convolution nonlinearity are introduced, see Lemmas 2.1–2.4; then we construct a saddle point structure with respect to the fibre for , see Lemma 2.8; finally, based on these constructions, we obtain the minimax characterization of m, see Lemma 2.9. In step 2, we first choose a minimizing sequence of Φ on , and show that is bounded in ; then, with the help of the key inequality (1.18) and a concentration-compactness argument, we prove that there exist and such that in , up to translations and extraction of a subsequence, and is a minimizer of , see Lemmas 2.14 and 2.15. This step is most difficult since there is no global compactness and not any information on . Finally, in step 3, inspired by [38, Lemma 2.13], we use the key inequality (1.18), the deformation lemma and intermediary theorem for continuous functions, which overcome the difficulty that may not be a -manifold of , due to the lack of the smoothness of , see Lemma 2.15.
Throughout the paper we make use of the following notations:
denotes the usual Sobolev space equipped with the inner product and norm
() denotes the Lebesgue space with the norm .
For any , for .
For any and , .
denote positive constants possibly different in different places.
The rest of the paper is organized as follows. In Section 2, we study the existence of ground state solutions for (1.1) by using the Nehari–Pohoz̆aev manifold , and give the proof of Theorem 1.3. In Section 3, based on Jeanjean’s monotonicity trick, we consider the existence of ground state solutions for (1.1), and complete the proof of Theorem 1.2. In Section 4, we study the non-existence of solutions for problem (1.12) and present the proof of Theorem 1.5.
2 Proof of Theorem 1.3
In this section, we give the proof of Theorem 1.3. To this end, we give some useful lemmas. Since satisfies (V1)–(V3), all conclusions on Φ are also true for in this paper. For (1.4), we always assume that . First, by a simple calculation, we can verify the following lemma.
which implies that for all and . ∎
which implies that for all . This shows that (2.4) holds. ∎
According to the Hardy inequality, we have
This shows that (2.9) holds. ∎
From Lemma 2.5, we have the following two corollaries.
Note that (F1) implies that for any , there exists such that
Next we claim that is unique for any . In fact, for any given , let be such that . Then . Jointly with (2.9), we have
Letting and in (2.1), we have, respectively,
as desired. ∎
there exists such that for all ,
(ii) Let be such that . There are two possible cases. Case 1: . In this case, by (2.14), one has
Case 2: . By (2.23), passing to a subsequence, we have
By (V1), there exists such that for . This implies
Making use of the Hölder inequality and the Sobolev inequality, we get
Cases 1 and 2 show that . ∎
This contradiction shows that . ∎
This contradiction shows that is bounded. Hence, is bounded in . Passing to a subsequence, we have in . Then in for and a.e. in . There are two possible cases: (i) and (ii) . Case (i) , i.e., in . Then in for and a.e. in . Using (V1) and (2.21), it is easy to show that
By (F1), for some and any , there exists such that
Therefore, there exists such that, passing to a subsequence,
For , we let
If there exists a subsequence of such that , then we have
which is a contradiction due to . Hence, . In view of Lemma 2.8, there exists such that . By (1.11), (2.4), (2.12), (2.13), (2.32), (2.35), (2.39), the weak semicontinuity of norm, Fatou’s lemma and Lemma 2.13, we have
This shows that m is achieved at . Case (ii) . In this case, analogous to the proof of (2.41), by using Φ and J instead of and , we can deduce that
Hence, the proof is complete. ∎
Following the idea of [38, Lemma 2.13], we use the deformation lemma and intermediary theorem for continuous functions to prove this lemma. Assume that . Then there exist and such that
By [37, equation (2.47)], one has Thus, there exists such that
The rest of the proof is similar to the proof of [38, Lemma 2.13]. Indeed, we can obtain the desired conclusion by using
instead of [38, (2.40) and ε], respectively. ∎
3 Proof of Theorem 1.2
In this section, we give the proof of Theorem 1.2. Without loss of generality, we consider that .
Proposition 3.1 ().
Let X be a Banach space and let be an interval. We consider a family of -functionals on X of the form
where for all , and such that either or as . We assume that there are two points in X such that
Then, for almost every , there is a bounded (PS) sequence for , that is, there exists a sequence such that
is bounded in X,
in , where is the dual of X.
Moreover, is nonincreasing and left continuous on .
Lemma 3.2 ().
We set . Then
for . Correspondingly, we also let
for . Set
By Corollary 2.6, we have the following lemma.
In view of Theorem 1.2, has a minimizer on , i.e.,
Since (1.10) is autonomous, and but , there exist and such that
there exists independent of λ such that for all ;
there exists a positive constant independent of λ such that for all ,
and are nonincreasing on .
The proof of Lemma 3.4 is standard, so we omit it.
Then , defined by Lemma 3.4 (ii). Moreover,
In both cases, we obtain for . ∎
exists, in and ;
and for ;
We agree that in the case , the above holds without .
By Lemma 3.6, there exist a subsequence of , still denoted by , and such that exists, in and , and there exist and such that for ,
Since , we have the following Pohoz̆aev identity:
If case (i) of (V2) holds, then it follows from the Hardy inequality that
If case (ii) of (V2) holds, then it follows from the Sobolev embedding inequality that
which, together with Lemma 3.5, implies that and . Hence, in and . ∎
In view of Lemma 3.7, there exist two sequences and , denoted by , such that
Proof of Theorem 1.2.
This shows that is a ground state solution of (1.1). ∎
4 Proof of Theorem 1.5
From Lemma 3.2, if , then u satisfies the following Pohoz̆aev type identity:
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About the article
Published Online: 2018-09-20
Published in Print: 2019-03-01
Funding Source: National Natural Science Foundation of China
Award identifier / Grant number: 11571370
Award identifier / Grant number: 11871199
Xianhua Tang was partially supported by the National Natural Science Foundation of China (No. 11571370). Binlin Zhang was partially supported by the National Natural Science Foundation of China (No. 11871199).
Citation Information: Advances in Nonlinear Analysis, Volume 9, Issue 1, Pages 148–167, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2018-0147.
© 2020 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0