In the present work we study the multiplicity and concentration of positive solutions for the following class of Kirchhoff problems:
where is a small parameter, are constants, , V is a continuous positive potential with a local minimum, and f is a superlinear continuous function with subcritical growth. The main results are obtained through suitable variational and topological arguments. We also provide a multiplicity result for a supercritical version of the above problem by combining a truncation argument with a Moser-type iteration. Our theorems extend and improve in several directions the studies made in (Adv. Nonlinear Stud.14 (2014), 483–510; J. Differ. Equ.252 (2012), 1813–1834; J. Differ. Equ.253 (2012), 2314–2351).
In this paper we focus our attention on the multiplicity and concentration of positive solutions for the following class of Kirchhoff problems:
where is a small parameter, are constants, and . Throughout the paper we will assume that the potential is a continuous function satisfying the following hypotheses introduced by del Pino and Felmer [14]:
there exists such that ,
there exists a bounded open set such that
We suppose that is a continuous function such that for and fulfills the following conditions:
as ,
if then there exists such that as , whereas if then we suppose that there exist , such that
there exists such that
the map is increasing on .
When and is replaced by the more general space , equation (1.1) reduces to a nonlinear Schrödinger equation of the type
which has been widely investigated in the last thirty years. The main motivation for studying (1.2) arises from seeking standing wave solutions, namely functions of the form , with constant, for the time-dependent Schrödinger equation
An interesting class of solutions of (1.2), sometimes called semi-classical states, are families of solutions which concentrate and develop a spike shape around one (or more) special points in , while vanishing elsewhere as . We refer the interested reader to [3,13,14,17,19,31] and their references for several existence and multiplicity results obtained by applying different variational and topological methods.
On the other hand, problem (1.1) is related to the stationary analogue of the Kirchhoff equation
which was proposed in 1883 by Kirchhoff [24] as an extension of the classical D’Alembert wave equations for free vibration of elastic strings. The Kirchhoff model takes into account changes in the length of the string produced by transverse vibrations. In (1.3), denotes the transverse string displacement at the spatial coordinate x and time t, L is the length of the string, h is the area of the cross section, E is Young’s modulus of the material, ρ is the mass density, and is the initial tension. We refer to [10,30] for the early classical studies dedicated to (1.3). We also note that nonlocal boundary value problems like (1.3) model several physical and biological systems where u describes a process which depends on the average of itself, as for example, the population density (see [2,12]). However, only after the pioneering work of Lions [25], where a functional analysis approach was proposed to attack (1.3), problem (1.1) began to attract the attention of several mathematicians (see [2,4,16,18,21–23,34], and also [6–8,36–38] for some interesting results for fractional Kirchhoff problems). In particular, He and Zou [22] obtained the existence and multiplicity of concentrating solutions for small of the following perturbed Kirchhoff equation
assuming that is a continuous potential satisfying the assumption introduced by Rabinowitz [31]:
and g is a subcritical nonlinearity. Subsequently, Wang et al. [34] investigated the multiplicity and concentration phenomenon for (1.4) when , f is a continuous subcritical nonlinearity and λ is sufficiently large. Figueiredo and Santos Júnior [18] proved a multiplicity result for a subcritical Schrödinger–Kirchhoff equation via the generalized Nehari manifold method, when the potential V has a local minimum. He et al. [23] considered the existence and multiplicity of solutions for (1.4) when , is a subcritical nonlinearity which does not satisfies the Ambrosetti–Rabinowitz condition [5].
Motivated by the above works, in this paper we study the multiplicity and concentration of solutions for (1.1) under conditions – on the potential V, and assuming – for the continuous nonlinearity f. In order to state our main result more precisely, we recall that if Y is a given closed set of a topological space X, we denote by the Ljusternik–Schnirelmann category of Y in X, this is the smallest number of closed contractible sets in X which cover Y (see [28,35] for more details). We are able to prove the following main result:
Assume that conditions–and–hold. Then for anysuch thatthere existssuch that for any, problem (
1.1
) admits at leastpositive solutions. Moreover, ifdenotes one of these solutions andis a global maximum point of, thenand there existsuch that
Our proof of Theorem 1.1 is obtained by applying appropriate variational arguments. First, motivated by [14], we overcome the lack of information about the behavior of potential V at infinity by making a suitable modification on the nonlinearity, solve the modified problem and then check that, for small enough, the solutions of the modified problem are indeed solutions of the original one. Due to the fact that f is only continuous, the Nehari manifold associated with the modified problem is not differentiable, so we cannot apply standard variational arguments for -Nehari manifolds developed, for example, in [3,13,22,23]. For this reason we use certain versions of critical point theorems due to Szulkin and Weth [33]. We also note that the presence of the Kirchhoff term creates some difficulties in getting the compactness of the modified functional . Indeed, it is not clear that weak limits of bounded sequences are critical points of . Moreover, when , problem (1.1) presents an extra difficulty due to the presence of the critical exponent, and in order to recover some compactness properties for , we invoke the Concentration–Compactness Lemma [27]. Since we are interested in obtaining multiple critical points, we use a technique introduced by Benci and Cerami [9], which consists in making precise comparisons between the category of some sublevel sets of and the category of the set M. Then we apply Ljusternik–Schnirelmann theory to deduce a multiplicity result for the modified problem. Finally, we show that the solutions of the modified problem are also solutions for (1.1), when is small enough, by using the Moser iteration technique [29].
In the last part of this paper we consider a supercritical version of problem (1.1). In this case, we deal with the sum of two homogeneous nonlinearities and add a new positive parameter μ. More precisely, we consider the following problem:
where and the exponents satisfy . Our multiplicity result for the supercritical case can be stated as follows.
Assume that conditions–hold. Then there existssuch that for anysatisfyingand for any, there existssuch that for any, problem (
1.5
) admits at leastpositive solutions. Moreover, ifdenotes one of these solutions andis a global maximum point of, then
The main difficulty in the study of (1.5) is due to the fact that is supercritical, and we cannot directly use variational techniques because the corresponding functional is not well-defined on the Sobolev space . In order to overcome this obstacle, we use some arguments inspired by [11,17,32] which can be summarized as follows. We first truncate in a suitable way the nonlinearity on the right hand side of (1.5), so we deal with a new problem but with subcritical growth. In the light of Theorem 1.1, we know that a multiplicity result for this truncated problem is available. Then we deduce a priori bound (independent of μ) for these solutions and by using an appropriate Moser iteration technique [29], we show that, for sufficiently small, the solutions of the truncated problem also solve the original one.
We stress that our theorems complement and improve the main results in [22,23,34], in the sense that we are considering multiplicity results for subcritical, critical and supercritical Kirchhoff problems involving continuous nonlinearities and imposing local conditions on the potential V.
The paper is organized as follows. In Section 2 we collect some notations and basic results. We also modify the nonlinearity and prove some useful lemmas to overcome the non differentiability of the Nehari manifold. In Section 3 we provide our first existence result. In Section 4 we deal with the autonomous problems. In Section 5 we introduce some tools which are needed to establish a multiplicity result. Section 6 is devoted to the proof of Theorem 1.1. In Section 7 we study the multiplicity of positive solutions for the supercritical problem.
The functional setting
Notations and basic results
We start by giving some notations and collecting useful preliminary results. If and , we denote by the -norm of a function . Let us define as the completion of with respect to the norm
Then we can consider the Sobolev space
endowed with the norm
We have the following well-known main Sobolev embeddings.
Ifis a bounded sequence inandfor some, theninfor all.
The modified problem
In order to study (1.1), we use the change of variable and we look for positive solutions to
In what follows, we introduce a penalized function [14] which will be useful to obtain our results. Let and be such that
and define
and
It is easy to check that g satisfies the following properties:
uniformly with respect to ,
for all , ,
for all , ,
for all , ,
for each the function is increasing on , and for each , the function is increasing on .
Let us consider the following modified problem
It is clear that if u is a positive solution of (2.3) with for all , then u is also a positive solution for (2.1), where .
The energy functional associated with (2.3) is given by
which is well-defined on the space
endowed with the norm
Clearly, is a Hilbert space with inner product
It is easy to check that and its differential is given by
for any . Let us introduce the Nehari manifold associated with (2.3), that is,
and we denote
where is the unit sphere in . Note that is a non-complete -manifold of codimension one, modelled on and contained in the open (see [33]). Then we have that for all , where .
Now we prove that possesses a mountain-pass geometry [5]:
The functionalsatisfies the following properties:
there existsuch thatwith;
there existswithsuch that.
(a) By assumptions and , we deduce that for any there exists such that
Then we can find such that with .
(b) Using -(i), we deduce that for any and
for some constants . Recalling that , we can conclude that as . □
Since f is only continuous, the next results will be very useful to overcome the non-differentiability of and the incompleteness of .
Assume that conditions–and–hold. Then the following assertions are true.
For each, letbe defined by. Then, there is a uniquesuch that
There existsindependent of u, such thatfor any. Moreover, for each compact setthere is a positive constantsuch thatfor any.
The mapgiven by, is continuous andis a homeomorphism betweenand. Moreover,.
If there is a sequencesuch that, thenand.
(i) Let us observe that . By Lemma 2.2, we can infer that , for small enough and for sufficiently large. Then there exists such that and is a global maximum for . Hence we can deduce that . Now we can prove the uniqueness of . Assume by contradiction that there are two positive numbers and such that and . Hence
and
Exploiting (2.5), (2.6), and , we can see that
where
and
Now we estimate each , . Considering , from the definition of g and using -(ii), we have
From the definition of g and using , we can infer
Finally, let us observe that by and from , it follow that . Thus we have
from which, multiplying both sides by and using assumption and (2.2), we obtain
Since and , we get a contradiction.
(ii) Let . By (i), there exists such that , that is
Using assumptions and , (2.7) and Theorem 2.1, given , there exists a positive constant such that
This implies that there exists , independent of u, such that . Now, let be a compact set. We prove that for any . Assume to the contrary, that there exists a sequence such that . Since is compact, there exists such that in . It follows from (2.4) that . Now, fix and using and , we can deduce that
Taking in (2.8) and using the facts and , we get
for n large, and this gives a contradiction.
(iii) First, we note that , and are well defined. Indeed, by (i), for each there exists a unique . On the other hand, if then . Otherwise, if , we have
which together with -(ii) implies that
and this yields a contradiction because and . As a consequence, , is well defined and continuous. Moreover, for all we have
from which we deduce that is a bijection. Now we prove that is a continuous function. Let and be such that in . Since for any , we may assume that . Then by (ii), there exists such that . Since , we obtain
and passing to the limit as , we get
which yields . This shows that
Therefore, and are continuous functions.
(iv) Let be such that . Since for each and we have
it follows that
Hence, by , and Theorem 2.1, there is a constant such that
Using , and -(ii), we can infer that, for each
from which,
Recalling the definition of and using (2.10) we get
which implies that
Since and is arbitrary, we obtain that and as . This completes the proof of the lemma. □
Now, we define the maps
by and . The next result is a direct consequence of Lemma 2.3 and Corollary 2.3 in [33].
Assume that conditions–and–hold. Then the following assertions are true.
andfor every,.
andfor every.
Ifis asequence for, thenis asequence for. Ifis a boundedsequence for, thenis asequence for the functional.
u is a critical point ofif, and only if,is a nontrivial critical point for. Moreover, the corresponding critical values coincide and
As in [33], we have the following variational characterization of the infimum of over :
Let us note that if , it follows from – that
which implies that for some r independent of u.
An existence result for the modified problem
In this section we focus our attention on the existence of positive solutions to (2.3) for sufficiently small . We begin by showing that the functional satisfies the Palais–Smale condition at any level if , and for some suitable depending on , when . This last fact is motivated by the following result:
Let. Thenfor all.
One can argue as in the proof of Lemma 2.1 in [23]. □
In view of Lemma 2.2, we can apply a version of the mountain-pass theorem without (PS) condition (see [35]) to obtain a sequence such that
We start with the following result:
Arguing as in the proof of Lemma 2.3-(ii) (see formula (2.8) there), we can deduce that
Since and , we can conclude that is bounded in . □
There is a sequenceandsuch that
Assume to the contrary, that the conclusion of lemma is not true. By Lemma 2.1, we then have
so, in view of and , we get
Since is bounded in , we may assume that in .
If , then we can use and (3.2) to deduce that , which in turn implies that , and this is impossible because .
Now assume that . Using the definition of g and (3.2), we can deduce that
and
From we have
Let be such that
and
Note that , otherwise (3.5) would yield as and then , which contradicts . Hence, putting together (3.5), (3.6) and (3.7), we have
By (3.3), (3.6), (3.7), (3.8) and , it follows that
On the other hand, from the definition of we can see that
and
This, together with (3.6), (3.7) and (3.8), implies that
which yields
Consequently,
Combining (3.9), (3.10), (3.11), it follows that
and by Lemma 3.1, this is a contradiction. □
For any , let be such that in and in , with and , where C is a constant independent of ρ. Since is bounded in , it follows that , namely
Take such that . Then, using -(ii) and Lemma 3.2, we get
which implies that
Now, if is unbounded, it follows by Lemma 3.3 and (3.12), that as , which gives a contradiction. □
The next results will be essential for obtaining the compactness of bounded Palais–Smale sequences.
Letbe asequence for. Then for each, there existssuch that
Let be such that , and such that in and in , with and , where C is a constant independent of R. Since is bounded in , we have that , and using -(ii), we get
Accordingly,
from which the assertion follows. □
The functionalsatisfies thecondition at any levelif, andif.
By Lemma 3.2, we know that any sequence is bounded, so we may assume that in and in for all . Let us start by proving that
Using (3.13), , , and the Sobolev embedding, we have, for n large,
On the other hand, choosing R sufficiently large, we may assume that
From this and (3.16), we get for n large enough,
Taking into account the definition of g, we know that
Since is bounded, from , , , the strong convergence in for , and by the dominated convergence theorem, it follows that
Next, we aim to prove that
Indeed, if (3.19) holds, we can infer from , , , the strong convergence in for and the dominated convergence theorem that
Therefore (3.15) follows by the above limit, (3.17) and (3.18).
At this point, we show the validity of (3.19). Since is bounded in , we may assume that
where μ and ν are bounded nonnegative measures in . By the Concentration Compactness Principle [27], we obtain an at most countable index set I, sequence and such that
Now we prove that . Assume to the contrary, that for some . For any , define the function , where is such that in , in , and . Assume that ρ is chosen in a such way that . Then as , that is
Note that by the boundedness of , the Hölder inequality, and since is compactly contained in , it follows that
and we have the following relations of limits
and
On the other hand, since has compact support and f has subcritical growth, we obtain
which gives
Therefore
which together with (3.20), gives
Now, using -(ii), we obtain
Taking the limit and using (3.20) and (3.22), we get
which yields a contradiction. This completes the proof of (3.15).
At this point, we know that , that is
On the other hand, by the weak convergence, it is easy to see that u is a weak solution to
where . Hence
Taking into account (3.15), (3.23) and (3.24), we can infer that in as . □
The functionalsatisfies thecondition onat any levelif, andif.
Let be a sequence for at level d. Then we have
Using Proposition 2.1-(c), we can see that is a sequence for in . Then, we can deduce from Lemma 3.6, that fulfills the condition in , so there exists such that, up to a subsequence,
Applying Lemma 2.3-(iii), we can infer that in . □
Now, we give the proof of the main result of this section:
Assume that conditions–and–hold. Then problem (
2.3
) admits a positive ground state for all.
In view of Lemma 2.2, we can apply a version of the mountain-pass theorem without (PS) condition (see [35]) to obtain a sequence such that and . By Lemma 3.2, is bounded in , so we may assume that in . Taking into account Lemma 3.3 and Lemma 3.4, we can assume that u is nontrivial. Now, we can prove that u is a critical point of . Indeed, for all we see that
where . Taking in the above identity and noting that (by Fatou’s Lemma), we obtain that . Let us prove that . Suppose to the contrary, that . Then there exists a unique such that . Therefore, by , and , it follows that
which gives a contradiction. Hence, and . From the above argument we can also deduce that so that . Since , where , and for , it is easy to check that in . Standard arguments (see [22,23,34]) show that for some , and using the Harnack inequality [20] we deduce that in . □
The autonomous problem
In this section we consider the limit problem associated with (2.3). More precisely, we deal with the following autonomous Kirchhoff problem:
The Euler–Lagrange functional associated with (4.1) is given by
which is well defined on the Hilbert space endowed with the inner product
The norm induced by the inner product is
The Nehari manifold associated with is given by
We denote by the open subset of defined as
and , where is the unit sphere of . We note that is a incomplete -manifold of codimension 1 modelled on and contained in . Thus for each , where . As in Section 2, we can see that the following results hold.
Assume that conditions–hold. Then the following assertions are true.
For each, letbe defined by. Then there is a uniquesuch that
There existsindependent of u such thatfor any. Moreover, for each compact setthere is a positive constantsuch thatfor any.
The mapgiven by, is continuous andis a homeomorphism betweenand. Moreover,.
If there is a sequencesuch that, thenand.
Let us define the maps
by and .
Assume that conditions–hold. Then the following assertions are true.
andfor everyand.
andfor every.
Ifis asequence for, thenis asequence for. Ifis a boundedsequence for, thenis asequence for the functional.
u is a critical point ofif, and only if,is a nontrivial critical point for. Moreover, the corresponding critical values coincide and
As in Section 2, we have the following variational characterization of the infimum of over :
Letbe asequence for, withifandif, and. Then only one of the alternatives below holds:
in;
there exist a sequenceand constantssuch that
Let us observe that, if is a sequence at level for the functional such that , then we can assume that . Otherwise, if and, if does not occur, in view of Lemma 4.2 we can find and such that
Setting , we can see that is a sequence for at the level , is bounded in and there exists such that and .
Now, we prove the following existence result for the autonomous problem:
Problem (
4.1
) admits a positive ground state solution.
Since has a mountain pass geometry, we can find (see [35]) a -sequence for at level . It is easy to see that is bounded in , so we may assume that in . By Remark 4.2, we may suppose that u is nontrivial. Now, we prove that u is a critical point of . Indeed, for all we can see that
where (by Fatou’s Lemma). Taking we have . Let us prove that . Suppose to the contrary that , then there exists a unique such that . Therefore, by and , it follows that
which gives a contradiction. Hence, and . From the above argument we can also deduce that so that . Since , where , and for , it is easy to check that in . Using the arguments in [22,23,34], for some , and from Harnack inequality [20] we obtain that in . Finally, we use a comparison argument to show the exponential decay of u. Since as and using , we can find such that
Let , and define with and for all . It is easy to check that
Since , we have
Set . Taking into account (4.2) and (4.3), we get
The maximum principle [20] implies that in and we deduce that for all . This completes the proof of theorem. □
The next lemma is a compactness result for the autonomous problem which will be useful later.
Letbe a sequence such that. Thenhas a convergent subsequence in.
Since and , we can apply Lemma 4.1-(iii) and Proposition 4.1-(d) and Remark 4.1 to infer that
and
Let us introduce the map defined as follows
We note that
Hence, applying the Ekeland variational principle [15] to , we can find such that is a sequence for on and . Then, using Proposition 4.1, Theorem 4.1 and arguing as in the proof of Corollary 3.1, the assertion follows. □
Finally, we prove the following useful relation between and :
.
Let ω be a positive ground state given by Theorem 4.1, and set , where with , , if and if . For simplicity, we assume that . Invoking the dominated convergence theorem, we see that
For each there exists such that
Then and this implies that
By , , and , it is easy to check that as . On the other hand, from the definition of and (4.5) we can see that
and using the fact that in and , we have
Taking into account that is bounded on the support of , as and (4.4), we can deduce that
On the other hand, in view of and , we know that for all , so we can conclude that as . □
The barycenter map and multiplicity of solutions to (1.1)
In this section, our main purpose is to apply the Ljusternik–Schnirelmann category theory to obtain a multiplicity result for problem (2.3). We begin proving the following technical results.
Letandbe such that. Then there existssuch that the translated sequencehas a subsequence which converges in. Moreover, up to a subsequence,is such that.
Since and , we can argue as in Lemma 3.2 to show that is bounded in . Let us observe that since . Therefore, proceeding as in Lemma 3.3, we can find a sequence and constants such that
Set . Then is bounded in and we may assume that
for some . Let be such that (see Lemma 4.1-(i)), and set . Then, using and , we can see that
which gives
In particular, (5.2) yields that is bounded in , so we may assume that . Obviously, is bounded and we may assume that . If , we get from the boundedness of , that , that is , in contrast with the fact that . Hence, . By the uniqueness of the weak limit, we have that and . Using Lemma 4.3, we deduce that
which implies that in and
Now, we show that admits a subsequence, still denoted by the same, such that . Assume to the contrary, that is not bounded, that is there exists a subsequence, still denoted by , such that . Since , we can see that
Take such that , and assume that . Then for any we get . From the definition of g we can deduce that
Since in , we can apply the dominated convergence theorem to get
Hence
which yields
Since and , we get a contradiction. Thus is bounded and, up to a subsequence, we may assume that . If , then there exists such that for any n large enough. Reasoning as before, we get a contradiction. Hence . Now, we show that . Assume to the contrary, that . Taking into account (5.3), Fatou’s Lemma and the invariance of by translations, we have
which is impossible. □
Now, we aim to relate the number of positive solutions of (2.3) to the topology of the set Λ. For this reason, we take such that
and consider a smooth non increasing function η defined in such that if , if , , and for some .
For any , we define
where is a positive ground state solution to (4.1) (such a solution exists by virtue of Theorem 4.1).
Let be the unique number such that
Finally, we consider defined by setting
The functionalsatisfies the following limit
Assume to the contrary, that there exist , and such that
Let us observe that using the change of variable , if , it follows that and then .
Then, recalling that for and for , we have
Now, we verify that the sequence satisfies as . It follows from the definition of that , namely,
Since in for all n large enough, we get from (5.6)
By the continuity of w we can find a vector such that
so that, using , we can deduce that
Now, assume to the contrary, that . Let us observe that the dominated convergence theorem yields
Hence, by , (5.7) and (5.8), we obtain a contradiction.
Therefore is bounded and, up to subsequence, we may assume that for some . Let us prove that . Suppose to the contrary, that . Then, taking into account (5.8) and assumptions and , we can see that (5.6) yields
which is impossible in view of Remark 2.2. Hence . Thus, by passing to the limit as in (5.6), we deduce from (5.8) that
Taking into account and using we can infer that . Then, letting in (5.5) and using and (5.8) we obtain
which contradicts (5.4). □
At this point, we are in a position to define the barycenter map. For any , we take such that , and we consider given by
We define the barycenter map as follows
Applying the dominated convergence theorem, it is easy to check that the function fulfills the following limit:
Now, we introduce a subset of by taking a function such that as , and setting
It follows from Lemma 5.2 that as . By the definition of , for any and , and . Moreover, we can prove a very interesting relation between and :
Let as . For any there exists such that
Therefore, it is suffices to prove that there exists such that
We note that , from which we deduce that
This yields . Using Lemma 5.1, we can find a sequence such that for n sufficiently large. Set and we see that
Since , we can deduce that , that is (5.9) indeed holds. □
In order to prove that (2.3) admits at least positive solutions, we recall the following useful abstract result whose proof can be found in [9,28].
Let I,andbe closed sets with, and letandbe two continuous maps such thatis homotopically equivalent to the embedding. Then.
Since is not a complete metric space, we cannot directly apply standard Ljusternik–Schnirelmann theory [28,35]. However, in the light of results in Section 2, we can make use of some abstract category results contained in [33].
Assume that conditions–and–hold. Then, given, there existssuch that, for any, problem (2.3) has at leastpositive solutions.
For any , we consider the map defined by .
Using Lemma 5.2, we see that
Set
where as by (5.10). Since , we have that for all .
By Lemma 2.3-(iii), Lemma 5.2, Lemma 5.3 and Lemma 5.4, we can find such that the following diagram
is well defined for any .
Thanks to Lemma 5.3, and decreasing if necessary, we see that for all , for some function such that uniformly in and for all . Then it is easy to check that with is a homotopy between and the inclusion map . This fact together with Lemma 5.5 implies that
Using Corollary 3.1, Lemma 4.4 and Theorem 27 in [33] with and , we can deduce that has at least critical points on . Taking into account Proposition 2.1-(d) and (5.11), we can infer that admits at least critical points in . □
This section is devoted to the proof of Theorem 1.1. Here we prove that the solutions obtained in Section 5 are indeed solutions of the original problem (1.1) for small enough.
First, we use a Moser iteration argument [29] to prove the following useful -estimate for the solutions of the modified problem (2.3).
Letandbe a solution to (
2.3
). Then, up to a subsequence, the translated sequence, and there existssuch thatwhereis given in Lemma
5.1
. Furthermore,uniformly in.
Since with , we can argue as in the proof of (5.1) to prove that . From Lemma 5.1 we can deduce that there is a sequence such that in , for some , , and .
Note that is a solution of the following problem
where and .
For any , , let such that , in , in and . For each and for , let
where , and to be determined later. Choosing as a test function in (6.1) we have
namely
Set . Since in with , we get for some positive constant C. Hence,
By assumptions and , for any there exists such that
Hence, using and choosing , we have
For each we can use Young’s inequality to obtain
and taking , we get
On the other hand, using the Sobolev inequality and the Hölder inequality, we can infer
Gathering (6.2) and (6.3), we have
At this point we can argue as in Lemma 4.5 in [3] to deduce the assertion. □
Now, we are ready to give the proof of our main multiplicity result:
Take such that . We begin by proving that there exists such that for any and any solution of (2.3),
Assume to the contrary that for some subsequence such that , we can find such that and
Since and , we can argue as in the proof of (5.1) to deduce that . In view of Lemma 5.1, we can find such that in and . Now, if we choose such that , we have that . Then for any
Therefore, for any n big enough,
Applying Lemma 6.1, there exists such that
from which
On the other hand, there exists such that for any we have
Consequently, for any and . which is impossible in view of (6.5).
Now, let be given by Theorem 5.1, and we fix where . In light of Theorem 5.1, we know that problem (2.3) admits at least nontrivial solutions. Let us denote by one of these solutions. Since satisfies (6.4), it follows by definition of g that is a solution of (2.1). Then is a solution to (1.1), and we can conclude that (1.1) has at least solutions.
Finally, we study the behavior of the maximum points of solutions for problem (2.1). Take and consider a sequence of solutions for (2.1) as above. Let us observe that implies that we can find such that
Arguing as before, we can find such that
Moreover, up to extract a subsequence, we may assume that
Indeed, if (6.9) does not hold, in view of (6.8), we see that . Then, using and (6.7), we can infer
which yields , and this is impossible. Hence (6.9) holds. Taking into account (6.8) and (6.9), we can deduce that if is a global maximum point of , then . Therefore for some . As a consequence, is a global maximum point of . Since for any and (in view of Lemma 5.1), we can infer from the continuity of V that
In what follows we prove the exponential decay of solutions of (1.1). Since as uniformly in , and using , we can find such that
Let , and define with and for all . It is easy to verify that
On the other hand, by , we have
Set . Putting together (6.10) and (6.11), we get
The maximum principle [20] implies that in and we deduce that for all and . Since solves (1.1), we obtain the desired estimate. This completes the proof of Theorem 1.1. □
Supercritical Kirchhoff problems
In this section we deal with the multiplicity of positive solutions for (1.5). After rescaling, we study the following Kirchhoff problem
where and the powers are such that . In what follows, we truncate the nonlinearity in a suitable way.
Let be a real number, whose value will be fixed later, and set
It is clear that satisfies the assumptions – ( with ). Moreover,
Therefore, we can consider the following truncated problem
It is easy to see that weak solutions of (7.3) are critical points of the energy functional defined by
where . We also consider the autonomous functional
Using Theorem 1.1, we know that for any and , there exists such that, for any , problem (7.3) admits at least positive solutions . Now, we prove that it is possible to estimate the -norm of these solutions uniformly with respect to μ. More precisely:
There existssuch thatfor anysufficiently small and uniformly in μ.
A simple inspection of the proof of Theorem 1.1 shows that any solution of (7.3) satisfies the following inequality
where is the mountain pass level related to the functional , and as . Then, decreasing if necessary, we can assume that
for any . Using the fact that for any , we can deduce that
for any . We can also note that
where in the last inequality we have used assumption . Putting together (7.5) and (7.6), we can infer that
for any . □
Now, our plan is to prove that is a solution of the original problem (7.1). To this end, we will show that we can find such that for any , there exists such that
In order to achieve our goal, we use a version of the Moser iteration technique [29]. For simplicity, we set . For any , we define , where will be chosen later, and let . Taking in (7.3), we see that
Putting together (7.8), (7.2) and , we get
where . On the other hand, by Theorem 2.1 and , we have
Taking into account (7.9) and (7.10), and using the Hölder inequality, we can deduce that
where and . In view of Lemma 7.1 and Lemma 2.1, we can see that
where
Now, we observe that if , we obtain from the definition of , , and (7.12),
Passing to the limit as in (7.13), the Fatou Lemma yields
whenever .
Now, we set , and observe that, since , the above inequality holds for this choice of β. Then, using the fact that , it follows that (7.14) holds with β replaced by . Therefore,
Iterating this process and recalling that , we can infer that for every ,
Taking the limit in (7.15) as and using Lemma 7.1, we get
where and
Next, we will find some suitable values of K and μ such that the following inequality holds
or equivalently,
Take such that
and fix satisfying
Then, thanks to (7.16), we obtain that
that is is a solution of (7.1). This completes the proof of Theorem 1.2.
Footnotes
Acknowledgements
The first author was partly supported by Gnampa project “Studio Di Problemi Frazionari Nonlocali Tramite Tecniche Variazionali”. The second author was partly supported by Slovenian research agency grants P1-0292, N1-0114, N1-0064 and J1-8131.
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