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ABSTRACT
We consider an anisotropic -equation, with a parametric and superlinear reaction term. We show that for all small values of the parameter the problem has at least five nontrivial smooth solutions, four with constant sign and the fifth nodal (sign-changing). The proofs use tools from critical point theory, truncation and comparison techniques, and critical groups.
2010 Mathematics Subject Classifications:
1. Introduction
Let be a bounded domain with a
-boundary
. In this paper, we study the following anisotropic
-equation
In this problem, the exponent
is Lipschitz continuous and
. By
, we denote the variable exponent (anisotropic) p-Laplacian, defined by
The reaction of the problem is parametric, with
being the parameter. The function
is measurable in
, continuous in
. We assume that
is
-superlinear as
(
) but without satisfying the usual in such cases Ambrosetti-Rabinowitz condition (the AR-condition for short). Our goal is to prove a multiplicity theorem for problem (
) providing sign information for all the solutions produced. Using variational tools from the critical point theory, together with suitable truncation and comparison techniques and also Morse Theory (critical groups), we show that for all small values of the parameter
the problem has at least five nontrivial smooth solutions (four of constant sign and the fifth nodal (sign-changing)).
Theorem 1.1
If hypotheses
hold, then there exists
such that for all
problem (
) has at least five nontrivial smooth solutions
nodal.
Remark 1.1
The hypotheses and spaces
are defined in the next section. We stress that the above multiplicity theorem provides sign information for all the solutions.
Anisotropic equations arise in a variety of models of physical processes. We mention the works of Bahrouni-Rădulescu-Repovš [Citation1] (transonic flow problems), Ržička [Citation2] (electrorheological and magnetorheological fluids), Zhikov [Citation3] (nonlinear elasticity theory), and Agarwal-Alghamdi-Gala-Ragusa [Citation4], Ragusa-Tachikawa [Citation5] (double phase problems). Recently there have been some existence and multiplicity results for various types of -equations with nonstandard growth. We refer to the works of Gasiński-Papageorgiou [Citation6], Rădulescu-Repovš [Citation7], Rădulescu [Citation8], Papageorgiou-Rădulescu-Repovš [Citation9], Papageorgiou-Scapellato [Citation10], Papageorgiou-Vetro [Citation11], Zhang-Rădulescu [Citation12]. They produce at most three nontrivial solutions, but no nodal solutions. We also mention the recent isotropic works of Li-Rong-Liang [Citation13], Papageorgiou-Vetro-Vetro [Citation14] producing two positive solutions for
- and
-equations, respectively, and the recent work of Papageorgiou-Scapellato [Citation15] who consider a different class of parametric equations (superlinear perturbations of the standard eigenvalue problem) and produce seven solutions, all with sign information.
2. Mathematical background – hypotheses
The analysis of problem () requires the use of Lebesgue and Sobolev spaces with variable exponents. A comprehensive treatment of such spaces can be found in the book of Diening-Hajulehto-Hästö-Ržička [Citation16].
Given , we define
Let
and
(as usual we identify two measurable functions which differ only on a Lebesgue null set). Given
, we define the variable exponent Lebesgue space
as follows
This vector space is equipped with the so-called ‘Luxemburg norm’
defined by
Then
becomes a separable, uniformly convex (hence also reflexive) Banach space. The reflexivity of these spaces leads to the reflexivity of the corresponding Sobolev spaces, which we introduce below. In reflexive Banach spaces bounded sequences have w-convergent subsequences (Eberlein-Šmulian theorem). We will be using this fact repeatedly. The dual
is given by
with
defined by
for all
(that is,
for all
). Also we have the following version of Hölder's inequality
If
and
for all
, then
continuously.
Now that we have variable exponent Lebesgue spaces, we can define variable exponent Sobolev spaces. So, if , then we define
with
being the weak gradient of u. This Sobolev space is equipped with the norm
When
is Lipschitz continuous (that is,
), then we define the Dirichlet anisotropic Sobolev space
by
Both spaces
and
are separable and uniformly convex (hence reflexive) Banach spaces.
If , then we define the critical Sobolev exponent corresponding to
by setting
Suppose that
,
and
for all
(resp.
for all
). Then the anisotropic Sobolev embedding theorem says that
The same embedding theorem remains true also for
provided
. Moreover, in this case the Poincaré inequality is true, namely, we can find
such that
This means that on the anisotropic Sobolev space
we can consider the equivalent norm
The following modular function is very helpful in the study of the anisotropic Lebesgue and Sobolev spaces. So, let
. We define
For
, we define
.
The modular function and the Luxemburg
are closely related.
Proposition 2.1
If and
then
(a) | for all | ||||
(b) |
| ||||
(c) | |||||
(d) |
Suppose that . We have
Then we introduce the operator
defined by
The next proposition summarizes the main properties of this operator (see Gasiński-Papageorgiou [Citation17], Proposition 2.5, and Rădulescu-Repovš [Citation7], p. 40).
Proposition 2.2
If and
is defined as above, then
is bounded
maps bounded sets to bounded sets
continuous, strictly monotone
hence also maximal monotone
and of type
that is, if
in
and
then
in
Given , we set
. Then for
, we define
for all
. We know that
If
are measurable functions such that
for a.a.
, then we define
and
.
We write if and only if for every compact
, we have
for a.a.
. Evidently, if
and
for all
, then
.
Besides the anisotropic Lebesgue and Sobolev spaces, we will also use the ordered Banach space . The positive (order) cone of
is
. This cone has a nonempty interior given by
with
being the outward unit normal on
.
Suppose X is a Banach space and . We set
We say that
satisfies the ‘C-condition’, if it has the following property:
‘Every sequence such that
admits a strongly convergent subsequence’.
Given , we set
.
Suppose . For every
, by
we denote the
-singular homology group with integer coefficients for the pair
. Let
be isolated and
. Then the critical groups of φ at u are defined by
where U is an open neighborhood of u such that
. The excision property of singular homology implies that this definition is independent of the choice of the isolating neighborhood U.
In the sequel, for economy in the notation, by we will denote the norm of the Sobolev space
(
). On account of the Poincaré inequality mentioned earlier, we have
Now we are ready to introduce our hypotheses on the data of problem (
).
:
and
for all
.
:
is a Carathéodory function such that
for a.a.
, and
for a.a.
, all
, with
,
with
for all
;
if
, then
there exists
such that
there exists
such that
for every
, there exists
such that for a.a.
, the function
is nondecreasing on
and for every s>0, we have
for a.a.
, all
.
Remark 2.1
Hypotheses imply that for a.a.
,
is
-superlinear. However, this superlinearity condition on
is not formulated using the AR-condition which is common in the literature when dealing with superlinear problems (see, for example, Fan-Deng [Citation18], Theorem 1.3). Instead we use condition
which incorporates in our framework superlinear nonlinearities with slower growth as
, which fail to satisfy the AR-condition. Consider for example the function
with
. This function satisfies hypotheses
, but fails to satisfy the AR-condition. Hypothesis
implies the presence of a concave term near zero.
3. Constant sign solutions - multiplicity
In this section, we show that for small, problem (
) has solutions of constant sign (positive and negative solutions). First we look for positive solutions. To this end, we introduce the
-functional
defined by
Working with
, we can produce multiple positive smooth solutions when
is small.
Proposition 3.1
If hypotheses
hold, then there exists
such that for all
problem (
) has at least two positive solutions
.
Proof.
On account of hypotheses , we have
(1)
(1)
Then for every
, we have
If
, then by Proposition 2.1 and the Poincaré inequality, we have
. Also recall that
and
continuously. Therefore, for
with
, we have
(2)
(2)
Let
and consider
with
. Then from (Equation2
(2)
(2) ) we have
(3)
(3)
The choice of
and since
, imply that
Hence we can find
such that
Then from (Equation3
(3)
(3) ) we see that
(4)
(4)
Let
denote the principal eigenvalue of the Dirichlet Laplacian and
the corresponding positive,
-normalized (that is,
) eigenfunction. We know that
(see for example, Gasiński-Papageorgiou [Citation19], p. 739). On account of hypothesis
, given
, we can find
such that
(5)
(5)
Since
, we can find
small such that
for all
. Then
(6)
(6)
Note that
Therefore from (Equation6
(6)
(6) ), we have
Since
(see hypothesis
), choosing
even smaller if necessary, we have
(7)
(7)
Using the anisotropic Sobolev embedding theorem (see Section 2), we infer that
is sequentially weakly lower semicontinuous. The ball
is sequentially weakly compact (recall that
is a reflexive Banach space and use the Eberlein-Šmulian theorem). So, by the Weierstrass-Tonelli theorem, we can find
such that
(8)
(8)
From (Equation7
(7)
(7) ) and (Equation8
(8)
(8) ), it follows that
Moreover, from (Equation4
(4)
(4) ) and (Equation8
(8)
(8) ), we infer that
(9)
(9)
From (Equation9
(9)
(9) ) we see that
is an interior point in
and a minimizer of
. Hence
(10)
(10)
In (Equation10
(10)
(10) ) we choose
and obtain
From (Equation10
(10)
(10) ), we have that
is a positive solution of problem (
) with
. From Fan-Zhao [Citation20, Theorem 4.1] (see also Gasiński-Papageorgiou [Citation6, Proposition 3.1]), we have that
. Then from Tan-Fang [Citation21, Corollary 3.1] (see also Fukagai-Narukawa [Citation22, Lemma 3.3]), we have that
. Finally, the anisotropic maximum principle of Zhang [Citation23] implies that
.
Now let and consider
. From the previous analysis, we know that problem
has a positive solution
. We will show that we can have
(11)
(11)
First we will show that we can have a solution
of
such that
. To this end let
(12)
(12)
This is a Carathéodory function. We set
and consider the
-functional
defined by
From Proposition 2.1 and (Equation12
(12)
(12) ), we see that
is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find
such that
(13)
(13)
As before, using hypothesis
and choosing
small so that we also have
(see Papageorgiou-Rădulescu-Repovš [Citation24], Proposition 4.1.22, p. 274 and recall that
), we will have
From (Equation13
(13)
(13) ) we have
(14)
(14)
In (Equation14
(14)
(14) ), we choose
and have
Next in (Equation14
(14)
(14) ) we choose
. We have
So, we have proved that
(15)
(15)
As before, from the anisotropic regularity theory and the anisotropic maximum principle, imply that
. So, we have produced a solution
of
such that
(see (Equation15
(15)
(15) )).
Now, let and let
be as postulated by hypothesis
. We have
(16)
(16)
Recall that
. So, on account of hypothesis
, we have
Then from (Equation16
(16)
(16) ) and Proposition 2.4 of Papageorgiou-Rădulescu-Repovš [Citation9], we infer that (Equation11
(11)
(11) ) is true.
Using , we introduce the following truncation of
(17)
(17)
We set
and consider the
-functional
defined by
From (Equation17
(17)
(17) ), we see that
(18)
(18)
From the first part of the proof, we know that
is a local minimizer of
. Then (Equation11
(11)
(11) ) and (Equation18
(18)
(18) ) imply that
(19)
(19)
(see Tan-Fang [Citation21], Theorem 3.2 and Gasiński-Papageorgiou [Citation6], Proposition 3.3). Using (Equation17
(17)
(17) ), we can easily check that
(20)
(20)
This implies that we may assume that
(21)
(21)
(otherwise we already have a whole sequence of distinct positive smooth solutions of (
) and so we are done). Then (Equation21
(21)
(21) ), (Equation19
(19)
(19) ) and Theorem 5.7.6, p. 449, of Papageorgiou-Rădulescu-Repovš [Citation24], imply that we can find
small such that
(22)
(22)
If
, then from hypothesis
we have
(23)
(23)
Moreover, (Equation18
(18)
(18) ) and Proposition 4.1 of Gasiński-Papageorgiou [Citation6], implies that
(24)
(24)
From (Equation22
(22)
(22) )–(Equation24
(24)
(24) ), we see that we can use the mountain pass theorem and obtain
such that
(25)
(25)
From (Equation25
(25)
(25) ) and (Equation17
(17)
(17) ), it follows that
is a positive solution of problem (
) (
),
.
In a similar fashion, we can generate two negative smooth solutions when is small. In this case, we start with the
-functional
defined by
Using this functional and reasoning as in the ‘positive’ case, we have the following multiplicity result.
Proposition 3.2
If hypotheses
hold, then there exists
such that for all
problem (
) has at least two negative solutions
.
4. Extremal constant sign solutions
Let be the set of positive solutions of (
) and
be the set of negative solutions of (
). We know that:
In this section, we show that
has a smallest element
, that is,
for all
and
has a biggest element
, that is,
for all
. We call
and
the ‘extremal’ constant sign solutions of (
). In Section 5 these solutions will be used to produce a nodal (sign-changing) solution of (
). Indeed, if we can produce a nontrivial solution of (
) in the order interval
distinct from
and
, on account of the extremality of
and
, this solution will be nodal.
To produce the extremal constant sign solutions, we need some preparation. Let and let
. On account of hypotheses
, we can find
such that
(26)
(26)
This unilateral growth restriction on
, leads to the following auxiliary anisotropic
-problem
For this problem, we have the following result
Proposition 4.1
If hypotheses hold, then for every
problem (
) has a unique positive solution
, and since problem (
) is odd,
is the unique negative solution.
Proof.
Consider the -functional
defined by
Since
, from this last inequality, we infer that
is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find
such that
(27)
(27)
Let
. We have
From the choice of η, we see that
Therefore, we can write that
Since
, taking
even smaller if necessary, we have
From (Equation27
(27)
(27) ) we have
(28)
(28)
In (Equation28
(28)
(28) ), we choose
and obtain
Then from (Equation27
(27)
(27) ), we see that
is a positive solution of (
). As before (see the proof of Proposition 3.1), the anisotropic regularity theory and the anisotropic maximum principle, imply that
Next we show the uniqueness of this positive solution. To this end, we consider the integral functional
defined by
From Theorem 2.2 Taká-Giacomoni [Citation25], we know that
is convex. Let
(the effective domain of
) and suppose
is another positive solution of (
). Again we have that
. Hence using Proposition 4.1.22, p. 274, of Papageorgiou-Rădulescu-Repovš [Citation24], we have
Let
. Then for
small, we have
Thus the convexity of
implies the Gateaux differentiability of
at
and at
in the direction h. Moreover, a direct calculation using Green's identity (see also [Citation25], Theorem 2.5), gives
The convexity of
implies the monotonicity of
. Hence
This proves the uniqueness of the positive solution
of (
). Since the equation is odd, it follows that
is the unique negative solution of (
),
.
The solution (resp.
), will provide a lower bound (resp. an upper bound) for the solution set
(resp.
). These bounds are important in generating the extremal constant sign solutions.
Proposition 4.2
If hypotheses
hold, then
for all
and
for all
.
Proof.
Let and consider the Carathéodory function
defined by
(29)
(29)
We set
and consider the
-functional
defined by
It follows that
is coercive. Also, it is sequentially weakly lower semicontinuous. So, we can find
such that
(30)
(30)
As before (see the proof of Proposition 4.1), for
small, we will have
From (Equation30
(30)
(30) ), we have
(31)
(31)
In (Equation31
(31)
(31) ), first we choose
. We obtain
Next in (Equation31
(31)
(31) ), we choose
. We have
So, we have proved that
(32)
(32)
From (Equation32
(32)
(32) ), (Equation29
(29)
(29) ) and (Equation31
(31)
(31) ), we see that
is a positive solution of (
). Then Proposition 4.1 implies that
Similarly we show that
for all
.
Now we are ready to produce the extremal constant sign solutions for problem (). Let
.
Proposition 4.3
If hypotheses
hold and
, then problem (
) has a smallest positive solution
and a biggest negative solution
.
Proof.
From Papageorgiou-Rădulescu-Repovš [Citation26] (see the proof of Proposition 4.3), we have that is downward directed (that is, if
, then we can find
such that
,
). Therefore using Lemma 3.10, p. 178, of Hu-Papageorgiou [Citation17], we can find
decreasing such that
We have
(33)
(33)
(34)
(34)
In (Equation33
(33)
(33) ) we choose
and obtain
(35)
(35)
From (Equation34
(34)
(34) ), (Equation35
(35)
(35) ) and hypothesis
, we infer that
So, we may assume that
(36)
(36)
In (Equation33
(33)
(33) ) we choose
, pass to the limit as
and use (Equation36
(36)
(36) ). We obtain
(37)
(37)
Therefore, if in (Equation33
(33)
(33) ), we pass to the limit as
and use (Equation37
(37)
(37) ), we obtain
For the negative solutions, we know that
is upward directed (that is, if
, then we can find
such that
,
). Reasoning as above, we produce
such that
for all
.
5. Nodal solutions
In this section using the extremal constant sign solutions and following the approach outlined in the beginning of Section 4, we will produce a nodal (sign-changing) solution for problem (),
.
Let and
be the two extremal constant sign solutions produced in Proposition 4.3. We introduce the Carathéodory function
defined by
(38)
(38)
We also consider the positive and negative truncations of
, namely, the Carathéodory functions
(39)
(39)
We set
and
, and then introduce the
-functionals
defined by
Using (Equation38
(38)
(38) ) and (Equation39
(39)
(39) ) and arguing as in the proof of Proposition 4.2, since
are the extremal constant sign solutions, we obtain the following proposition.
Proposition 5.1
If hypotheses
hold and
then
,
,
.
The next result will allow the use of the mountain pass theorem.
Proposition 5.2
If hypotheses
hold and
then the two extremal constant sign solutions
and
are local minimizers of
.
Proof.
From (Equation38(38)
(38) ), (Equation39
(39)
(39) ) and hypothesis
, we have
Therefore
Also, it is sequentially weakly lower semicontinuous. So, we can find
such that
Since
, from Proposition 5.1, we infer that
But from (Equation38
(38)
(38) ) and (Equation39
(39)
(39) ), it is clear that
It follows that
Similarly using this time
, we show that
is a local minimizer of
.
It is clear from Proposition 5.1, that we may assume that
(40)
(40)
Otherwise, we already have a sequence of distinct smooth nodal solutions of (
) and so we are done.
Also, we may assume that
(41)
(41)
The reasoning is similar if the opposite inequality holds.
Proposition 5.3
If hypotheses
hold and
then the problem (
) has a nodal solution
.
Proof.
From (Equation40(40)
(40) ), (Equation41
(41)
(41) ) and Theorem 5.7.6, p. 449, of Papageorgiou-Rădulescu-Repovš [Citation24], we can find
small such that
(42)
(42)
Also
is coercive (see (Equation38
(38)
(38) )). Hence Proposition 5.1.15, p. 369, of Papageorgiou-Rădulescu-Repovš [Citation24], implies that
(43)
(43)
Then (Equation42
(42)
(42) ) and (Equation43
(43)
(43) ) permit the use of the mountain pass theorem. So, we can find
such that
(44)
(44)
From (Equation44
(44)
(44) ), (Equation38
(38)
(38) ) and (Equation42
(42)
(42) ), we infer that
From Theorem 6.5.8, p. 527, of Papageorgiou-Rădulescu-Repovš [Citation24], we know that
(45)
(45)
On the other hand, hypothesis
and Proposition 4.2 of Leonardi-Papageorgiou [Citation27] imply that
(46)
(46)
Comparing (Equation45
(45)
(45) ) and (Equation46
(46)
(46) ), we conclude that
and so
is a nodal solution of (
).
This also proves Theorem 1.1.
Acknowledgements
The authors wish to thank a knowledgeable referee for his/her constructive remarks and criticisms.
Disclosure statement
No potential conflict of interest was reported by the author(s).
Additional information
Funding
References
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