Abstract
In this article, we prove the existence of a nontrivial positive solution for the elliptic system
where denotes the pLaplacian operator, and Ω is a smooth bounded domain in (). The weight functions ω and ρ are continuous, nonnegative and not identically null in Ω, and the nonlinearities f and g are continuous and satisfy simple hypotheses of local behavior, without involving monotonicity hypotheses or conditions at ∞. We apply the fixed point theorem in a cone to obtain our result.
MSC: 35B09, 35J47, 58J20.
1 Introduction
Coupled systems involving quasilinear operators as the pLaplacian have been a theme of interest for researchers of partial differential equations. In this paper we prove the existence of a nontrivial positive solution for the elliptic system
where denotes the pLaplacian operator defined by , and Ω denotes a smooth bounded domain in (). In other words, we will prove the existence of a pair such that satisfies (P), with u and v strictly positive in Ω. The weight functions are continuous, nonnegative in Ω and positive in for some . The nonlinearities are continuous, g is positive in for some , and both satisfy simple hypotheses of local behavior.
We suppose that the nonlinearity f is superlinear at origin and f, g are allowed to be sub or superlinear at ∞. Moreover, there is no monotonicity hypotheses on these nonlinearities. We suppose the existence of positive constants such that
where the constants and depend only on ω, ρ and Ω (see Figure 1). These constants will be defined later on in this paper and, as proved in [1], , where is the first eigenvalue of the pLaplacian operator.
Figure 1. The nonlinearitiesfandgsatisfy (H1) and (H2).
Elliptic problems concerning the existence of positive solutions for equations and systems of equations related to Dirichlet problems have been studied in several papers during the last decades. In this way, many existence results for systems involving the pLaplacian operator in general bounded domains in have been considered in recent articles. In particular, systems as (P) have been studied in articles in [26] for example.
The main interest of this paper is studying systems whose nonlinearities present some kind of coupling as (P). A paper which deals with this sort of problem is [2], where problem (P) is considered under the assumptions , . In this paper, among other hypotheses, the nonlinearities f and g are supposed to be at least continuous if and locally Holder continuous with exponent if . Moreover, both are supposed to be nondecreasing in , nonnegative at origin and satisfying (for ) the fundamental condition
Schauder’s fixed point theorem, the LeraySchauder degree and a variant of Krasnoselskii’s method are applied to guarantee the existence of a positive solution for (P).
The studies of [2] were extended by Hai and Shivaji in [4] (for ), [5] (for ) and by Hai in [3] (for ). In these papers, the authors deal with problem (P), and (in [4] and [5], ), with no sign conditions on or and without monotonicity conditions on f or g. In this way, semipositone cases were also considered in these papers (for more details about semipositone problems, see [7] and the references therein).
In [4], the nonlinearities f and g in (P) are supposed to be , monotone and satisfying the following conditions at ∞
in addition to condition (1). The existence of a positive solution is guaranteed for large λ by applying the subsupersolution method.
The paper [5] deals with problem (P) in the particular case and a positive solution is guaranteed by applying the sub and supersolution method and Schauder’s fixed point theorem. The nonlinearities considered are continuous and there exist positive numbers L, K such that and for . Moreover, the authors considered, as it has been done in [2], condition (1) with .
In [3], the author obtains necessary and sufficient conditions for the existence of positive solutions for problem (P). The nonlinearities f and g are positive, continuous and nondecreasing in , with g strictly positive for , and
sublinear at 0 and ∞. In this paper, the maximum principle and fixed point arguments are applied to guarantee the existence of a solution.
Another paper dealing with the existence of a positive solution for a class of coupled systems is [6]. In this paper, the authors studied problem (P), with and , in which λ is a positive parameter. The existence of a solution is guaranteed via the method of sub and supersolution if, among other assumptions, the functions a and b considered are signchanging functions that may be negative near the boundary and the positive nonlinearities f and g are supposed to be and nondecreasing.
Recently, many articles have applied fixed point results to prove the existence of positive solutions of partial differential equations or systems (see, for example, [1,812]). In this paper we study problem (P) in general domains, assuming that (H1) and (H2) hold. As system (P) has no variational structure, our main arguments are based on fixedpoint index and comparison theorems, following the ideas of [8,10] and [11]. In particular, our assumptions on the nonlinearities do not involve monotonicity hypotheses or sublinearity conditions at ∞.
Our strategy is as follows. At first, we show an existence result for the radial case when , applying a fixed point theorem in a cone. Afterwards, we utilize this result to prove our main existence result for (P), when is a bounded smooth domain. In this case, we do a symmetrization of weigh functions and combine comparison theorems with a new application of the fixed point theorem.
For completeness, we will consider concrete examples of coupled systems for which it is possible to apply our method to guarantee the existence of at least one positive solution. It will be clear in some of these examples that conditions (1), (2) and (3) are not required in our method.
2 The radial case
Let us consider the radial version of problem (P)
where and the weight functions are radial, continuous, nonnegative and not identically null functions. The positive functions f and g are supposed to be continuous and satisfying local conditions that depend on the positive constants defined in (6) and (7).
Let (the inverse of the wellknown function ), and let us define
Consider the real number such that
Moreover, define the positive constants and by
and
It is easy to see that . In fact,
Remark 1 If () is the unitary ball and are the weight functions in problem (P), it is easy to see that and are given by
and
in which and are the conjugate exponents of p and q, respectively.
Finally we assume that the nonlinearities f and g satisfy the local conditions.
Now, we are in a position to state the main result of this section: the existence of a positive radial solution for (P_{r}).
Theorem 2Suppose thatare positive, continuous nonlinearities satisfying (H1) and (H2) (with constantsanddefined in (6) and (7)), and let the radial weight functionsbe continuous, nonnegative and not identically null. Then problem (P_{r}) has at least a nontrivial positive solution. Moreover, ifis a positive solution for (P_{r}), then
To prove the last theorem, we will apply a wellknown result of the fixedpoint index theory, known as a fixed point cone theorem (see, for example, [13]).
Lemma 3LetEbe a Banach space andbe its norm. LetKbe a cone inE. For, defineand denote its boundary by, that is, . Suppose that
is completely continuous.
then
In what follows, we will consider with the norm
Proof of Theorem 2 It is easy to see that is a solution of (P_{r}) if and only if is a fixed point of given by
where
and
Moreover, it is straightforward that the operator T is completely continuous.
In fact, by (H1) we obtain
and
As a consequence of Lemma 3, it follows that
Now, we will prove that . In order to prove that, we will show that exists such that
and let
We claim that
In fact, let us suppose that there are and such that
(Note that it is immediate that . In fact, if and if , we have , which contradicts .)
As , we have for all . If , by (11) and (H2) we obtain
which contradicts (12).
Then
and, consequently,
Thus, T has a nontrivial fixed point in . The regularity of the solution follows from classical results of Lieberman and Tolksdorf (see [14] and [15]). □
3 The general case
Now we will establish the main result of this paper: the existence of a nontrivial positive solution for (P) when is a smooth bounded domain.
3.1 The constants and in Ω
For the general problem
where
It is well known in pLaplacian operator theory that T is completely continuous. Moreover, a simple maximum principle argument guarantees that in Ω for .
As it has been done in the radial case, in order to obtain a result of existence for (P), we will apply Lemma 3.
Let us denote by the solution of
where and is such that . As ρ and w satisfy the same hypotheses as h, we can define
Fix such that , let be such that and
(where ξ is defined similarly as it has been done in (4) and (5)).
3.2 Main theorem
Theorem 4Suppose thatare positive, continuous nonlinearities satisfying (H1) and (H2) (with constantsanddefined in (14) and (16), respectively), and let the weight functionsbe continuous, nonnegative and not identically null. Then problem (P) has at least a nontrivial positive solution. Moreover, ifis a positive solution for (P), then
Proof Let be such that . It follows from (H1) and (13) that
and
Therefore, by the maximum principle, we conclude that
and, consequently,
In the same way, we obtain
and, as a consequence of the last two inequalities, we have
It follows from Lemma 3 that
We claim that . To prove our claim, we will show, as it has been done in the radial case, that there exists such that
For fixed , let be such that and
Consider
We claim that
In fact, suppose that there exist and such that
where
Then
and by the maximum principle, we have
Thus, by (17) we obtain
As , we conclude from (18) that there is such that
we have
by the fact that if . Repeating the same ideas of the radial case, we conclude that
which contradicts (19). By the additivity of index, it follows that
proving the theorem. As in the radial case, the regularity follows from [14] and [15]. □
Remark 5 Due to the hypotheses on the nonlinearities and on the weight functions, simple applications of the maximum principle allow us to guarantee that if is a solution of problem (P), then both u and v are strictly positive in Ω.
4 Examples
Example 6 Let us consider the problem
where and () is a smooth domain. If and , problem (20) has at least one positive solution.
Considering constants (8) and (9), it is enough to consider such that
and
Choosing δ and M as above, hypotheses (H1) and (H2) are verified and, as a consequence of Theorem 2, we guarantee the existence of a positive solution for coupled system (20). Furthermore, according to Theorem 4, it is easy to see that if is the considered positive solution, we have as large as α is.
One of the advantages of our method is that conditions (1), (2) and (3) are not required. Let us see examples of these situations.
Consider the problem
where , () is a smooth domain and is a nonlinearity satisfying (H1) and (H2). Examples of will be presented in the following examples.
Example 7 Consider the nonlinearity in problem (23) given by
where , and M is the positive constant whose existence is guaranteed in Example 6. If and , the same arguments as those applied in Example 6 can guarantee the existence of a positive solution to (23).
It is clear that according to the constant k, condition (1) is not satisfied by the nonlinearities of problem (23). In fact, if and , simple calculations show that
and condition (1) does not hold.
Example 8 Now, let us consider given by (24) as the nonlinearity of problem (23). In this case, we have
In this way, can be either sub or superlinear at +∞ according to the constant k. Therefore, we have an example in which we guarantee the existence of a positive solution even if condition (3) is not satisfied.
Example 9 Finally, let us consider problem (23), with the nonlinearity given by
in which M is the positive constant whose existence is guaranteed in Example 6, and . As a consequence of previous examples, it is straightforward to guarantee the existence of a positive solution to this problem. Furthermore, it is clear that condition (2) does not hold.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
The authors thank for the support of FAPEMIG and Universidade Federal de Ouro Preto.
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