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
we define by
where
(13)
It is well known in p-Laplacian 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
(14)
Fix such that , let be such that and
(15)
Furthermore, let us define by
(16)
(where ξ is defined similarly as it has been done in (4) and (5)).
3.2 Main theorem
Theorem 4 Suppose that are positive, continuous nonlinearities satisfying (H1) and (H2) (with constants and defined in (14) and (16), respectively), and let the weight functions be continuous, nonnegative and not identically null. Then problem (P) has at least a nontrivial positive solution. Moreover, if is 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
where .
We claim that
In fact, suppose that there exist and such that
(17)
where .
Since , it follows that
Let be the solution of
where
(18)
Then
Note that given , we obtain
and by the maximum principle, we have
Thus, by (17) we obtain
Moreover, if , we have
As , we conclude from (18) that there is such that
Noting that and considering
(19)
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.