Abstract
This paper deals with the existence of positive solutions for the elliptic problems with sublinear and superlinear nonlinearities in , in , on , where is a real parameter, . is a bounded domain in , and and are some given functions. By means of variational method and super-subsolution method, we obtain some results about existence of positive solutions.
1. Introduction
In this paper, we consider the elliptic problems with sublinear and superlinear nonlinearities where is a real parameter, . is a bounded domain in , and and are some given functions which satisfies the following assumptions:(), , , where , are positive constants,
or(), , , where is a positive constant.
For convenience, we denote with hypothesis or by and , respectively.
Such problems occur in various branches of mathematical physics and population dynamics, and sublinear analogues or superlinear analogues of have been considered by many authors in recent years (see [1–9] and their references). But most of such studies have been concerned with equations of the type involving sublinear nonlinearity (see [3–6, 8, 9]), with only few references dealing with the elliptic problems with sublinear and superlinear nonlinearities. In [1], Ambrosetti et al. deal with the analogue of with . It is known from [2] that there exist , such that problem has a solution if and has no solution if , provided on .
Our goal in this paper is to show how variational method and super-subsolution method can be used to establish some existence results of problem . We work on the Sobolev space equipped with the norm . For we define by
Let be the first eigenvalue of
denotes the corresponding eigenfunction satisfying . , , denotes Lebesgue spaces, and the norm in is denoted by .
2. The Existence of Positive Solution of
It is well known that Define ; from (2.1) we know , so we can split the domain into two parts: and , where . Let ; we obtain that by the positivity of in , and is nonempty when is small enough.
Theorem 2.1. Let , satisfy assumption , and , where is the limiting exponent in the Sobolev embedding. Then there exists a constant such that possesses at least a weak positive solution for .
Proof. Let denote the positive solution of the following equation:
Here and hereafter we use the following notations: , , . Since , for all , there exists satisfying
Observing that , as a consequence, the function verifies
and hence it is a supersolution of . Let , , . For , we have or . We will discuss it from two conditions.
(I) For all , observing that and when is small enough, we have
Since , then it follows that , . From (2.5) we infer
Multiplying (2.6) with , we get
It follows that
(II) For all , there exists , such that for all , and we have
Since , then we have (and ). From (2.9), it follows that
From (2.8) and (2.10), we derive that there exists such that for all , for all ,
that is, is a subsolution of . Taking as sufficiently large, we also have by minimal principle. Define , and let , then is closed and convex (and weakly closed). Let , for all . We consider the function
Observe that , ; we infer that is coercive, bounded, since it is blow and weakly lower semicontinuous. Using this fact, we conclude that there exists , such that (see [10]). In the following, we will prove that is a solution of problem .
For , define , such that
Clearly, achieves its minimum at , and
For all , , define
Obviously, , and inserting (2.15) into (2.14), we find
Since and are supersolution and subsolution, respectively, then
Observe that meas, meas, as ,
Since , it follows that
Similar to (2.19), we have
Similar to (2.18), as , it follows that
As , we also have
Inserting (2.17), (2.19), and (2.20) into (2.16), we find
Dividing by and letting , using (2.18), (2.21), and (2.22), we derive
Noting that is arbitrary, this holds equally for , and it follows that is indeed a weak solution of , and the strong maximum principle yields , in . Therefore it is a weak positive solution of .
3. The Existence of Positive Solution of
Theorem 3.1. Let , satisfy assumption , and . Then there exists , , such that(i)for all problem has a minimal solution such that . Moreover is increasing with respect to ;(ii)for problem has at least one weak solution ;(iii)for all problem has no solution.
To prove Theorem 3.1, let us define First of all we prove a useful lemma.
Lemma 3.2. One has .
Proof. Let denote the solution of the following equation:
Since , we can find such that for all there exists satisfying
As a consequence, the function verifies
and hence it is a supersolution of . Moreover, let denote the solution of the following problem:
(From [3] we know that exists.) Then is a subsolution of , provided
which is satisfied for all small enough and all . Taking as possibly smaller, we also have
It follows that has a solution , whenever , and thus .
Next, let be such that
If is such that has a solution , multiplying by and integrating over we find
This and (3.5) immediately imply that and show that , hence .
We are now ready to give the proof of Theorem 3.1.
Proof. (i) From the proof of lemma, it follows that, for all , problem has a solution . Let satisfy (3.5); the iteration
satisfies by making use of Lemma 3.3 of [1] and maximum principle. It is easy to check that is a minimal solution of . Indeed, if is any solution of , then and is a supersolution of . Thus , for all , by induction, and . Next, we will prove that . Indeed,
Since is a solution of we have
From Lemma 3.5 of [1], we know
In particular with , we infer
Combining (3.12) and (3.14), we obtain
To complete the proof of (i), it remains to show that
Indeed, if then is a supersolution of . Since, for small, is a subsolution of and , then possesses a solution , with
Since is the minimal solution of , we infer that . Moreover
Since (because ), then the Hopf Maximum principle yields .
(ii) Let be a sequence such that ; then from we deduce that there exists such that
Then there exists such that a.e. in , strongly in and weakly in . Such a is thus a weak solution of for .
(iii) This follows from the definition of .
Acknowledgment
This work supported by the Physics and Mathematics Foundation of Changzhou University (ZMF10020065).