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Mathematical Problems in Engineering
Volume 2010, Article ID 640841, 10 pages
http://dx.doi.org/10.1155/2010/640841
Research Article

Positive Solution for the Elliptic Problems with Sublinear and Superlinear Nonlinearities

College of Physics and Mathematics, Changzhou University, Changzhou, Jiangsu 213164, China

Received 8 October 2010; Revised 13 December 2010; Accepted 13 December 2010

Academic Editor: Jyh Horng Chou

Copyright © 2010 Chunmei Yuan et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 [19] and their references). But most of such studies have been concerned with equations of the type involving sublinear nonlinearity (see [36, 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).

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