Mathematical Problems in Engineering

Volume 2010 (2010), Article ID 640841, 10 pages

http://dx.doi.org/10.1155/2010/640841

## 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 [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).

#### References

- A. Ambrosetti, H. Brezis, and G. Cerami, “Combined effects of concave and convex nonlinearities in some elliptic problems,”
*Journal of Functional Analysis*, vol. 122, no. 2, pp. 519–543, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - Z. Tan and Z. Yao, “Existence of multiple solutions for semilinear elliptic equation,”
*Advances in Mathematics*, vol. 31, no. 4, pp. 343–354, 2002. View at Google Scholar - H. Brezis and S. Kamin, “Sublinear elliptic equations in ${\mathbb{R}}^{N}$,”
*Manuscripta Mathematica*, vol. 74, no. 1, pp. 87–106, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - H. Brezis and L. Oswald, “Remarks on sublinear elliptic equations,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 10, no. 1, pp. 55–64, 1986. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - K. Taira and K. Umezu, “Positive solutions of sublinear elliptic boundary value problems,”
*Nonlinear Analysis: Theory, Methods & Applications*, vol. 29, no. 7, pp. 761–771, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Ghergu and V. Rădulescu, “Nonradial blow-up solutions of sublinear elliptic equations with gradient term,”
*Communications on Pure and Applied Analysis*, vol. 3, no. 3, pp. 465–474, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Ghergu and V. D. Rădulescu,
*Singular Elliptic Problems: Bifurcation and Asymptotic Analysis*, vol. 37 of*Oxford Lecture Series in Mathematics and Its Applications*, The Clarendon Press, Oxford University Press, Oxford, UK, 2008. - R. Kajikiya, “Comparison theorem and uniqueness of positive solutions for sublinear elliptic equations,”
*Archiv der Mathematik*, vol. 91, no. 5, pp. 427–435, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - R. Kajikiya, “A priori estimates of positive solutions for sublinear elliptic equations,”
*Transactions of the American Mathematical Society*, vol. 361, no. 7, pp. 3793–3815, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH - M. Struwe,
*Variational Methods*, Springer, Berlin, Germany, 1990.