Research Article | Open Access
Multiple Positive Solutions for Semilinear Elliptic Equations with Sign-Changing Weight Functions in
Existence and multiplicity of positive solutions for the following semilinear elliptic equation: in , , are established, where if if , , satisfy suitable conditions, and maybe changes sign in . The study is based on the extraction of the Palais-Smale sequences in the Nehari manifold.
In this paper, we deal with the multiplicity of positive solutions for the following semilinear elliptic equation: where , ( if , if ) and are measurable functions and satisfy the following conditions: , where , and there exist and such that , , is bounded and has a compact support in . There exist , and such that
Semilinear elliptic equations with concave-convex nonlinearities in bounded domains are widely studied. For example, Ambrosetti et al.  considered the following equation: where , . They proved that there exists such that () admits at least two positive solutions for all , has one positive solution for and no positive solution for . Actually, Adimurthi et al. , Damascelli et al. , Korman , Ouyang and Shi , and Tang  proved that there exists such that () in the unit ball has exactly two positive solutions for , has exactly one positive solution for and no positive solution exists for . For more general results of () (involving sign-changing weights) in bounded domains; see, the work of Ambrosetti et al. in , of Garcia Azorero et al. in , of Brown and Wu in , of Brown and Zhang in , of Cao and Zhong in , of de Figueiredo et al. in , and their references.
However, little has been done for this type of problem in . We are only aware of the works [13–17] which studied the existence of solutions for some related concave-convex elliptic problems (not involving sign-changing weights). Furthermore, we do not know of any results for concave-convex elliptic problems involving sign-changing weight functions except [18, 19]. Wu in  have studied the multiplicity of positive solutions for the following equation involving sign-changing weights: where the parameters . He also assumed that is sign chaning and , where and satisfy suitable conditions and proved that () has at least four positive solutions.
In a recent work , Hsu and Lin have studied () in with a sign-changing weight function. They proved there exists such that () has at least two positive solutions for all provided that , satisfy suitable conditions and maybe changes sign in .
In order to describe our main result, we need to define where , and is the best Sobolev constant for the imbedding of into .
This paper is organized as follows. In Section 2, we give some notations and preliminary results. In Section 3, we establish the existence of a local minimum. In Section 4, we prove the existence of a second solution of ().
At the end of this section, we explain some notations employed. In the following discussions, we will consider with the norm . We denote by the best constant which is given by The dual space of will be denoted by . denote the dual pair between and . We denote the norm in by for . is a ball in centered at with radius . denotes as . , will denote various positive constants, the exact values of which are not important.
2. Preliminary Results
Associated with (1.3), the energy functional defined by for all is considered. It is well-known that and the solutions of () are the critical points of .
Lemma 2.1. is coercive and bounded from below on .
Proof. If , then by , (2.3), and the Hölder and Sobolev inequalities, one has Since , it follows that is coercive and bounded from below on .
The Nehari manifold is closely linked to the behavior of the function of the form for . Such maps are known as fibering maps and were introduced by Drábek and Pohozaev in  and are also discussed by Brown and Zhang in . If , we have It is easy to see that and so, for and , if and only if that is, the critical points of correspond to the points on the Nehari manifold. In particular, if and only if . Thus, it is natural to split into three parts corresponding to local minima, local maxima, and points of inflection. Accordingly, we define and note that if , that is, , then
We now derive some basic properties of , , and .
Lemma 2.2. Suppose that is a local minimizer for on and , then in .
Proof. See the work of Brown and Zhang in [10, Theorem 2.3].
Lemma 2.3. If , then .
Proof. We argue by contradiction. Suppose that there exists such that . Then for by (2.9) and the Sobolev inequality, we have and so Similarly, using (2.10), Hölder and Sobolev inequalities, we have which implies Hence, we must have which is a contradiction.
In order to get a better understanding of the Nehari manifold and fibering maps, we consider the function defined by Clearly, if and only if . Moreover, and so it is easy to see that if , then . Hence, (or ) if and only if (or ).
Let . Then, by (2.17), has a unique critical point at , where and clearly is strictly increasing on and strictly decreasing on with . Moreover, if , then
Therefore, we have the following lemma.
Lemma 2.4. Let and . (i)If , then there exists a unique such that , is inceasing on and decreasing on . Moreover, (ii)If , then there exist unique such that , , is decreasing on , inceasing on and decreasing on (iii).(iv)There exists a continuous bijection between and . In particular, is a continuous function for .
Proof. See the work of Hsu and Lin in [19, Lemma 2.5].
(i) If , then we have .
(ii) If , then for some .
In particular, for each , we have .
Proof. See the work of Hsu and Lin in [19, Theorem 3.1].
3. Existence of a Positive Solution
First, we define the Palais-Smale (simply by ) sequences, -values, and -conditions in for as follows.
Definition 3.1. (i) For , a sequence is a -sequence in for if and strongly in as .
(ii) is a -value in for if there exists a -sequence in for .
(iii) satisfies the -condition in if any -sequence in for contains a convergent subsequence.
Now we will ensure that there are -sequence and -sequencein on and , respectively, for the functional .
Proposition 3.2. If , then (i)there exists a -sequence in for .(ii)there exists a -sequence in for .
Proof. See Wu [21, Proposition 9].
Now, we establish the existence of a local minimum for on .
Proof. From Proposition 3.2(i) it follows that there exists satisfying
By Lemma 2.1 we infer that is bounded on . Passing to a subsequence (Still denoted by ), there exists such that as
By , Egorov theorem and Hölder inequality, we have
By (3.1) and (3.2), it is easy to see that is a solution of (). From and (2.4), we deduce that
Let in (3.4). By (3.1), (3.3) and , we get
Thus, is a nonzero solution of ().
Next, we prove that strongly in and . From the fact and applying Fatou's lemma, we get This implies that and . Standard argument shows that strongly in . By Theorem 2.5, for all we have that and which implies . Since and , by Lemma 2.2 we may assume that is a nonzero nonnegative solution of (). By Harnack inequality  we deduce that in . Finally, by (2.10), Hölder and Sobolev inequlities, and thus we conclude the proof.
4. Second Positive Solution
Lemma 4.1. Assume that and hold. If is a -sequence for , then is bounded in .
Proof. See the work of Hsu and Lin in [19, Lemma 4.1].
Let us introduce the problem at infinity associated with (): We state some known results for problem (). First of all, we recall that by Lions  has studied the following minimization problem closely related to problem (): where . Note that a minimum exists and is attained by a ground state in such that where . Gidas et al.  showed that for every , there exist positive constants , such that for all , We define Clearly, .
Lemma 4.2. Let be a domain in . If satisfies then
Proof. We know . Then, Since as , then the lemma follows from the Lebesgue dominated convergence theorem.
Lemma 4.3. Under the assumptions , - and . Then there exists a number such that for In particular, for all .
Proof. (i) First, since for all and is continuous in and , we infer that there exists such that
(ii) Since , there exists such that if , we get for . Then, for Thus, there exists such that for any and we get
(iii) By (i) and (ii), we need to show that there exists such that for We know that . Then, , we have Suppose satisfies , we get for all and some positive constant . By (4.3) and Lemma 4.3, there exists such that for any By and (4.3), we get By , (4.3) and Lemma 4.3, we have Since and and using (4.13)–(4.16), we have there exists such that for all , then This implies that if , then for all we get From for all and (4.17), we have Combining this with Lemma 2.4(ii), from the definition of and , for all , we obtain that there exists such that and
Proof. Let be a -sequence for with . We know from Lemma 4.1 that is bounded in , and then there exist a subsequence of (still denoted by ) and such that
It is easy to see that and by , Egorov theorem and Hölder inequality, we have
Next we verify that . Arguing by contradiction, we assume . By , for any , there exists such that for all . Since strongly in for , is a bounded sequence in , therefore . Setting , then , we have We set Since and is bounded, then by (4.22), we can deduce that that is,
If , then we get , which contradicts to . Thus we conclude that . Furthermore, by the definition of we obtain Then, as , we have which implies that Hence, from (4.2) and (4.22)–(4.29), we get This is a contradiction to . Therefore, is a nonzero solution of ().
Now, we establish the existence of a local minimum of on .
Proof. If , then by Theorem 2.5(ii), Proposition 3.2(ii) and Lemma 4.3(ii), there exists a -sequence in for with . From Lemma 4.4, there exist a subsequence still denoted by and a nonzero solution of () such that weakly in .
First, we prove that . On the contrary, if , then by is closed in , we have . From (2.9) and for all , we get By Lemma 2.4(ii), there exists a unique such that . If , then it is easy to see that From (3.1), and (4.32), we can deduce that which is a contradiction. Thus, .
Next, by the same argument as that in Theorem 3.3, we get that strongly in and for all . Since and , by Lemma 2.2 we may assume that is a nonzero nonnegative solution of (). Finally, by the Harnack inequality  we deduce that in .
Now, we complete the proof of Theorem 1.1. By Theorems 3.3, 4.5, we obtain () has two positive solutions and such that , . Since , this implies that and are distinct. It completes the proof of Theorem 1.1.
- A. Ambrosetti, H. Brézis, 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.
- Adimurthi, F. Pacella, and S. L. Yadava, “On the number of positive solutions of some semilinear Dirichlet problems in a ball,” Differential and Integral Equations, vol. 10, no. 6, pp. 1157–1170, 1997.
- L. Damascelli, M. Grossi, and F. Pacella, “Qualitative properties of positive solutions of semilinear elliptic equations in symmetric domains via the maximum principle,” Annales de l'Institut Henri Poincaré. Analyse Non Linéaire, vol. 16, no. 5, pp. 631–652, 1999.
- P. Korman, “On uniqueness of positive solutions for a class of semilinear equations,” Discrete and Continuous Dynamical Systems, vol. 8, no. 4, pp. 865–871, 2002.
- T. Ouyang and J. Shi, “Exact multiplicity of positive solutions for a class of semilinear problem. II,” Journal of Differential Equations, vol. 158, no. 1, pp. 94–151, 1999.
- M. Tang, “Exact multiplicity for semilinear elliptic Dirichlet problems involving concave and convex nonlinearities,” Proceedings of the Royal Society of Edinburgh, vol. 133, no. 3, pp. 705–717, 2003.
- A. Ambrosetti, J. Garcia Azorero, and I. Peral, “Multiplicity results for some nonlinear elliptic equations,” Journal of Functional Analysis, vol. 137, no. 1, pp. 219–242, 1996.
- J. Garcia Azorero, J. J. Manfredi, and I. Peral Alonso, “Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations,” Communications in Contemporary Mathematics, vol. 2, no. 3, pp. 385–404, 2000.
- K. J. Brown and T. F. Wu, “A fibering map approach to a semilinear elliptic boundary value problem,” Electronic Journal of Differential Equations, vol. 2007, no. 69, pp. 1–9, 2007.
- K. J. Brown and Y. Zhang, “The Nehari manifold for a semilinear elliptic equation with a sign-changing weight function,” Journal of Differential Equations, vol. 193, no. 2, pp. 481–499, 2003.
- D. M. Cao and X. Zhong, “Multiplicity of positive solutions for semilinear elliptic equations involving critical Sobolev exponents,” Nonlinear Analysis, vol. 29, no. 4, pp. 461–483, 1997.
- D. G. de Figueiredo, J. P. Gossez, and P. Ubilla, “Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity,” Journal of the European Mathematical Society, vol. 8, no. 2, pp. 269–286, 2006.
- J. Chabrowski and J. M. Bezzera do Ó, “On semilinear elliptic equations involving concave and convex nonlinearities,” Mathematische Nachrichten, vol. 233-234, pp. 55–76, 2002.
- K. J. Chen, “Combined effects of concave and convex nonlinearities in elliptic equation on ,” Journal of Mathematical Analysis and Applications, vol. 355, no. 2, pp. 767–777, 2009.
- J. V. Goncalves and O. H. Miyagaki, “Multiple positive solutions for semilinear elliptic equations in involving subcritical exponents,” Nonlinear Analysis, vol. 32, no. 1, pp. 41–51, 1998.
- Z. Liu and Z. Q. Wang, “Schrödinger equations with concave and convex nonlinearities,” Zeitschrift fur Angewandte Mathematik und Physik, vol. 56, no. 4, pp. 609–629, 2005.
- T. F. Wu, “Multiplicity of positive solutions for semilinear elliptic equations in ,” Proceedings of the Royal Society of Edinburgh, vol. 138, no. 3, pp. 647–670, 2008.
- T. F. Wu, “Multiple positive solutions for a class of concave-convex elliptic problems in involving sign-changing weight,” Journal of Functional Analysis, vol. 258, no. 1, pp. 99–131, 2010.
- T. S. Hsu and H. L. Lin, “Multiple positive solutions for semilinear elliptic equations in involving concave-convex nonlinearities and sign-changing weight functions,” Abstract and Applied Analysis, vol. 2010, Article ID 658397, 21 pages, 2010.
- P. Drábek and S. I. Pohozaev, “Positive solutions for the -Laplacian: application of the fibering method,” Proceedings of the Royal Society of Edinburgh, vol. 127, no. 4, pp. 703–726, 1997.
- T. F. Wu, “On semilinear elliptic equations involving concave-convex nonlinearities and sign-changing weight function,” Journal of Mathematical Analysis and Applications, vol. 318, no. 1, pp. 253–270, 2006.
- N. S. Trudinger, “On Harnack type inequalities and their application to quasilinear elliptic equations,” Communications on Pure and Applied Mathematics, vol. 20, pp. 721–747, 1967.
- P. L. Lions, “On positive solutions of semilinear elliptic equations in unbounded domains,” in Nonlinear Diffusion Equations and Their Equilibrium States II, vol. 13 of Mathematical Sciences Research Institute, pp. 85–122, Springer, New York, NY, USA, 1988.
- B. Gidas, W. M. Ni, and L. Nirenberg, “Symmetry and related properties via the maximum principle,” Communications in Mathematical Physics, vol. 68, no. 3, pp. 209–243, 1979.
Copyright © 2011 Tsing-San Hsu. 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.