Multiple Positive Solutions for Classes of p-laplacian Equations

Transcription

1 K $ Multiple Positive Solutions for Classes of p-laplacian quations Mythily Ramaswamy and Ratnasingham Shivaji We study positive Abstract solutions to classes of boundary value problems of the form on where denotes the p-laplacian operator defined by div! #$ is a parameter and is a bounded domain in./ ; 024 with of class (If 0 + in! &%('*),+#- ) and connected., we assume that is a bounded open interval.) In particular, we establish existence of three positive solutions for classes of nondecreasing, p-sublinear functions belonging to 65 -&786. Our proofs are based on sub-super solution techniques. Introduction We consider weak solutions to classes of boundary value problems of the form 9;: =< > A@CD<A in F (.) < > on HIF AMS subject classification 5J70, 5J55 J TATA Institute for Fundamental Research Centre, IISc Campus, angalore ,India, mythily@math.tifrbng.res.in Department of Mathematics and Statistics, Mississippi State University, Mississippi State, MS 9762, USA, e- mail: shivaji@ra.msstate.edu

2 2 $ where : denotes the p-laplacian operator defined by : > div #$ positive parameter and F is a bounded domain in / with HIF in class, is a and connected. (If >, we assume that F is a bounded open interval.) y a weak solution of (.), we mean, a function <! F that satisfies $# #$ # <% <'&( *) D<A+)-,/.0) 2! F However, in this paper, we in fact study the existence and multiplicity of 4 F solutions, that are strictly positive in F. Throughout this paper our classes of satisfy: 76 8, 9 is a nondecreasing function such : and C # > ( - For such positone -sublinear nonlinearities, it is easy to establish that there is a positive solution for every D #. Further is nondecreasing, uniqueness of the positive solution for every # follows from [DS]. In this paper we will consider the case C is not monotonic. In particular, we for which there exists F and such that IH F H and J F, K > F is sufficiently large. For such classes of nonlinearities we discuss the existence of three positive solutions for a certain range of. Our work extends the multiplicity result of [IS], where the authors study S-shaped bifurcation curves for the Laplacian case ( >N ). In [IS] the reen s function played a crucial role in the proof. However, here in the -Laplacian case new ideas are required to overcome the non availability of the reen s function. We now state our main result. Theorem. There exists a positive constant > OP,QR, F such that if J F, S for some points F and, F H, then the equation. has at least three positive solutions for a certain range of. Remark : Recently in [COS], the authors study a multiplicity result for a class of positone -sublinear problems via the antimaximum principle. However this requires rather restrictive assumptions for small < and also do not extend the work in [IS] in a natural way.

3 H We establish Theorem. by the method of sub-super solutions. y a super solution mean a function F (4 F such that > on HIF and # # #$ &( I) and by a sub solution we mean a function # #$ # &( I) +)-,/. ) (.2) F (4 F such that > on HIF +)-,/.0)D, (.) where >! F 2 in F. Then by the weak comparison principle (see [FT] or [DKT]), if there exist sub and super solutions and respectively such that in F then (.) has a 4 F solution < such that <. We prove multiplicity by a sub-super solution result for the -Laplacian case discussed in [COS]. This result extends the corresponding result for the Laplacian (p=2) case proved in [A] and [S]. The result is as follows: Lemma. be nonnegative and nondecreasing and suppose there exist a sub solution a strict super solution, a strict sub solution $ and a super solution $ for (.) such that H $, H $ H $ and $. Then (.) has at least three distinct solutions <=>,, such that < H < $-H < $. We will prove Theorem., for the case when F is a ball in Section 2. Here the proof depends heavily on the construction of a crucial positive subsolution. In Section, we extend the theorem for general domains, by using a simple variant of this subsolution. Finally in Section 4 we will discuss a popular example arising in combustion theory., 2 Case when is a ball In this section we shall prove the theorem in the case when F is a ball of radius. Lemma 2. There exists a positive constant > &OP,QR, such that for any positive number if then there exists a subsolution of the equation (.) on, with.

4 4 Proof of Lemma 2. : Let us define, for some, and, A > 9 9 H and let A >. Denoting > 9 9, $( > we have that for H *H, 9 8> 9 $ &, and hence Then define as the radially symmetric solution of 9 9;: A A % & in > on H (2.) Then satisfies 9 / & > & D > 8 > 8, where for any real, >5 $ Integrating once, we get for H IH, Observe that being monotone 9 & > 9 > /! also is continuous and monotone. Hence, /! &! (2.2) We claim that. (2.)

5 >, 5 Then from 2. it follows that that is a subsolution is monotone. oth at. Thus, in order to show 2., it is enough to show that and vanish. (2.4) from equation (2.2) while > for, it is enough to verify equation(2.4) in the range. For * we have! A and hence from (2.2), using the monotonicity of 9 / Thus (2.4) will hold also for all, if / S 9 which is the same as / 9 > # 9 Thus if / / 9 then 2.4 holds. To get the best possible lower bound for, we can take > 4>, since is for any > 8>. Define > < / 9 / # Then < 9 # / This infimum is attained for the value > 9 which gives P,Q,. This proves the lemma.

6 6 H Theorem 2. There exists a positive constant > OP,QR, such that if J F, S for some points F and, F H, then the equation (.) on has at least three positive solutions for a certain range of. Proof: We shall construct supersolutions and $ and subsolutions and $ as in Lemma.. Clearly > is a subsolution for every Let > F where (4 F is the solution of 9: > in F, > on H F. Then 9: > F and hence a supersolution if F # > (say). Note that > F. Next let $> Then by Lemma 2., $ is a subsolution such that $ # if (say). Note that if J F, : where > OP,, > then H. Finally, let $ > A. Then 9;: $ A& A@ $, and hence a supersolution for any given, if A is chosen sufficiently large so that # # This is possible since the is -sublinear. Here since H can also choose A large enough so that $- $ and $- solutions for in on HIF, we Ḧ. Hence there exist three positive Proof of Theorem. In this section we will prove Theorem. in general domains. First we construct a positive subsolution A in F with. Let be the largest inscribed ball in F and OP,, be as in Lemma 2.. Assume J # F,, and let be the subsolution constructed in in Lemma 2.. Now define A > % if and A > if F 9. Then F (4 F and > on HIF. Further, on we have 9;: A> 9: Q % A@C % A, while outside we have 9;: A > C A& ). Hence A is a subsolution in F for with. The rest of the proof of Theorem. is identical to that of the proof of Theorem 2. except that here we define $ >.

7 9 7 4 Application in Combustion Theory Here we consider the example 9: < > P6 < < in F < > on H F. The D<A> 6 < < arises in the theory of combustion and it was discussed in [IS] and many references cited within for the case when > (Laplacian case). In [IS] the authors prove that a necessary condition for multiple positive solutions is. Further they prove that if is large enough then there are at least three positive solutions for a ceratin range of. Here we will establish similar results for the - Laplacian case. satisfies hypothesis < #. Also a simple calculation shows < is nondecreasing if I 9 (. Hence a necessary condition for multiplicity is O 9 (. Further choosing F > and > we have J F, > # F # 8> 6 Hence given any positive constant > O,Q, F, for large we have J,, and thus there exists at least three positive solutions for a certain range of by Theorem.. References [A] H. Amman, Fixed point equations and nonlinear eigenvalue problems in ordered anach spaces, SIAM Rev., 8(4) (976), pp [IS] K. J. rown, M. M. A. Ibrahim and R. Shivaji, S-shaped bifurcation curves, J. of Nonlinear Analysis, TMA, Vol. 5, No. 5 (98) pp [COS] C. Maya, S. Oruganti and R. Shivaji, Positive solutions for classes of -Laplacian quations, To appear in Diff. Int. qns.

8 8 [DKT] P. Drabek, P. Krejci and P. Takac, Nonlinear Differential quations, Chapman&Hall/CRC, 999. [DS] J. I. Diaz and J.. Saa, xistence et unicite de solutions positives pour certainesequations elliptiques quasilineaires, Comptes Rendus Acad. Sc. paris, Serie I, 05 (98) pp [FT] J. Fleckinger and P. Takac, Uniqueness of positive solutions for nonlinear cooperative systems with the p-laplacian, Indiana Univ. Math. J. 4(4) (994) pp [S] R. Shivaji, A remark on the existence of three solutions via sub-super solutions, Nonlinear Analysis and Application, Lecture notes in pure and applied mathematics, Vol. 09 (987), pp , d. V. Lakshmikantham.