Báo cáo hóa học: "Research Article Positive Solutions for Multiparameter Semipositone Discrete Boundary Value Problems via Variational Method" pptx

Hindawi Publishing Corporation Advances in Difference Equations Volume 2008, Article ID 840458, 15 pages doi:10.1155/2008/840458 Research Article Positive Solutions for Multiparameter Semipositone Discrete Boundary Value Problems via Variational Method Jianshe Yu, 1, 2 Benshi Zhu, 1 and Zhiming Guo 2 1 College of Mathematics and Econometrics, Hunan University, Changsha 410082, China 2 College of Mathematics and Information Sciences, Guangzhou University, Guangzhou 510006, China Correspondence should be addressed to Jianshe Yu, jsyu@gzhu.edu.cn Received 13 March 2008; Accepted 24 August 2008 Recommended by Kanishka Perera We study the existence, multiplicity, and nonexistence of positive solutions for multiparameter semipositone discrete boundary value problems by using nonsmooth critical point theory and subsuper solutions method. Copyright q 2008 Jianshe Yu 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. 1. Introduction Let Z and R be the set of all integers and real numbers, respectively. For a, b ∈ Z, define Za{a, a  1, }, Za, b{a, a  1, ,b}, when a ≤ b. In this paper, we consider the multiparameter semipositone discrete boundary value problem −Δ 2 ut − 1λfut  μgut,t∈ Z1,N, u00,uN  10, 1.1 where λ, μ > 0 are parameters, N ≥ 4 is a positive integer, Δutut1−ut is the forward difference operator, Δ 2 utΔΔut, f : 0, ∞ → R is a continuous positive function satisfying f0 > 0, and g : 0, ∞ → R is continuous and eventually strictly positive with g0 < 0. We notice that for fixed μ>0, λf0μg0 < 0 whenever λ>0issufficiently small. We call 1.1 a semipositone problem. Semipositone problems are derived from 1, where Castro and Shivaji initially called them nonpositone problems, in contrast 2 Advances in Difference Equations with the terminology positone problems, put forward by Keller and Cohen in 2, where the nonlinearity was positive and monotone. Semipositone problems arise in bulking of mechanical systems, design of suspension bridges, chemical reactions, astrophysics, combustion, and management of natural resources; for example, see 3–6. In general, studying positive solutions for semipositone problems is more difficult than that for positone problems. The difficulty is due to the fact that in the semipositone case, solutions have to live in regions where the nonlinear term is negative as well as positive. However, many methods have been applied to deal with semipositone problems, the usual approaches are quadrature method, fixed point theory, subsuper solutions method, and degree theory. We refer the readers to the survey papers 7, 8 and references therein. Due to its importance, in recent years, continuous semipositone problems have been widely studied by many authors, see 9–15. However, we noticed that there were only a few papers on discrete semipositone problems. One can refer to 16–18. In these papers, semipositone discrete boundary value problems with one parameter were discussed, and subsuper solutions method and fixed point theory were used to study them. To the authors’ best knowledge, there are no results established on semipositone discrete boundary value problems with two parameters. Here we want to present a different approach to deal with this topic. In 11, Costa et al. applied the nonsmooth critical point theory developed by Chang 19 to study the existence and multiplicity results of a class of semipositone boundary value problems with one parameter. We think it is also an efficient tool in dealing with the semipositone discrete boundary value problems with two parameters. Our main objective in this paper is to apply the nonsmooth critical point theory to deal with the positive solutions of semipositone problem 1.1. More precisely, we define the discontinuous nonlinear terms f 1 s ⎧ ⎨ ⎩ 0ifs ≤ 0, fs if s>0, g 1 s  0ifs ≤ 0, gs if s>0. 1.2 Now we consider the slightly modified problem −Δ 2 ut − 1λf 1 ut  μg 1 ut,t∈ Z1,N, u00,uN  10. 1.3 Just to be on the convenient side, we define hsλfsμgs, h 1 sλf 1 sμg 1 s, HsλFsμGs, H 1 sλF 1 sμG 1 s, where Fs  s 0 fτdτ, Gs  s 0 gτdτ, F 1 s  s 0 f 1 τdτ   0ifs ≤ 0, Fs if s>0, G 1 s  s 0 g 1 τdτ   0ifs ≤ 0 Gs if s>0. 1.4 Jianshe Yu et al. 3 We will prove in Section 3 that the sets of positive solutions of 1.1 and 1.3 do coincide. Moreover, any nonzero solution of 1.3 is nonnegative. Our main results are as follows. Theorem 1.1. Suppose that there are constants C 1 > 0, α>1, and β>2 such that when s>0 is large enough, fs <C 1 s α , 1.5 sfs ≥ βFs > 0, 1.6 lim s → ∞ gs s  0. 1.7 Then for fixed μ>0,thereisa λ>0 such that for λ ∈ 0, λ, problem 1.3 has a nontrivial nonnegative solution. Hence problem 1.1 has a positive solution. Remark 1.2. By 1.6, there are constants C 2 ,C 3 > 0 such that for any s ≥ 0, Fs ≥ C 2 s β − C 3 . 1.8 Equations 1.6 and 1.8 imply that lim s → ∞ fs s ∞, 1.9 which shows that f is superlinear at infinity. Remark 1.3. Equation 1.7 implies that g is sublinear at infinity. Moreover, it is easy to know that lim s → ∞ Gs s 2  0. 1.10 Hence G is subquadratic at infinity. Theorem 1.4. Suppose that the conditions of Theorem 1.1 hold. Moreover, g is increasing on 0, ∞. Then there is a μ ∗ > 0 such that for μ>μ ∗ , problem 1.1 has at least two positive solutions for sufficiently small λ. Theorem 1.5. Suppose that the conditions of Theorem 1.1 hold. Moreover, f is nondecreasing on 0, ∞. Then for fixed μ>0, problem 1.1 has no positive solution for sufficiently large λ. 2. Preliminaries In this section, we recall some basic results on variational method for locally Lipschitz functional I : X → R defined on a real Banach space X with norm ·. I is called locally 4 Advances in Difference Equations Lipschitzian if for each u ∈ X, there is a neighborhood V  V u of u and a constant B  Bu such that |Ix − Iy|≤Bx − y, ∀x, y ∈ V. 2.1 The following abstract theory has been developed by Chang 19. Definition 2.1. For given u, z ∈ X, the generalized directional derivative of the functional I at u in the direction z is defined by I 0 u; zlim sup k → 0 t → 0 1 t Iu  k  tz − Iu  k. 2.2 The following properties are known: i z → I 0 u; z is subadditive, positively homogeneous, continuous, and convex; ii |I 0 u; z|≤Bz; iii I 0 u; −z−I 0 u; z. Definition 2.2. The generalized gradient of I at u, denoted by ∂Iu, is defined to be the subdifferential of the convex function I 0 u; z at z  0, that is, w ∈ ∂Iu ⊂ X ∗ ⇐⇒  w,z≤I 0 u; z, ∀z ∈ X. 2.3 The generalized gradient ∂Iu has the following main properties. 1 For all u ∈ X, ∂Iu is a nonempty convex and w ∗ -compact subset of X ∗ ; 2 w X ∗ ≤ B for all w ∈ ∂Iu. 3 If I, J : X → R are locally Lipschitz functional, then ∂I  Ju ⊂ ∂Iu∂Ju. 2.4 4 For any λ>0, ∂λIuλ∂Iu. 5 If I is a convex functional, then ∂Iu coincides with the usual subdifferential of I in the sense of convex analysis. 6 If I is G ˆ ateaux differential at every point of v of a neighborhood V of u and the G ˆ ateaux derivative is continuous, then ∂Iu{I  u}. 7 The function ζu min w∈∂Iu w X ∗ 2.5 exists, that is, there is a w 0 ∈ ∂Iu such that w 0  X ∗  min w∈∂Iu w X ∗ . 8 I 0 u; zmax{w,z|w ∈ ∂Iu}. Jianshe Yu et al. 5 9 If I has a minimum at u 0 ∈ X, then 0 ∈ ∂Iu 0 . Definition 2.3. u ∈ X is a critical point of the locally Lipschitz functional I if 0 ∈ ∂Iu. Definition 2.4. I is said to satisfy Palais-Smale condition PS condition for short if any sequence {u n } such that Iu n  is bounded and ζu n min w∈∂Iu n  w X ∗ → 0 has a convergent subsequence. Lemma 2.5 see 19, Mountain Pass Theorem. Let X be a real Hilbert space and let I be a locally Lipschitz functional satisfying (PS) condition. Suppose t hat I00 and that the following hold. i There exist constants ρ>0 and a>0 such that Iu ≥ a if u  ρ. ii Thereisane ∈ X such that e >ρand Ie ≤ 0. Then I possesses a critical value c ≥ a. Moreover, c can be characterized as c  inf γ∈Γ max s∈0,1 Iγs, 2.6 where Γ{g ∈ C0, 1,X | γ00,γ1e}. 2.7 Next we give the definitions of the subsolution and the supersolution of the following boundary value problem: −Δ 2 ut − 1μgut,t∈ Z1,N, u00,uN  10. 2.8 Definition 2.6. If u 1 t,t∈ Z0,N 1 satisfies the following conditions: −Δ 2 u 1 t − 1 ≤ μgu 1 t,t∈ Z1,N, u 1 0 ≤ 0,u 1 N  1 ≤ 0, 2.9 then u 1 is called a subsolution of problem 2.8. Definition 2.7. If u 2 t,t∈ Z0,N 1 satisfies the following conditions: −Δ 2 u 2 t − 1 ≥ μgu 2 t,t∈ Z1,N, u 2 0 ≥ 0,u 2 N  1 ≥ 0, 2.10 then u 2 is called a supersolution of problem 2.8. 6 Advances in Difference Equations Lemma 2.8. Suppose that there exist a subsolution u 1 and a supersolution u 2 of problem 2.8 such that u 1 t ≤ u 2 t in Z1,N. Then there is a solution ˇ u of problem 2.8 such that u 1 t ≤ ˇ ut ≤ u 2 t in Z1,N. Remark 2.9. If 2.8 is replaced by 1.1, then we have similar definitions and results as Definitions 2.6, 2.7,andLemma 2.8 3. Proof of main results Let E be the class of the functions u : Z0,N 1 → R such that u0uN  10. Equipped with the usual inner product and the usual norm u, v N  t 1 ut,vt, u   N  t 1 u 2 t  1/2 , 3.1 E is an N-dimensional Hilbert space. Define the functional J on E as Ju 1 2 N1  t 1  Δut − 1 2 − 2H 1 ut   1 2 u T Au − N  t 1 H 1 ut  Ku − N  t 1 H 1 ut, 3.2 where u  {u1,u2, ,uN}, Ku1/2u T Au and A  ⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ 2 −10··· 00 −12−1 ··· 00 0 −12··· 00 ··· ··· ··· ··· ··· ··· 000··· 2 −1 000··· −12 ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ N×N . 3.3 Clearly, H 1 is a locally Lipschitz function and Ju is a locally Lipschitz functional on E.Bya simple computation, we obtain ∂ ∂ut Ku2ut − ut  1 − ut − 1−Δ 2 ut − 1. 3.4 By 19, Theorem 2.2, the critical point of the functional Ju is a solution of the inclusion −Δ 2 ut − 1 ∈  h 1 ut, h 1 ut  ,t∈ Z1,N, 3.5 where h 1 sminh 1 s  0,h 1 s − 0, h 1 smaxh 1 s  0,h 1 s − 0. Jianshe Yu et al. 7 Remark 3.1. We can show that h 1 sh 1 sλfsμgs for s>0, h 1 sh 1 s0fors< 0. For fixed μ and sufficiently small λ, λf0μg0 < 0. Then h 1 0λf0μg0, h 1 00. Remark 3.2. If u>0, then the above inclusion becomes −Δ 2 ut − 1λfut  μgut,t∈ Z1,N. 3.6 It is clear that A is a positive definite matrix. Let η max > 0,η min > 0 be the largest and smallest eigenvalue of A, respectively. Denote by u −  max{−u, 0}.LetP 1  {t ∈ Z1,N | ut ≤ 0},P 2  {t ∈ Z1,N | ut > 0}.Noticethatu − t0fort ∈ P 2 and f 1 ut  0for t ∈ P 1 . Then N  t 1 f 1 utu − t  t∈P 1 f 1 utu − t  t∈P 2 f 1 utu − t0. 3.7 Similarly, g 1 ut  0fort ∈ P 1 . Hence N  t 1 g 1 utu − t  t∈P 1 g 1 utu − t  t∈P 2 g 1 utu − t0. 3.8 Lemma 3.3. If u is a solution of 1.3,thenu ≥ 0. Moreover, either u>0 in Z1,N,oru  0 everywhere. Proof. It is not difficult to see that Δu − tΔutΔu − t ≤ 0fort ∈ Z0,N. In fact, no matter that Δut ≥ 0orΔut < 0, the former inequality holds. Hence Δu − t·Δut ≤−Δu − t 2 . If u is a solution of 1.3, then we have 0  N  t 1  Δ 2 ut − 1λf 1 ut  μg 1 ut  u − t  − N1  t 1 Δut − 1Δu − t − 1 N  t 1 λf 1 ut  μg 1 utu − t ≥ N1  t 1 Δu − t − 1 2 u −  T Au − ≥ η min u −  2 . 3.9 So u −  0. Hence u ≥ 0. If ut0, then ut  1ut − 1Δ 2 ut − 1−λf 1 ut − μg 1 ut  −λf 1 0 − μg 1 00. 3.10 Therefore ut  1ut − 10. It follows that u  0 everywhere. Lemma 3.4. If 1.6 and 1.7 hold, then h 1 ss ≥ β 0 H 1 s for large s>0,whereβ 0 ∈ 2,β. 8 Advances in Difference Equations Proof. Notice that h 1 ss ≥ β 0 H 1 s is equivalent to hss ≥ β 0 Hs if s>0. To prove that hss ≥ β 0 Hs for large s>0, it suffices to show that lim s → ∞ hss β 0 Hs > 1. 3.11 By 1.6,forlarges>0, we have β 0 Fs fss ≤ β 0 β . 3.12 Hence, if s>0 is large, then hss β 0 Hs  λfss  μgss β 0 λFsμGs  1  μgs/λfs β 0 Fs/fss  β 0 μGs/λfss ≥ 1  μgs/λfs β 0 /β  β 0 μGs/λfss . 3.13 Taking inferior limit on both sides of the above inequality, we have lim s → ∞ hss β 0 Hs ≥ lim s → ∞ 1  μgs/λfs β 0 /β  β 0 μGs/λfss ≥ lim s → ∞ 1  μgs/λfs lim s → ∞ β 0 /β  β 0 μGs/λfss . 3.14 Since f is superlinear and g is sublinear, lim s → ∞ μgs/λfs  0. Then lim s → ∞ 1  μgs/λfs  lim u → ∞ 1  μgs/λfs  1. Moreover, since G is subquadratic and f is superlinear, lim u → ∞ Gs/fsslim s → ∞ Gs/s 2 /fss/s 2   0. Therefore, lim s → ∞ β 0 /ββ 0 μGs/λfsslim s → ∞ β 0 /ββ 0 μGs/λfssβ 0 /β. From the above results, we can conclude that lim s → ∞ hss/β 0 Hs ≥ β/β 0 > 1. Lemma 3.5. If 1.6 and 1.7 hold, then J satisfies (PS) condition. Proof. Notice that E ∗  E.LetLu  N t1 H 1 ut.From19, Theorem 2.2, for any given w ∈ ∂Lu ⊂ E ∗ , we have wt ∈ h 1 ut, h 1 ut. Then wtλf 1 ut  μg 1 ut if ut /  0,wt ∈ λf0μg0, 0 if ut0. 3.15 Therefore w, u  N  t 1 h 1 utut, ∀w ∈ ∂Lu. 3.16 Jianshe Yu et al. 9 By Lemma 3.4, there is a constant M>0 such that Lu ≤ 1/β 0 w, u  M for u ∈ R N . Suppose that {u n } is a sequence such that Ju n  is bounded and ζu n  → 0asn →∞. Then by Properties 3 and 7 in Definition 2.2, there are C>0andw n ∈ ∂Lu n  such that |Ju n |≤C and    ∂K  u n  − w n ,u n    ≤   u n   for sufficiently large n. 3.17 It implies that u T n Au n −  w n ,u n  ≥−   u n   . 3.18 Hence C ≥ 1 2 u T n Au n − L  u n  ≥ 1 2 u T n Au n − 1 β 0  w n ,u n  − M   1 2 − 1 β 0  u T n Au n  1 β 0  u T n Au n −  w n ,u n  − M ≥  1 2 − 1 β 0  η min   u n   2 − 1 β 0   u n   − M. 3.19 This implies that {u n } is bounded. Since E is finite dimensional, {u n } has a convergent subsequence in E. Lemma 3.6. For fixed μ>0,thereexistρ>0 and λ>0 such that if λ ∈ 0, λ,thenJu ≥ η min M 2 1 /16λ −2/α−1 for u  ρ. Proof. By 1.5 and 1.7, there are C 4 ,C 5 > 0 such that F 1 s ≤ C 1 |s| α1 α  1  C 4 , ∀s ∈ R, 3.20 G 1 s ≤ η min 4μ |s| 2  C 5 , ∀s ∈ R. 3.21 The equivalence of norm on E implies that there exists C 6 > 0 such that u α1 ≤ C 6 u, where u α1   N t1 |ut| α1  1/α1 .LetM 1 η min α  1/8C 1 C α1 6  1/α−1 and ρ  M 1 λ −1/α−1 .Let 10 Advances in Difference Equations u  ρ. It follows from 3.20 and 3.21 that there is λ>0 such that if λ ∈ 0, λ, then Ju 1 2 u T Au − N  t 1 H 1 ut ≥ 1 2 η min u 2 − λC 1 α  1 N  t 1 |ut| α1 − λC 4 N − η min 4μ ·μ N  t 1 |ut| 2 − μC 5 N ≥ 1 4 η min u 2 − λC 1 C α1 6 α  1 u α1 − λC 4 N − μC 5 N  λ −2/α−1  η min M 2 1 8 − λ α1/α−1 C 4 N − λ 2/α−1 μC 5 N  ≥ η min M 2 1 16 λ −2/α−1 . 3.22 Lemma 3.7. Thereisane ∈ E such that e >ρand Je < 0. Proof. It follows from Remark 1.2 that Fs ≥ C 2 s β − C 3 for s>0. By the equivalence of the norms on E, there exists C 7 > 0 such that u β ≥ C 7 u, where u β   N t1 |ut| β  1/β .Letv 1 be the eigenfunction to the principal eigenvalue η 1 of −Δ 2 ut − 1ηut,t∈ Z1,N, u00,uN  10 3.23 with v 1 > 0andv 1   1. Let G m  min{Gu | u ∈ 0, ∞}. 3.24 Clearly G m < 0. Since β>2, for k>0, Jkv 1  1 2 k 2 v T 1 Av 1 − λ N  t 1 F  kv 1 t  − μ N  t 1 G  kv 1 t  ≤ η max 2 k 2 − λC 2  C 7 k  β  λC 3 N − μG m N −→ − ∞ as k −→ ∞. 3.25 Hence there is a k 1 >ρsuch that Jk 1 v 1  < 0. Let e  k 1 v 1 . Then e >ρand Je < 0. The second condition of Mountain Pass theorem is verified. Proof of Theorem 1.1. Clearly, J00. Lemma 3.5 implies that J satisfies PS condition. It follows from Lemmas 3.6, 3.7,and2.5 that J has a nontrivial critical point u such that Ju ≥ η min M 2 1 /16λ −2/α−1 .ByLemma 3.3 and Remark 3.2, u is a positive solution of 1.1. The proof is complete. [...]... 2, pp 207–215, 2000 17 R P Agarwal, S R Grace, and D O’Regan, Discrete semipositone higher-order equations,” Computers & Mathematics with Applications, vol 45, no 6–9, pp 1171–1179, 2003 18 D Jiang, L Zhang, D O’Regan, and R P Agarwal, “Existence theory for single and multiple solutions to semipositone discrete Dirichlet boundary value problems with singular dependent nonlinearities,” Journal of Applied... positive solutions for a class of p-Laplacian systems,” Nonlinear Analysis: Theory, Methods & Applications, vol 56, no 7, pp 1007–1010, 2004 14 D D Hai and R Shivaji, “Uniqueness of positive solutions for a class of semipositone elliptic systems,” Nonlinear Analysis: Theory, Methods & Applications, vol 66, no 2, pp 396–402, 2007 15 N Yebari and A Zertiti, “Existence of non-negative solutions for nonlinear... “On a variational approach to existence and multiplicity results for semipositone problems, ” Electronic Journal of Differential Equations, vol 2006, no 11, pp 1–10, 2006 12 E N Dancer and Z Zhang, “Critical point, anti-maximum principle and semipositone p-Laplacian problems, ” Discrete and Continuous Dynamical Systems, supplement, pp 209–215, 2005 13 D D Hai and R Shivaji, “An existence result on positive. .. model for a two-species predator-prey system with harvesting and stocking,” Journal of Mathematical Biology, vol 30, no 4, pp 389–411, 1992 5 J F Selgrade, “Using stocking or harvesting to reverse period-doubling bifurcations in discrete population models,” Journal of Difference Equations and Applications, vol 4, no 2, pp 163–183, 1998 6 Q Yao, “Existence of n solutions and/or positive solutions to a semipositone. .. Theorem 1.4 We will apply the subsuper solutions method to prove the multiplicity results Firstly, we will prove that there exists μ∗ > 0 such that if μ > μ∗ , then the following boundary value problem −Δ2 u t − 1 u 0 t ∈ Z 1, N , μg u t , 0, u N 1 3.26 0 has a positive solution u In fact, since g u is increasing on 0, ∞ and eventually strictly positive, g u ≥ −C8 for u ≥ 0 and some C8 > 0 Let r1 be the... supersolution of 1.1 Thus, by Remark 2.9, problem 1.1 has a solution u such that u ≤ u ≤ u for μ > μ∗ and λ small, which is positive for t ∈ Z 1, N Now we are going to find the second positive solution of problem 1.1 Notice that u and u are independent of λ Since f is positive on 0, ∞ , by the definition of f1 we have N ≥ 0 Then for u ∈ u, u , t 1 F1 u t J u N 1 T u Au − λ F1 u t 2 t 1 N 1 ≤ uT Au − μ G1 u t... /16 λ−2/ α−1 > J0 1 for ∈ u ρ Hence by Theorem 1.1, J u > J0 So u/ u, u and u / u, which shows that u and u are two different positive solutions of 1.1 The proof is complete Proof of Theorem 1.5 Just to be on the contradiction side, let u be a positive solution of 1.1 Since f is superlinear and increasing, f 0 > 0, there are C11 , C12 > 0 such that for s ≥ 0, f s ≥ C11 s C12 Hence for λ > 0 and s ≥... Acknowledgments The authors would like to thank the referees for valuable suggestions This project is supported by National Natural Science Foundation of China no 10625104 and Research Fund for the Doctoral Program of Higher Education of China Grant no 20061078002 References 1 A Castro and R Shivaji, “Nonnegative solutions for a class of nonpositone problems, ” Proceedings of the Royal Society of Edinburgh... t Δr1 t 2 2μ1 r1 t − Δr1 t On the other hand, for t ∈ Q1 , we have Δr1 t that C8 2 2μ1 r1 t − Δr1 t C9 2 2 2 2 − r1 t − Δr1 t − 1 − Δr1 t − 1 2 Δr1 t − 1 − Δr1 t − 1 2 2 2 3.28 2 − 2μ1 r1 t ≥ C9 , which implies −g ψ t ≤ 0 3.29 Then for t ∈ Q1 , −Δ2 ψ t − 1 ≤ μg ψ t Next, for t ∈ Z 1, N \ Q1 , we have r1 t ≥ r for some 2 2 r > 0 and C8 /C9 r1 t ≥ C10 for some C10 C8 /C9 r 2 > 0 Hence ψ t μC8 /C9 r1... non-negative solutions for nonlinear equations in the semipositone case,” in Proceedings of the Oujda International Conference on Nonlinear Analysis (Oujda 2005), vol 14 of Electronic Journal of Differential Equations Conference, pp 249–254, Southwest Texas State University, San Marcos, Tex, USA, 2006 16 R P Agarwal and D O’Regan, “Nonpositone discrete boundary value problems, ” Nonlinear Analysis: Theory, Methods . Equations Volume 2008, Article ID 840458, 15 pages doi:10.1155/2008/840458 Research Article Positive Solutions for Multiparameter Semipositone Discrete Boundary Value Problems via Variational Method Jianshe. multiplicity, and nonexistence of positive solutions for multiparameter semipositone discrete boundary value problems by using nonsmooth critical point theory and subsuper solutions method. Copyright. general, studying positive solutions for semipositone problems is more difficult than that for positone problems. The difficulty is due to the fact that in the semipositone case, solutions have to