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Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 621621, 19 pages doi:10.1155/2008/621621 Research Article The Method of Subsuper Solutions for Weighted pr-Laplacian Equation Boundary Value Problems Qihu Zhang, 1, 2 Xiaopin Liu, 2 and Zhimei Qiu 2 1 Department of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, Henan 450002, China 2 School of Mathematics Science, Xuzhou Normal University, Xuzhou, Jiangsu 221116, China Correspondence should be addressed to Zhimei Qiu, zhimeiqiu@yahoo.com.cn Received 23 May 2008; Accepted 21 August 2008 Recommended by Marta Garcia-Huidobro This paper investigates the existence of solutions for weighted pr-Laplacian ordinary boundary value problems. Our method is based on Leray-Schauder degree. As an application, we give the existence of weak solutions for px-Laplacian partial differential equations. Copyright q 2008 Qihu Zhang 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 In this paper, we consider the existence of solutions for the following weighted pr-Laplacian ordinary equation with right-hand terms depending on the first-order derivative: − wr u pr−2 u f r, u, wr 1/pr−1 u 0, ∀r ∈ T 1 ,T 2 , P with one of the following boundary value conditions: u T 1 c, u T 2 d, 1.1 g u T 1 , w T 1 1/pT 1 −1 u T 1 0,u T 2 d, 1.2 g u T 1 , w T 1 1/pT 1 −1 u T 1 0,h u T 2 , w T 2 1/pT 2 −1 u T 2 0, 1.3 u T 1 u T 2 ,w T 1 u T 1 pT 1 −2 u T 1 w T 2 u T 2 pT 2 −2 u T 2 , 1.4 where p ∈ CT 1 ,T 2 , R and pr > 1; w ∈ CT 1 ,T 2 , R satisfies 0 <wr, ∀r ∈ T 1 ,T 2 , and wr −1/pr−1 ∈ L 1 T 1 ,T 2 ; −wr|u | pr−2 u is called the weighted pr-Laplacian; the 2 Journal of Inequalities and Applications notation wT 1 1/pT 1 −1 u T 1 means lim r → T 1 wr 1/pr−1 u r exists and w T 1 1/pT 1 −1 u T 1 : lim r → T 1 wr 1/pr−1 u r, 1.5 similarly w T 2 1/pT 2 −1 u T 2 : lim r → T − 2 wr 1/pr−1 u r; 1.6 where gx, y and hx, y are continuous and increasing in y for any fixed x, respectively. The study of differential equations and variational problems with nonstandard pr- growth conditions is a new and interesting topic. Many results have been obtained on these kinds of problem, for example, 1–18.Ifwr ≡ pr ≡ p a constant, P is the well-known p-Laplacian problem. Because of the nonhomogeneity of px-Laplacian, px-Laplacian problems are more complicated than those of p-Laplacian, many methods and results for p-Laplacian problems are invalid for px-Laplacian problems. For example, 1 if Ω ⊂ R n is an open bounded domain, then the Rayleigh quotient λ px inf u∈W 1,px 0 Ω\{0} Ω 1/px |∇u| px dx Ω 1/px |u| px dx 1.7 is zero in general, and only under some special conditions λ px > 0 see 4, but the fact that λ p > 0 is very important in the study of p-Laplacian problems. In 19, the author considers the existence and nonexistence of positive weak solution to the following quasilinear elliptic system: −Δ p u λfu, vλu α v γ in Ω, −Δ q v λgu, vλu δ v β in Ω, u v 0on∂Ω, S the fi rst eigenfunction is used to constructing the subsolution of problem S successfully. On the px-Laplacian problems, maybe px-Laplacian does not have the first eigenvalue and the first eigenfunction. Because of the nonhomogeneity of px-Laplacian, the first eigenfunction cannot be used to construct the subsolution of px-Laplacian problems, even if the first eigenfunction of px-Laplacian exists.On the existence of solutions for px- Laplacian equations Dirichlet problems via subsuper solution methods, we refer to 13, 14; 2 if wr ≡ pr ≡ p a constant and −Δ p u>0, then u is concave, this property is used extensively in the study of one-dimensional p-Laplacian problems, but it is invalid for −Δ pr . It is another difference on −Δ p and −Δ pr : −|u | pr−2 u ; 3 on the existence of solutions of the typical pr-Laplacian problem: − u pr−2 u |u| qr−2 u C, r ∈ 0, 1, 1.8 because of the nonhomogeneity of pt-Laplacian, when we use critical point theory to deal with the existence of solutions, we usually need the corresponding functional is coercive or satisfy Palais-Smale conditions. If 1 ≤ max r∈0,1 qr < min r∈0,1 pr, then the corresponding functional is coercive, if max r∈0,1 pr < min r∈0,1 qr, then the corresponding functional Qihu Zhang et al. 3 satisfies Palais-Smale conditions see 3. But if min r∈0,1 pr ≤ qr ≤ max r∈0,1 pr, one can see that the corresponding functional is neither coercive nor satisfying Palais-Smale conditions, the results on this case are rare. There are many papers on the existence of solutions for p-Laplacian boundary value problems via subsuper solution method see 20–24. But results on the sub-super-solution method for px-Laplacian equations and systems are rare. In this paper, when pr is a general function, we establish several sub-super-solution theorems for the existence of solutions for weighted pr-Laplacian equation with Dirichlet, Robin, and Periodic boundary value conditions. Moreover, the case of min r∈0,1 pr ≤ qr ≤ max r∈0,1 pr is discussed. Our results partially generalize the results of 13, 14, 20, 25. Let T 1 <T 2 and I T 1 ,T 2 , the function f : I × R × R → R is assumed to be Caratheodory, by this we mean the following: i for almost every t ∈ I, the function ft, ·, · is continuous; ii for each x, y ∈ R × R, the function f·,x,y is measurable on I; iii for each ρ>0, there is a α ρ ∈ L 1 I,R such that, for almost every t ∈ I and every x, y ∈ R × R with |x|≤ρ, |y|≤ρ, one has ft, x, y ≤ α ρ t. 1.9 We set C CI,R, C 1 {u ∈ C | u is continuous in T 1 ,T 2 , lim r → T 1 wr|u | pr−2 u r and lim r → T − 2 wr|u | pr−2 u r exist}. Denote u 0 sup r∈T 1 ,T 2 |ur| and u 1 u 0 wr 1/pr−1 u 0 . The spaces C and C 1 will be equipped with the norm · 0 and · 1 , respectively. We say a function u : I → R is a solution of P,ifu ∈ C 1 and wr|u | pr−2 u r is absolutely continuous and satisfies P almost every on I. Functions α, β ∈ C 1 are called subsolution and supersolution of P,if|α | pr−2 α r and |β | pr−2 β r are absolutely continuous and satisfy − wr α pr−2 α f r, α, wr 1/pr−1 α ≤ 0, a.e. on I, − wr β pr−2 β f r, β, wr 1/pr−1 β ≥ 0, a.e. on I. 1.10 Throughout this paper, we assume that α ≤ β are subsolution and supersolution, respectively. Denote Ω 0 t, x | t ∈ I, x ∈ αt,βt , Ω 1 t, x, y | t ∈ I, x ∈ αt,βt ,y∈ R . 1.11 We also assume that H 1 |ft, x, y|≤A 1 t, xK 1 t, x, yA 2 t, xK 2 t, x, y, for all t, x, y ∈ Ω 1 , where A i t, xi 1, 2 are positive value and continuous on Ω 0 , K i t, x, yi 1, 2 are positive value and continuous on Ω 1 . H 2 There exist positive numbers M 1 and M 2 such that K 1 t, x, y ≤|y|φ|y|, K 2 t, x, y ≤ M 1 φ|y|, for |y|≥M 2 , where φ ∈ C1, ∞, 1, ∞ is increasing and satisfies ∞ 1 1/φy 1/p − −1 dy ∞, where p − min r∈I pr. 4 Journal of Inequalities and Applications Our main results are as the following theorem. Theorem 1.1. If f is Caratheodory and satisfies (H 1 ) and (H 2 ), α and β satisfy αT 1 ≤ c ≤ βT 1 , αT 2 ≤ d ≤ βT 2 ,thenP with 1.1 possesses a solution. Theorem 1.2. If f is Caratheodory and satisfies (H 1 ) and (H 2 ), α and β satisfy αT 2 ≤ d ≤ βT 2 , and g α T 1 , w T 1 1/pT 1 −1 α T 1 ≥ 0 ≥ g β T 1 , w T 1 1/pT 1 −1 β T 1 , 1.12 then P with 1.2 possesses a solution. Theorem 1.3. If f is Caratheodory and satisfies (H 1 ) and (H 2 ), α and β satisfy g α T 1 , w T 1 1/pT 1 −1 α T 1 ≥ 0 ≥ g β T 1 , w T 1 1/pT 1 −1 β T 1 , h α T 2 , w T 2 1/pT 2 −1 α T 2 ≤ 0 ≤ h β T 2 , w T 2 1/pT 2 −1 β T 2 , 1.13 then P with 1.3 possesses a solution. Theorem 1.4. If f is Caratheodory and satisfies (H 1 ) and (H 2 ), α and β satisfy α T 1 α T 2 <β T 1 β T 2 , w T 1 α T 1 pT 1 −2 α T 1 ≥ w T 2 α T 2 pT 2 −2 α T 2 , w T 1 β T 1 pT 1 −2 β T 1 ≤ w T 2 β T 2 pT 2 −2 β T 2 , 1.14 then P with 1.4 possesses a solution. As an application, we consider the existence of weak solutions for the following px- Laplacian partial differential equation: −div |∇u| px−2 ∇u f x, u, |x| n−1/px−1 |∇u| 0, ∀x ∈ Ω, 1.15 where Ω is a bounded symmetric domain in R n , p ∈ CΩ; R is radially symmetric. We will write pxp|x|pr,andpr satisfies 1 <pr ∈ C, f ∈ C Ω × R × R, R is radially symmetric with respect to x, namely, fx, u, vf|x|,u,vfr, u, v,andf satisfies the Caratheodory condition. 2. Preliminary Denote ϕr, x|x| pr−2 x, ∀r, x ∈ I × R. Obviously, ϕ has the following properties. Lemma 2.1. ϕ is a continuous function and satisfies i for any r ∈ T 1 ,T 2 , ϕr, · is strictly increasing; ii ϕr, · is a homeomorphism from R to R for any fixed r ∈ I. Qihu Zhang et al. 5 For any fixed r ∈ I, denote ϕ −1 r, · as ϕ −1 r, x|x| 2−pr/pr−1 x, for x ∈ R \{0},ϕ −1 r, 00. 2.1 It is clear that ϕ −1 r, · is continuous and send bounded sets into bounded sets. Let us now consider the simple problem wrϕ r, u r fr, 2.2 with boundary value condition 1.1, where f ∈ L 1 .Ifu is a solution of 2.2 with 1.1,by integrating 2.2 from T 1 to r,wefindthat wrϕ r, u r w T 1 ϕ T 1 ,u T 1 r T 1 ftdt. 2.3 Denote Ffr r T 1 ftdt, a w T 1 ϕ T 1 ,u T 1 , 2.4 then uru T 1 r T 1 ϕ −1 r, wr −1 a Ffr dr. 2.5 The boundary conditions imply that T 2 T 1 ϕ −1 r, wr −1 a Ffr dr d − c. 2.6 For fixed h ∈ C, we denote Λ h a T 2 T 1 ϕ −1 r, wr −1 a hr dr c − d. 2.7 We have the following lemma. Lemma 2.2. The function Λ h has the following properties. i For any fixed h ∈ C, the equation Λ h a0 2.8 has a unique solution ah ∈ R. ii The function a : C → R, defined in (i), is continuous and sends bounded sets to bounded sets. Proof. i Obviously, for any fixed h ∈ C, Λ h · is continuous and strictly increasing, then, if 2.8 has a solution, it is unique. Since wr −1/pr−1 ∈ L 1 T 1 ,T 2 and h ∈ C,itiseasytoseethat lim a → ∞ Λ h a∞, lim a →−∞ Λ h a−∞. 2.9 6 Journal of Inequalities and Applications It means the existence of solutions of Λ h a0. In this way, we define a function ah : CT 1 ,T 2 → R, which satisfies T 2 T 1 ϕ −1 r, wr −1 ahhr dr 0. 2.10 ii We claim that ah ≤ |c − d| T 2 T 1 ϕ −1 r, wr −1 dr 1 p 1 h 0 , ∀h ∈ C. 2.11 If it is false. Without loss of generality, we may assume that there are some h ∈ C such that ah > |c − d| T 2 T 1 ϕ −1 r, wr −1 dr 1 p 1 h 0 , 2.12 then ahh> |c − d| T 2 T 1 ϕ −1 r, wr −1 dr 1 p 1 , T 2 T 1 ϕ −1 r, wr −1 ahhr dr d − c > |c − d| T 2 T 1 ϕ −1 r, wr −1 dr 1 T 2 T 1 ϕ −1 r, wr −1 dr d − c |c − d| T 2 T 1 ϕ −1 r, wr −1 dr d − c > 0. 2.13 It is a contradiction. Thus, 2.11 is valid. It mens that a sends bounded sets to bounded sets. Finally, to show the continuity of a,let{u n } be a convergent sequence in C and u n → u,asn → ∞. Obviously, {au n } is a bounded sequence, then it contains a convergent subsequence {au n j }.Letau n j → a 0 as j → ∞. Since T 2 T 1 ϕ −1 r, wr −1 a u n j u n j r dr 0, 2.14 letting j → ∞, we have T 2 T 1 ϕ −1 r, wr −1 a 0 ur dr 0, 2.15 from i,wegeta 0 au, it means a is continuous. This completes the proof. Qihu Zhang et al. 7 Now, we define a : L 1 → R is defined by aha Fh . 2.16 It is clear that a is a continuous function which send bounded sets of L 1 into bounded sets of R, and hence it is a complete continuous mapping. We continue now with our argument previous to Lemma 2.2. By solving for u in 2.3 and integrating, we find uru T 1 F ϕ −1 r, wr −1 afFfr r. 2.17 Let us define KhtF ϕ −1 r, wr −1 ahFh t, ∀t ∈ T 1 ,T 2 . 2.18 We denote by N f u : C 1 × T 1 ,T 2 → L 1 , the Nemytsky operator associated to f defined by N f urf r, ur, wr 1/pr−1 u r , a.e. on I. 2.19 It is easy to see the following lemma. Lemma 2.3. u is a solution of P with boundary value condition 1.1 if and only if u is a solution of the following abstract equation: u c K N f u . 2.20 Lemma 2.4. The operator K is continuous and sends equi-integrable sets in L 1 into relatively compact sets in C 1 . Proof. It is easy to check that Kht ∈ C 1 . Since wr −1/pr−1 ∈ L 1 ,and wt 1/pt−1 Kh tϕ −1 t, ahFh , ∀t ∈ T 1 ,T 2 , 2.21 it is easy to check that K is a continuous operator from L 1 to C 1 . Let now U be an equi-integrable set in L 1 , then there exists ρ ∈ L 1 , such that ut ≤ ρt a.e. in I, for any u ∈ U. 2.22 We want to show that KU ⊂ C 1 is a compact set. Let {u n } be a sequence in KU, then there exist a sequence {h n }∈U such that u n Kh n . For t 1 ,t 2 ∈ I, we have that F h n t 1 − F h n t 2 ≤ t 2 t 1 ρtdt . 2.23 Hence, the sequence {Fh n } is uniformly bounded and equicontinuous, then there exists a subsequence of {Fh n } which is convergent in C, and we name the same. Since the operator a is bounded and continuous, we can choose a subsequence of {ah n Fh n } which we still denote {ah n Fh n } that is convergent in C, then wtϕ t, Kh n t a h n F h n 2.24 is convergent in C. Since K h n tF wr −1/pr−1 ϕ −1 r, a h n F h n t, ∀t ∈ T 1 ,T 2 , 2.25 according to the continuous of ϕ −1 and the integrability of wr −1/pr−1 in L 1 , then Kh n is convergent in C. Then, we can conclude that {u n } convergent in C 1 . 8 Journal of Inequalities and Applications Lemma 2.5. Let α, β ∈ C 1 be subsolution and supersolution of P, respectively, which satisfies αt ≤ βt for any t ∈ T 1 ,T 2 , then there exists a positive constant L such that, for any solution x of P with 1.1 whichsatisfies αt ≤ xt ≤ βt,one has wt 1/pt−1 x 0 ≤ L. Proof. We denote μ 0 T 2 T 1 A 1 t, xt A 2 t, xt dt, a 0 max wr 1/pr−1 | r ∈ T 1 ,T 2 , σ max βs − αt | t, s ∈ T 1 ,T 2 , γ max wt 1/pt−1 A 1 t, x | t, x ∈ Ω 0 , 2.26 then there exists a t 0 ∈ T 1 ,T 2 such that w t 0 1/pt 0 −1 x t 0 ≤ a 0 x t 0 ≤ a 0 σ T 2 − T 1 . 2.27 From H 2 , there exist positive numbers σ 1 and N 1 such that N 1 ≥ σ 1 ≥ max r∈I M 2 a 0 σ T 2 − T 1 1 pr , N 1 σ 1 1 φ y 1/pr−1 dy > γσ M 1 μ 0 , for r ∈ T 1 ,T 2 uniformly. 2.28 Assume that our conclusion is not true, combining 2.27, then there exists t 1 ,t 2 ⊂ T 1 ,T 2 such that wr 1/pr−1 x keeps the same sign on t 1 ,t 2 , and w t 1 x pt 1 −2 x t 1 σ 1 ,w t 2 x pt 2 −2 x t 2 N 1 , 2.29 or inversely w t 1 x pt 1 −2 x t 1 −σ 1 ,w t 2 x pt 2 −2 x t 2 −N 1 . 2.30 For simplicity, we assume that the former appears. Hence, γσ M 1 μ 0 < N 1 σ 1 1 φ y 1/pr−1 dy t 2 t 1 wr x pr−1 φ wr x pr−1 1/pr−1 dr t 2 t 1 f r, x, wr 1/pr−1 x φ wr 1/pr−1 x dr ≤ t 2 t 1 wr 1/pr−1 A 1 r, xr x dr M 1 μ 0 ≤ γσ M 1 μ 0 , 2.31 which is impossible. T he proof is completed. Qihu Zhang et al. 9 Let us consider the auxiliary SBVP of the form wr u pr−2 u f r, Rr, u,R 1 wr 1/pr−1 u R 2 r, u def fr, u,r∈ T 1 ,T 2 , 2.32 where Rt, u ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ βt,ut >βt, u, αt ≤ ut ≤ βt, αt,ut <αt, R 1 y ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ L 1 ,y>L 1 , y, |y|≤L 1 , −L 1 ,y<−L 1 , 2.33 where L 1 1 max L, sup r∈T 1 ,T 2 wr 1/pr−1 β r , sup r∈T 1 ,T 2 wr 1/pr−1 α r , 2.34 where L is defined in Lemma 2.5,and R 2 t, u ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ et, u u − βt 1 u 2 ut >βt, 0,αt ≤ ut ≤ βt, et, u u − αt 1 u 2 ut <αt, 2.35 where et, u1 A 1 t, Rt, u A 2 t, Rt, u. Lemma 2.6. Let the conditions of Lemma 2.5 hold, and let ut be any solution of SBVP with 1.1 satisfies αT 1 ≤ c ≤ βT 1 and αT 2 ≤ d ≤ βT 2 ,thenαt ≤ ut ≤ βt, for any t ∈ T 1 ,T 2 . Proof. We will only prove that ut ≤ βt for any t ∈ T 1 ,T 2 . The argument of the case of αt ≤ ut for any t ∈ T 1 ,T 2 is similar. Assume that ut >βt for some t ∈ T 1 ,T 2 , then there exist a t 0 ∈ T 1 ,T 2 and a positive number δ such that ut 0 βt 0 δ, ut ≤ βtδ, for any t ∈ T 1 ,T 2 . Hence, w t 0 1/pt 0 −1 u t 0 w t 0 1/pt 0 −1 β t 0 . 2.36 There exists a positive number η such that ut >βt, for any t ∈ J :t 0 − η, t 0 η ⊂ T 1 ,T 2 . From the definition of β, u, and f we conclude that wr β pr−2 β ≤ f r, β, wr 1/pr−1 β fr, β < fr, u on t 0 − η 1 ,t 0 η 1 , 2.37 10 Journal of Inequalities and Applications where η 1 ∈ 0,η is small enough. For any r ∈ t 0 ,t 0 η 1 , we have r t 0 wr β pr−2 β dr < r t 0 fr, udr r t 0 wr u pr−2 u dr. 2.38 From 2.36 and 2.38, we have β pr−2 β < u pr−2 u on t 0 ,t 0 η 1 , 2.39 it means that β δ <u on t 0 ,t 0 η 1 . 2.40 It is a contradiction to the definition of t 0 ,sout ≤ βt, for any t ∈ T 1 ,T 2 . 3. Proofs of main results In this section, we will deal with the proofs of main results. Proof of Theorem 1.1. From Lemmas 2.5 and 2.6, we only need to prove the existence of solutions f or SBVP with 1.1. Obviously, u is a solution of SBVP with 1.1 if and only if u is a solution of u Φ f u : c K N f u . 3.1 We set C 1 c,d u ∈ C 1 | u T 1 c, u T 2 d . 3.2 Obviously, N f u sends C 1 into equi-integrable sets in L 1 . Similar to the proof of Lemma 2.4, we can conclude that K sends equi-integrable sets in L 1 into relatively compact sets in C 1 , then Φ f u is compact continuous. Obviously, for any u ∈ C 1 , we have Φ f u ∈ C 1 c,d ,andΦ f C 1 is bounded. By virtue of Schauder fixed point theorem, Φ f u has at least one fixed point u in C 1 c,d . Then, u is a solution of SBVP with 1.1. This completes the proof. Proof of Theorem 1.2. Let d with αT 2 ≤ d ≤ βT 2 be fixed. According to Theorem 1.1, P with the following boundary value condition: u 1 T 1 α T 1 ,u 1 T 2 d, 3.3 possesses a solution u 1 such that αt ≤ u 1 t ≤ βt, ∀t ∈ T 1 ,T 2 . 3.4 Since lim r → T 1 wr u 1 pr−2 u 1 r exists, we have u 1 r − u 1 T 1 r T 1 wt −1/pt−1 wt 1/pt−1 u 1 t dt w T 1 1/pT 1 −1 u 1 T 1 r T 1 wt −1/pt−1 1 o1 dt. 3.5 [...]... 3.43 From 3.41 , 3.42 , and 3.43 , we can conclude that P with 1.4 possesses a solution This completes the proof On the case of minr∈ R, R p r ≤ q r ≤ maxr∈ R, R p r , we consider − |u |p r −2 C|u|q r u u R −2 u r ∈ R, R , e r u R I 0, where q r , e r ∈ C R, R , R , minr∈ R, R p r ≤ q r ≤ maxr∈ R, R p r , C is a positive constant Denote p max p r , p r R, R min p r 3.44 r R, R We have the following... functional -Laplacian impulsive c equations,” Nonlinear Analysis: Theory, Methods & Applications, vol 67, no 3, pp 827–841, 2007 Qihu Zhang et al 19 23 Q Huang and Y Li, “Nagumo theorems of nonlinear singular boundary value problems,” Nonlinear Analysis: Theory, Methods & Applications, vol 29, no 12, pp 1365–1372, 1997 24 N S Papageorgiou and V Staicu, The method of upper-lower solutions for nonlinear second... 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Analysis: Theory, Methods & Applications, vol 52, no 2, pp 585–594, 2003 2 X.-L Fan, “Global C1,α regularity for variable exponent elliptic equations in divergence form,” Journal of Differential Equations, vol 235, no 2, pp 397–417, 2007 3 X.-L Fan and Q.-H Zhang, “Existence of solutions for p x -Laplacian Dirichlet problem,” Nonlinear Analysis: Theory, Methods & Applications, vol 52, no 8, pp 1843–1852,... calculus of variations and elasticity theory,” Mathematics of the USSR-Izvestiya, vol 29, no 1, pp 33–66, 1987 19 C Chen, “On positive weak solutions for a class of quasilinear elliptic systems,” Nonlinear Analysis: Theory, Methods & Applications, vol 62, no 4, pp 751–756, 2005 20 L E Bobisud and D O’Regan, “Positive solutions for a class of nonlinear singular boundary value problems at resonance,” Journal... following corollary Corollary 3.1 If p ∈ C R, 1, ∞ R 1 is even, R satisfies C − p −1 / p p− −1 max e r r∈ R, R , 3.45 then I possesses at least a nontrivial solution Proof It is easy to see that α ≡ 0 is a subsolution of I Denote 1− β r r |μs|1/ p s −1 −1 μs ds, 3.46 0 where μ is a positive constant satisfying β R see that 0 ≤ β r ≤ 1, r ∈ R, R , and − β p r −2 R β μ 0 Since p is even, then β R 1 p ξ |s|1/... behavior of positive solutions to p x -Laplacian equations with singular nonlinearities,” Journal of Inequalities and Applications, vol 2007, Article ID 19349, 9 pages, 2007 17 Q Zhang, Boundary blow-up solutions to p x -Laplacian equations with exponential nonlinearities,” Journal of Inequalities and Applications, vol 2008, Article ID 279306, 8 pages, 2008 18 V V Zhikov, “Averaging of functionals of the. .. |s|1/ p s −1 ds ≥ 0 ≥ R 0 R 0 It is easy to 1 p ξ |s|1/ p −1 ds 0 3.47 1 p |s|1/ p −1 ds ≥1 C max e r ≥ C|β|q r r∈ R, R −2 β e r , where ξ ∈ R, R Then, β is a supersolution of I From Theorem 1.1, one can see that I possesses at least a nontrivial solution 16 Journal of Inequalities and Applications 4 Applications in PDE Let Ω ⊂ Rn be an open bounded domain In this section, we always denote p max p. .. max p x , p min p x 4.1 x∈Ω x∈Ω Let us now consider 1.15 with one of the following boundary value conditions: 0, u|∂Ω ∇u 4.2 ∀x ∈ ∂Ω 0, 4.3 If u is a radial solution of 1.15 , then it can be transformed into − r n−1 u p r −2 r n−1 f r, u, |r| n−1 / p r u −1 u 0, r ∈ T1 , T2 , where T1 ≥ 0, 4.4 and the boundary value condition will be transformed into 1.1 , 1.2 , or 1.3 , respectively Theorem 4.1 If . R, R , u R u R 0, I where q r ,e r ∈ C R, R , R ,min r R, R p r ≤ q r ≤ max r R, R p r , C is a positive constant. Denote p max r R, R p r ,p − min r R, R p r . 3.44 We. well-known p- Laplacian problem. Because of the nonhomogeneity of p x -Laplacian, p x -Laplacian problems are more complicated than those of p- Laplacian, many methods and results for p- Laplacian problems. many papers on the existence of solutions for p- Laplacian boundary value problems via subsuper solution method see 20–24. But results on the sub-super-solution method for p x -Laplacian equations
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