Hindawi Publishing Corporation Boundary Value Problems Volume 2010, Article ID 584843, 20 pages doi:10.1155/2010/584843 Research Article Existence of Solutions for the px-Laplacian Problem with Singular Term Fu Yongqiang and Yu Mei Department of Mathematics, Harbin Institute of Technology, Harbin 15000, China Correspondence should be addressed to Yu Mei, yumei301796@gmail.com Received 19 August 2009; Accepted 7 March 2010 Academic Editor: Raul F. Manásevich Copyright q 2010 F. Yongqiang and Y. Mei. 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. We study the following px-Laplacian problem with singular term: −div|∇u|px−2 ∇u |u|px−2 u λ|u|αx−2 u fx, u, x ∈ Ω, u 0, x ∈ ∂Ω, where Ω ⊂ RN is a bounded domain, 1,px 1 < p− ≤ px ≤ p < N. We obtain the existence of solutions in W0 Ω. 1. Introduction After Kováčik and Rákosnı́k first discussed the Lpx Ω spaces and W k,px Ω spaces in 1, a lot of research has been done concerning these kinds of variable exponent spaces, for example, see 2–5 for the properties of such spaces and 6–9 for the applications of variable exponent spaces on partial differential equations. Especially in W 1,px Ω spaces, there are a lot of studies on px-Laplacian problems; see 8, 9. In the recent years, the theory of problems with px-growth conditions has important applications in nonlinear elastic mechanics and electrorheological fluids see 10–14. In this paper, we study the existence of the weak solutions for the following pxLaplacian problem: − div |∇u|px−2 ∇u |u|px−2 u λ|u|αx−2 u fx, u, u 0, x ∈ Ω, 1.1 x ∈ ∂Ω, where Ω ⊂ RN is a bounded domain, 0 < λ ∈ R, px is Lipschitz continuous on Ω, and 1 < p− ≤ px ≤ p < N. 2 Boundary Value Problems Let PΩ be the set of all Lebesgue measurable functions p : Ω → 1, ∞. For all px ∈ PΩ, we denote p supx∈Ω px, p− infx∈Ω px, and denote by p1 x p2 x the fact that inf{p2 x − p1 x} > 0. We impose the following condition on f: F fx, t bxgx, t, 0 < bx ∈ Lrx Ω, 1 rx ∈ CΩ and for gx, t ∈ CΩ × R, there exist M > 0 such that |gx, t| ≤ c1 c2 |t|qx−1 , whenever |t| ≥ M. A typical example of 1.1 is the following problem involving subcritical SobolevHardy exponents of the form − div |∇u|px−2 ∇u |u|px−2 u λ|u|αx−2 u u 0, λ |x|sx |u|qx−2 u, x ∈ Ω, 1.2 x ∈ ∂Ω, where 0 < λ ∈ R, sx, qx ∈ CΩ, 0 ≤ s− ≤ s < N, 1 qx < N − sx/Np∗ x, sx , gx, t |t|qx−2 t, and rx ≡ 1/21 N/s , then for all x ∈ Ω. In fact, take bx λ/|x| it is easy to verify that F is satisfied. Our object is to obtain the existence of solutions in the following four cases: 1 αx > px, qx > px; 2 αx < px, qx > px; 3 αx < px, qx < px; 4 αx > px, qx < px. When bx 1, the solution of the p-Laplacian equations without singularity has been studied by many researchers. The study of problem 1.1 with variable exponents is a new topic. The paper is organized as follows. In Section 2, we present some necessary preliminary knowledge of variable exponent Lebesgue and Sobolev spaces. In Section 3, we prove our main results. 2. Preliminaries In this section we first recall some facts on variable exponent Lebesgue space Lpx Ω and variable exponent Sobolev space W 1,px Ω, where Ω ⊂ RN is an open set; see 1–4, 8, 15 for the details. Let px ∈ PΩ and u px up inf λ > 0 : dx ≤ 1 . Ω λ 2.1 The variable exponent Lebesgue space Lpx Ω is the class of functions u such that |ux|px dx < ∞. Lpx Ω is a Banach space endowed with the norm 2.1. Ω Boundary Value Problems 3 For a given px ∈ PΩ, we define the conjugate function p x as px . px − 1 p Theorem 2.1. Let px ∈ PΩ. Then the inequality fx · gxdx ≤ rp f g p p 2.2 2.3 Ω holds for every f ∈ Lpx Ω and g ∈ Lp x Ω with the constant rp depending on px and Ω only. Theorem 2.2. The dual space of Lpx Ω is Lp x Ω if and only if p ∈ L∞ Ω. The space Lpx Ω is reflexive if and only if 1 < p− ≤ p < ∞. 2.4 Theorem 2.3. Suppose that px satisfies 2.4. Let meas Ω < ∞, p1 x, p2 x ∈ PΩ, then necessary and sufficient condition for Lp2 x Ω ⊂ Lp1 x Ω is that for almost all x ∈ Ω one has p1 x ≤ p2 x, and in this case, the imbedding is continuous. Theorem 2.4. Suppose that px satisfies 2.4. Let ρu Ω |ux|px dx. If u, uk ∈ Lpx Ω, then 1 up < 1 1; > 1 if and only if ρu < 1 1; > 1, p− p p p− 2 if up > 1, then up ≤ ρu ≤ up , 3 if up < 1, then up ≤ ρu ≤ up , 4 limk → ∞ uk p 0 if and only if limk → ∞ ρuk 0, 5 uk p → ∞ if and only if ρuk → ∞. We assume that k is a given positive integer. Given a multi-index α α1 , . . . , αn ∈ Nn , we set |α| α1 · · ·αn and Dα D1α1 · · · Dnαn , where Di ∂/∂xi is the generalized derivative operator. The generalized Sobolev space W k,px Ω is the class of functions f on Ω such that α D f ∈ Lpx for every multi-index α with |α| ≤ k, endowed with the norm Dα f . f k,p p 2.5 |α|≤k k,px By W0 Ω we denote the subspace of W k,px Ω which is the closure of C0∞ Ω with respect to the norm 2.5. In this paper we use the following equivalent norm on W 1,px Ω: u1,p ∇u px u px inf λ > 0 : dx ≤ 1 . λ λ Ω Then we have the inequality 1/2∇up up ≤ u1,p ≤ 2∇up up . 2.6 4 Boundary Value Problems k,px Theorem 2.5. The spaces W k,px Ω and W0 satisfies 2.4. Ω are separable reflexive Banach spaces, if px Theorem 2.6. Suppose that px satisfies 2.4. Let Iu W 1,px Ω, then Ω |∇ux|px |ux|px dx. If u, uk ∈ 1 u1,p < 1 1; > 1 if and only if Iu < 1 1; > 1, p− p p p− 2 if u1,p > 1, then u1,p ≤ Iu ≤ u1,p , 3 if u1,p < 1, then u1,p ≤ Iu ≤ u1,p , 4 limk → ∞ uk 1,p 0 if and only if limk → ∞ Iuk 0, 5 uk 1,p → ∞ if and only if Iuk → ∞. k,px We denote the dual space of W0 Ω by W −k,p x Ω, then we have the following. Theorem 2.7. Let p ∈ PΩ ∩ L∞ Ω. Then for every G ∈ W −k,p x Ω there exists a unique system of functions {gα ∈ Lp x Ω : |α| ≤ k} such that Dα fxgα xdx, G f |α|≤k Ω k,px f ∈ W0 Ω. 2.7 The norm of W −k,p x Ω is defined as G−k,p G f k,px sup : f ∈ W0 Ω \ {0} . f k,p 2.8 Theorem 2.8. Let Ω be a domain in Rn with cone property. If p : Ω → R is Lipschitz continuous and 1 < p− ≤ p < N/k, qx : Ω → R is measurable and satisfies px ≤ qx ≤ p∗ x : Npx/N − kpx a.e. x ∈ Ω, then there is a continuous embedding W k,px Ω → Lqx Ω. 1,px Theorem 2.9. Let Ω be a bounded domain. If px ∈ L∞ Ω and u ∈ W0 up ≤ C∇up , Ω, then 2.9 where C is a constant depending on Ω. Next let us consider the weighted variable exponent Lebesgue space. Let ax ∈ PΩ, and ax > 0 for x ∈ Ω. Define px Lax Ω u ∈ P Ω : px Ω ax|ux| dx < ∞ 2.10 Boundary Value Problems 5 with the norm |u|Lpx Ω up,a ax ux px dx ≤ 1 , inf λ > 0 : ax λ Ω 2.11 px then Lax Ω is a Banach space. Theorem 2.10. Suppose that px satisfies 2.4. Let ρu px Lax Ω, Ω ax|ux|px dx. If u,uk ∈ then 1 for u / 0, up,a λ if and only if ρu/λ 1, 2 up,a < 1 1; > 1 if and only if ρu < 1 1; > 1, p− p p p− 3 if up,a > 1, then up,a ≤ ρu ≤ up,a , 4 if up,a < 1, then up,a ≤ ρu ≤ up,a , 5 limk → ∞ uk p,a 0 if and only if limk → ∞ ρuk 0, 6 uk p,a → ∞ if and only if ρuk → ∞. Theorem 2.11. Assume that the boundary of Ω possesses the cone property and px ∈ CΩ. Suppose that 0 < ax ∈ Lrx Ω, and 1 rx ∈ CΩ, for x ∈ Ω. If qx ∈ CΩ and 1 ≤ qx < rx − 1/rxp∗ x, for all x ∈ Ω, then there is a compact embedding qx W 1,px Ω → Lax Ω. Theorem 2.12. Let Ω ⊂ Rn be a measurable subset. Let g : Ω × R → R be a Caracheodory function and satisfies gx, u ≤ ax b|u|p1 x/p2 x for any x ∈ Ω, t ∈ R, 2.12 where pi x ≥ 1, i 1, 2, ax ∈ Lp2 x Ω, ax ≥ 0, b ≥ 0 is a constant, then the Nemytsky operator from Lp1 x Ω to Lp2 x Ω defined by Ng ux gx, ux is a continuous and bounded operator. 3. Existence and Multiplicity of Solutions Let 1 |u|αx − Fx, udx, |∇u|px |u|px − λ αx Ω px t t Fx, t fx, sds, Gx, t gx, sds. Iu 0 0 3.1 6 Boundary Value Problems The critical points of Iu, that is, 0 I uϕ Ω |∇u|px−2 ∇u∇ϕ |u|px−2 uϕ − λ|u|αx−2 uϕ − fx, uϕ dx 3.2 1,px Ω, are weak solutions of problem 1.1. So we need only to consider the for all ϕ ∈ W0 existence of nontrivial critical points of Iu. Denote by c, ci , C, Ci , K, and Ki the generic positive constants. Denote by |Ω| the Lebesgue measure of Ω. To study the existence of solutions for problem 1.1 in the first case, we additionally impose the following conditions. A-1 αx, qx ∈ CΩ and px αx < p∗ x, px qx < rx − 1/rxp∗ x. B-1 There exists a function μx ∈ C1 Ω, such that px μx and 0 < μxGx, t ≤ tgx, t for x ∈ Ω, |t| ≥ M. C-1 there exist δ > 0 such that |gx, t| ≤ c3 |t|qx−1 for x ∈ Ω, |t| ≤ δ, where qx ∈ CΩ ∗ and px qx < rx − 1/rxp x. D-1 gx, −t −gx, t, for all x ∈ Ω, t ∈ R. Theorem 3.1. Under assumptions (F) and (A-1)–(C-1), problem 1.1 admits a nontrivial solution. 1,px Proof. First we show that any P Sc sequence is bounded. Let {un } ⊂ W0 Ω and c ∈ R, such that Iun → c and I un → 0 in W −1,p x Ω. By A-1 and B-1, inf{1/px − 1/αx} a1 > 0, and inf{1/px − 1/μx} a2 > 0. Let a0 min{a1 , a2 } and lx 1/px − a0 /2, then lx > 1/αx, lx > 1/μx, |∇lx| < C, 1/px − lx a0 /2 and lx − 1/αx ≥ a0 /2. Let Ω1 {x ∈ Ω : |ux| ≥ M} and Ω2 Ω \ Ω1 . Then Iun − I un , lxun 1 1 − lx |∇un |px |un |px dx λ lx − |un |αx dx px αx Ω Ω lxfx, un un − Fx, un dx − Fx, un dx Ω1 Ω2 Ω2 lxfx, un un dx − 3.3 Ω |∇un |px−2 ∇un ∇l · un dx. By B-1, we get Ω1 lxfx, un un − Fx, un dx ≥ Ω1 lx − 1 fx, un un dx > 0. μx 3.4 Boundary Value Problems 7 By F, we get gx, t ∈ CΩ × −M, M, so there exist K > 0 such that |gx, t| ≤ K on Ω × −M, M. Note fx, t bxgx, t, so we have Fx, un dx ≤ MKbxdx < ∞, Ω2 Ω2 fx, un un dx ≤ MKbxdx < ∞. Ω2 Ω2 3.5 By Young’s inequality, for ε1 ∈ 0, 1, we get Ω |∇un |px−2 ∇un ∇l · un dx ≤ Cε1 1−p Ω |∇un |px dx Cε1 Ω |un |px dx. 3.6 Take ε1 sufficiently small so that a0 /2 > Cε1 . Note that px αx, by Young’s inequality again and for ε2 ∈ 0, 1, we get Ω |∇un |px−2 ∇un ∇l · un dx ≤ Cε1 Ω |∇un | px dx 1−p Cε1 ε2 Ω 1−p Take ε2 sufficiently small so that λa0 /2 ≥ Cε1 From the above remark, we have Iun − I un , lxun |un | αx dx 1−p −p /α−p− Cε1 ε2 |Ω|. 3.7 ε2 . a0 − Cε1 |∇un |px |un |px dx − C1 . ≥ Ω 2 3.8 As lxun p ≤ l un p , lx∇un p ≤ l ∇un p and ∇lx · un p ≤ Cun p , we have lxun 1,p ≤ 4Cl un 1,p . Since Iun → c and I un → 0, by Theorem 2.6 we have c o1 4Cl un 1,p ≥ a 0 2 p− − Cε1 un 1,p − C1 , 3.9 when un 1,p ≥ 1 and n is sufficiently large. Then it is easy to see that {un } is bounded in 1,px W0 Ω. Next we show that {un } possesses a convergent subsequence still denoted by {un }. 8 Boundary Value Problems Note that |∇un |px−2 ∇un − |∇u|px−2 ∇u ∇un − ∇udx Ω ≤ I un , un − u I u, un − u Ω λ |un |αx−2 un − |u|αx−2 u un − udx Ω fx, un − fx, u un − udx I1 I2 I3 I4 . 3.10 1,px Because {un } is bounded in W0 Ω, there exists a subsequence {un } still denoted 1,px by {un }, such that un u weakly in W0 embeddings W 1,px Ω → L αx Ω and W 1,px Ω. By Theorem 2.11, there are compact qx Ω → Lbx Ω, then un → u in Lαx Ω qx and Lbx Ω. So we get I3 λ |un |αx−2 un − |u|αx−2 u un − udx Ω 3.11 ≤ 2λ |un |αx−1 un − uα 2λ |u|αx−1 un − uα . α α Hence I3 → 0 as n → ∞. By F, we have Ω fx, uun − udx ≤ ≤ Ω C2 bx|un − u|dx Ω2 Ω1 C2 bx|un − u|dx Ω bx|un − u||u| qx−1 bx|un − u||u|qx−1 dx 3.12 dx, and similarly for every n, Ω fx, un un − udx ≤ Ω C2 bx|un − u|dx Ω bx|un − u||un |qx−1 dx. 3.13 Since Ω bx|un − u||u| qx−1 dx Ω bx1/qx bx1/q x |un − u||u|qx−1 dx ≤ 2 bx1/q x |u|qx−1 bx1/qx |un − u| q 3.14 q Boundary Value Problems 9 qx and un → u in L1bx Ω and Lbx Ω, we obtain u||u|qx−1 dx → 0. Similarly, Ω Ω bx|un − u|dx → 0 and Ω bx|un − bx|un − u||un |qx−1 dx ≤ 2 bx1/q x |un |qx−1 bx1/qx |un − u| . q q 3.15 Because bx1/q x |un |qx−1 q is bounded,we get Ω bx|un − uun |qx−1 dx → 0, as n → ∞. From the above remark, we conclude I4 Ω |fx, un − fx, uun − u|dx → 0, as n → ∞. Thus I1 I2 I3 I4 → 0, as n → ∞. Then we get Ω |∇un |px−2 ∇un −|∇u|px−2 ∇u∇un − ∇udx → 0. As in the proof of Theorem 3.1 in 6, 7, we divide Ω into the following two parts: Ω3 x ∈ Ω : 1 < px < 2 , Ω4 x ∈ Ω : px ≥ 2 . 3.16 On Ω3 , we have |∇un − ∇u|px dx ≤ Ω3 Ω3 K1 px/2 |∇un |px−2 ∇un − |∇u|px−2 ∇u ∇un − ∇u 2−px/2 × |∇un |px |∇u|px dx px/2 px−2 px−2 ≤ 2K1 |∇un | ∇un − |∇u| ∇u ∇un − ∇u 2−px/2 × |∇un |px |∇u|px Then 2/px,Ω3 . 2/2−px,Ω3 |∇un − ∇u|px dx → 0, as n → ∞. On Ω4 , we have Ω3 Ω4 3.17 |∇un − ∇u|px dx ≤ K2 Ω4 |∇un |px−2 ∇un − |∇u|px−2 ∇u ∇un − ∇udx, 3.18 |∇un − ∇u|px dx → 0, as n → ∞. 1,px Ω. Thus we get Ω |∇un − ∇u|px dx → 0. Then un → u in W0 μx − c5 , for all x ∈ Ω, t ∈ R. So we get From F and B-1 we have Gx, t ≥ c4 |t| Fx, t ≥ c4 bx|t|μx − c5 bx, for all x ∈ Ω, t ∈ R. for all x0 ∈ Ω, take ε 1/6αx0 − px0 , then px0 < αx0 −3ε. Since px and αx ∈ CΩ, there exists δ > 0 such that |px−px0 | < so Ω4 10 Boundary Value Problems ε, and |αx − αx0 | < ε, for x ∈ Bx0 , δ. Let ϕ ∈ C0∞ Bx0 , δ, such that ϕx 1 for x ∈ Bx0 , δ/2, ϕx 0 for x ∈ Ω \ Bx0 , δ, and 0 ≤ ϕx ≤ 1 in Bx0 , δ. Then we have I sϕ ≤ px Ω s px ∇ϕ px ≤ px s px ∇ϕ ≤ spB Bx0 ,δ px c5 px px ∇ϕ px Bx0 ,δ px ϕ − λs px ϕ px px ϕ px αx αx ϕ − λs αx αx μx − c4 bxsμx ϕ c5 bxdx αx ϕ αx − dx c5 dx − sαB λ Bx0 ,δ bxdx Bx0 ,δ αx ϕ αx 3.19 dx bxdx, Bx0 ,δ where pB < α−B . So if s is sufficiently large, we obtain Isϕ < 0. cδ|t|qx , then |Fx, t| ≤ c3 bx|t|qx From F and C-1, we have |Gx, t| ≤ c3 |t|qx qx cδbx|t| . So we get Iu ≥ 1 p Ω λ − cδbx|u|qx dx. 3.20 |u|αx − c3 bx|u|qx |∇u|px |u|px dx − − α Ω Let θx min{αx, qx, qx}, then θx ∈ CΩ. By Theorem 2.11, uα ≤ c6 u1,p , uq,b ≤ c7 u1,p , and uq,b ≤ c8 u1,p . When u1,p is sufficiently small, uα < 1, uq,b < 1 and uq,b < 1. For any x ∈ Ω, as px, θx ∈ CΩ, for any ε > 0, we can find QR x {y y1 , . . . , yn : |yk − xk | < R, k 1, . . . , n} such that |py − px| < ε and |θy−θx| < ε whenever y ∈ QR x∩Ω. Take ε 1/4θx−px, then supy∈QR x∩Ω py < infy∈QR x∩Ω θy. {QR x}x∈Ω is an open covering of Ω. As Ω is compact, we can pick a finite ⊂ Ω we define subcovering {QRk xk }m k1 for Ω from the covering {QR x}x∈Ω . If QRk xk / u 0 on QRk xk \ Ω. We can use all the hyperplanes, for each of which there exists at least one hypersurface of some QRk xk lying on it, to divide m k1 QRk xk into finite open Q Q hypercubes {Qi }i1 which mutually have no common points. It is obvious that Ω ⊆ i1 Qi and for each Qi there exists at least one QRk xk such that Qi ⊆ QRk xk . Let Ωi Qi ∩ Ω, then pi supy∈Ωi py < infy∈Ωi θy θi− and we have Q 1 λ px px αx qx qx Iu ≥ dx − − |u| − c3 bx|u| − cδbx|u| dx |u| |∇u| p Ωi α Ωi i1 Q 1 λ α−i pi α−i qi− qi− qi− qi− ≥ u1,p,Ωi − − c6 u1,p,Ωi − c3 c7 u1,p,Ωi − cδc8 u1,p,Ωi . p α i1 3.21 Boundary Value Problems 11 If u1,p k ≥ u1,p,Ωi is sufficiently small such that 1 λ α− pi α−i q− qi− q− qi− − − c6 i u1,p,Ω − c3 c7i u1,p,Ω − cδc8i u1,p,Ω > 0, u1,p,Ω i i i i p α 3.22 we have Iu > 0. The mountain pass theorem guarantees that I has a nontrivial critical point u. 1,px 1,px W0 Since W0 Ω is a separable and reflexive Banach space, there exist {en }∞ n1 ⊂ −1,p x Ω and {en∗ }∞ Ω such that n1 ⊂ W 1,px W0 ei , ej∗ ⎧ ⎨1, i j, ⎩0, i/ j, 3.23 Ω span{en : n 1, 2, . . .}, W −1,p x Ω span{en∗ : n 1, 2, . . .}. For k 1, 2, . . . , denote Xk span{ek }, Yk ⊕kj1 Xj , Zk ⊕∞ X. jk j Theorem 3.2. Under assumptions (F), (A-1)–(D-1), problem 1.1 admits a sequence of solutions 1,px Ω such that Iun → ∞. {un } ∈ W0 Proof. Let ϕu Ω λ|u|αx /αx Fx, udx. We first show that ϕu is weakly strongly 1,px 1,px Ω. By the compact embedding W0 Ω → continuous. Let un u weakly in W0 αx on Ω. By the inequality |u ≤ Lαx Ω, we have Ω |un − u|αx dx → 0 and un → u a.e. n| α −1 αx αx αx αx |un − u| |u| and the Vitali Theorem, we get Ω |un | dx → Ω |u| dx. 2 Note that Ω |Fx, un − Fx, u|dx Ω bx|Gx, un − Gx, u|dx 3.24 ≤ 2bxr Gx, un − Gx, ur . When |u| ≥ M, q x |u|qx |u| |u|qx c1 |u|qx c2 ≤ c |Gx, u| ≤ c1 2 μx μx q x μxqx qx μx ≤ C3 1 |u|qx C3 1 |u|r xqx/r x . 3.25 When |u| ≤ M, Gx, u is bounded. So we get |Gx, u| ≤ C4 1 |u|r xqx/r x . 3.26 12 Boundary Value Problems 1,px r xqx Ω → Lr xqx Ω, we get Ω. So By the compact embedding W0 un → u in L by Theorem 2.12 we obtain |Gx, un − Gx, u|r → 0, that is, Ω |Fx, un − Fx, u|dx → 0. Hence we obtain that ϕu is weakly strongly continuous. By Proposition 3.5 in 8, βk βk r supu∈Zk ,u1,p ≤r |ϕu| → 0 as k → ∞ for r > 0. For all n, there exists a positive integer kn such that βk n ≤ 1 for all k ≥ kn . Assume kn < kn1 for each n. Define {rk : k 1, 2, . . .} in the following way: rk ⎧ ⎨n, kn ≤ k < kn 1, ⎩1, 1 ≤ k < k1 . 3.27 Note that rk → ∞ as k → ∞. Hence for u ∈ Zk with u1,p rk , we get Iu Ω 1 1 p− |∇u|px |u|px dx − ϕu ≥ u1,p − 1. px p 3.28 Iu −→ ∞ as k −→ ∞. 3.29 So inf u∈Zk ,u1,p rk Note that Fx, u bxGx, u ≥ c4 bx|u|μx −c5 bx and px, αx ∈ CΩ. Since the dimension of Yk is finite, any two norms on Yk are equivalent, then c9 |u|1,p ≤ |u|α ≤ c10 |u|1,p . If |u|1,p ≤ 1, it is immediate that |u|α ≤ c10 . If |u|1,p > 1/c9 , then |u|α > 1. As in the proof of Q Theorem 3.1 we can find hypercubes {Qi }i1 which mutually have no common points such Q that Ω ⊆ i1 Qi and pi supy∈Ωi py < infy∈Ωi αy α−i , where Ωi Qi ∩ Ω. Then we need only to consider the case: |u|1,p,Ωi > max{1, 1/c9 } for every i. We have Iu ≤ Ω ≤ Ω |∇u|px |u|px px px |∇u|px |u|px px px −λ |u|αx − c4 bx|u|μx c5 bxdx αx −λ |u|αx dx c5 αx Ω bxdx Q λ α−i 1 pi α−i ≤ bxdx u1,p,Ωi − c9 u1,p,Ωi c5 p− α Ω i1 ⎛ ⎞ Q λ α−i ⎝ α− pi α−i ⎠ c c − bxdx. u u 5 1,p,Ωi 1,p,Ωi α− α 9 Ω i1 p− λc i 3.30 9 α− − Let c0 α /p− λc9 i , fi t tαi − c0 tpi . Let fi si mint≥0 fi t < 0 and fi ti 0. − "Q Denote t i1 u1,p,Ωi . Let t be sufficiently large such that t > max{Q, Qti , Q2c0 1/αi −pi , Boundary Value Problems 13 i 1, 2, . . . , Q}. There at least exists one i0 such that u1,p,Ωi0 ≥ 1/Q We have "Q i1 u1,p,Ωi t/Q. Q Q pi α−i −pi fi u1,p,Ωi − c0 u1,p,Ω u1,p,Ω i i i1 i1 i / i0 ≥ pi u1,p,Ω i α−i −pi u1,p,Ω i fi si t Q fi si c0 t, Q i/ i0 ≥ Q i1 − c0 u1,p,Ωi pi 0 t Q pi 0 α−i −pi 0 0 α−i −pi 0 u1,p,Ωi − c0 0 0 0 3.31 − c0 "Q and i1 fi u1,p,Ωi → ∞ as t → ∞. Hence we obtain that Iu → −∞ as u1,p → ∞. Thus for each k, there exists ρk > rk such that Iu < 0 for u ∈ YK ∩ Sρk . From Theorem 3.1 Iu satisfies P Sc condition. In view of D-1, by Fountain Theorem see 16, we conclude the result. In the second case, we additionally impose the following condition: A-2 αx, qx ∈ CΩ and 1 ≤ αx px qx < rx − 1/rxp∗ x. Theorem 3.3. Under assumptions (F), (A-2), (B-1), and (C-1) there exist λ∗ > 0 such that when λ ∈ 0, λ∗ , problem 1.1 admits a nontrivial solution. Proof. It is obvious that α− < p− . Let ε0 > 0 be such that α− ε0 < p− . Since αx ∈ CΩ, there exists δ > 0 and x0 ∈ Ω such that |αx − α− | < ε0 , for all x ∈ Bx0 , δ. Thus αx ≤ α− ε0 for all x ∈ Bx0 , δ. Let ϕ be as defined in Theorem 3.1. By C-1, |Gx, t| ≤ c3 |t|qx , and qx |Fx, t| ≤ c3 bx|t| , when t < δ. Then for any t ∈ 0, 1 and t < δ, we have I tϕ Ω p− t px px ∇ϕ px px ∇ϕ ≤t px Bx0 ,δ − tq px ϕ px − λt αx αx ϕ αx − F x, tϕ dx px αx ϕ ϕ α dx − t dx λ px αx Bx0 ,δ qx c3 bxϕ dx Bx0 ,δ p− px ∇ϕ ≤t px Bx0 ,δ − tq px αx ϕ ϕ α− ε0 dx − t dx λ px αx Bx0 ,δ qx c3 bxϕ dx. Bx0 ,δ If t is sufficiently small, Itϕ < 0. 3.32 14 Boundary Value Problems cδ|t|qx and |Fx, t| ≤ c3 bx|t|qx From F and C-1, we have |Gx, t| ≤ c3 |t|qx qx cδbx|t| . By Theorems 2.8 and 2.11, there exist positive constants k1 , k2 , k3 such that ≤ k3 |u|1,p . When u1,p is sufficiently small, we have |u|α ≤ k1 |u|1,p , |u|q,b ≤ k2 |u|1,p , |u|q,b < 1, and u < 1. As in the proof of Theorem 3.1 we can find hypercubes uα < 1, uq,b q,b Q Q {Qi }i1 which mutually have no common points such that Ω ⊆ i1 Qi , pi supy∈Ωi py < infy∈Ωi qy qi− and pi < qi− , where Ωi Qi ∩ Ω. Then Iu ≥ Q i1 ≥ Ωi 1 px px − c3 bx|u|qx − cδbx|u|qx dx |u| |∇u| p λ − − α Ω |u|αx dx Q − 1 λ − pi qi− qi− qi− qi− − c k − cδk − − k1α uα1,p . u u u 3 3 2 1,p,Ωi 1,p,Ωi 1,p,Ωi p α i1 3.33 p Since u1,p ≥ u1,p,Ωi , for all i, when u1,p ρ, u1,p,Ωi ρi < ρ. Fix ρ such that 1/p ρi i − q− q− q− q− c3 k3 i ρi i − cδk2 i ρi i > 0. Then we have N 1 pi λ − − qi− qi− qi− qi− Iu ≥ ρ − c3 k3 ρi − cδk2 ρi − − k1α ρα l1 ρ − λl2 ρ . i p α i1 3.34 Let λ∗ l1 ρ/2l2 ρ. When λ ∈ 0, λ∗ , Iu ≥ l1 /2 > 0. As in the proof of Theorem 2.1 1,px Ω : u1,p < ρ}, we have inf∂Bρ 0 Iu > 0 and −∞ < in 17, denote Bρ 0 {u ∈ W0 c infBρ 0 Iu < 0. Let 0 < ε < inf∂Bρ 0 Iu − infBρ 0 Iu. Applying Ekeland’s variational principle to the functional Iu : Bρ 0 → R, we find uε ∈ Bρ 0 such that Iuε < infBρ 0 Iu ε, Iuε < Iuεu−uε 1,p , uε / u ∈ Bρ 0, and I uε −1,p ≤ ε. Thus we get a sequence {un } ⊂ 1,px Bρ 0 such that Iun → c and I un → 0. It is clear that {un } is bounded in W0 Ω. As in the proof of Theorem 3.1, we get a subsequence of {un }, still denoted by {un }, such that 1,px Ω. So Iu c < 0 and I u 0. un → u in W0 Theorem 3.4. Under assumptions (F), (A-2), and (B-1)–(D-1), problem 1.1 has a sequence of 1,px Ω such that Iun → ∞. solutions {un } ∈ W0 1,px Proof. First we show that any P Sc sequence is bounded. Let {un } ∈ W0 Ω and c ∈ R, −1,p x Ω. By B-1, there exist σ > 0 such that such that Iun → c and I un → 0 in W inf{1/px − 1 σ/μx} μ1 > 0. From F, A-2, and B-1–D-1, we have 1 fx, un un − Fx, un dx > 0, Ω1 μx 1 fx, u − Fx, u u n n n dx < C1 , μx Ω2 1 μx fx, un un dx C2 ≥ Fx, un dx ≥ c4 bx|un | dx − c5 bxdx. Ω μx Ω Ω Ω 3.35 Boundary Value Problems 15 Thus we have 1 |un |αx − Fx, un dx |∇un |px |un |px − λ αx Ω px 1σ 1 |un |αx − − Fx, un dx |∇un |px |un |px − λ μx αx Ω px 1σ 1σ 1σ αx λ|un | dx fx, un un |∇un |px |un |px μx Ω μx Ω μx Iun 1σ 1σ λ|un |αx − fx, un un dx μx μx 1σ 1σ 1 px px dx − − un dx ≥ |un | |∇un |px−2 ∇un ∇ |∇un | μx μx Ω px Ω |un |αx μx − dx λ σc4 bx|un | dx − σc5 bxdx αx Ω Ω Ω # $ 1σ I un , un − C3 . μx 3.36 − By Young’s inequality, for ε1 , ε2 , ε3 ∈ 0, 1, we get Ω |∇un | px−2 ≤ C1 ε1 1σ · un dx ∇un ∇ μx px Ω |∇un | Ω dx 1−p C1 ε1 ε2 |un |αx dx ≤ ε3 αx Ω 1−p −p /μ−p− ε2 |Ω|, |un |μx dx C1 ε1 −α /p−α− Ω |un |px dx ε3 3.37 |Ω|. 1−p Take ε1 , ε2 , and ε3 sufficiently small so that μ1 −C1 ε1 > 0, μ1 −λε3 ≥ 0 and σc4 bx−C1 ε1 0, then $ # 1σ un ≥ C4 |∇un |px dx − C5 . Iun − I un , μx Ω ε2 ≥ 3.38 1,px Therefore by Theorems 2.6 and 2.9, we get that {un } is bounded in W0 Ω. Then as in the proof of Theorem 3.1{un } possesses a convergent subsequence {un } still denoted by {un }. By Theorem 3.2, we can also have inf u∈Zk ,u1,p rk Iu −→ ∞ as k −→ ∞. 3.39 16 Boundary Value Problems Q As in the proof of Theorem 3.1 we can find hypercubes {Qi }i1 which mutually have Q no common points such that Ω ⊆ i1 Qi and pi supy∈Ωi py < infy∈Ωi μy μ−i , where Ωi Qi ∩ Ω. Since the dimension of Yk is finite, any two norms on Yk are equivalent. Then we need only to consider the cases |u|1,p,Ωi > 1 and uμ,b,Ωi > 1 for every i. We have Iu ≤ Ω ≤ Ω |∇u|px |u|px px px |∇u|px |u|px px px −λ |u|αx − c4 bx|u|μx c5 bxdx αx − c4 bx|u|μx dx c5 Ω 3.40 bxdx Q 1 pi μ−i bxdx. ≤ u1,p,Ωi − C4 u1,p,Ωi c5 p Ω i1 Hence Iu → −∞ as u1,p → ∞. As in the proof of Theorem 3.2, we complete the proof. In the third case, we additionally impose the following condition: A-3 αx, qx ∈ CΩ, and 1 ≤ αx, qx px, B-3 there exist θ > 0 such that Gx, t ≥ 0 for t ∈ 0, θ. Theorem 3.5. Under assumptions (F), (A-3), and (B-3), problem 1.1 admits a nontrivial solution. Proof. By Young’s inequality, for ε ∈ 0, 1, we get |un |αx /αx ≤ ε|un |px ε−α we have Fx, u ≤ Cbx|u|qx C1 bx. Thus Iu Ω |∇u|px |u|px px px −λ /p−α− . By F, |u|αx − Fx, udx αx |∇u|px |u|px ≥ dx − λε |u|px dx px px Ω Ω qx −α /p−α− bx|u| dx − λε bxdx −C |Ω| − C1 ≥ Ω Ω 1 − λε p |∇u|px |u|px dx − C 3.41 Ω Ω bx|u|qx dx − C2 . Take ε sufficiently small so that 1/p − λε > 0. From Theorem 2.11, uq,b ≤ cu1,p . If u1,p ≤ 1, uq,b is bounded. Then we need only to consider the case u1,p > 1. As in the proof of Boundary Value Problems 17 Q Theorem 3.1 we can find hypercubes {Qi }i1 which mutually have no common points such Q that Ω ⊆ i1 Qi and qi supy∈Ωi qy < infy∈Ωi py pi− , where Ωi Qi ∩ Ω. Then we have Iu ≥ M i1 pi− c1 u1,p,Ω i − qi c2 u1,p,Ω i J pj− qj− c1 u1,p,Ωj − c3 u1,p,Ωj C3 j1 3.42 M pi− qi C4 , ≥ c1 u1,p,Ω − c u 4 1,p,Ω i i i1 where Ωi bx|u|qx dx > 1, i 1, 2, . . . M, and Ωj bx|u|qx dx ≤ 1, i 1, 2, . . . J, M J Q. As in the proof of Theorem 3.2, we obtain that Iu is coercive, that is, Iu → ∞ as u1,p → ∞. Thus I has a critical point u such that Iu infu∈W 1,px Ω Iu and further u is a weak 0 solution of 1.1. Next we show that u is nontrivial. Let Bx0 , δ be the same as that in Theorem 3.3. By B-3, Fx, tϕ ≥ 0. Then I tϕ Ω p− ≤t ≤t p− t px px ∇ϕ px px ∇ϕ Ω px px ∇ϕ Ω px px ϕ px − λt αx αx ϕ αx − F x, tϕ dx px αx ϕ ϕ dx dx − tα λ px αx Ω0 3.43 px αx ϕ ϕ α− ε0 dx − t dx. λ px αx Ω0 If t is sufficiently small, Itϕ < 0. In the fourth case, we additionally impose the following condition: A-4 αx, qx ∈ CΩ and 1 ≤ qx px αx < p∗ x. Theorem 3.6. Under assumptions (F), (A-4), and (D-1), problem 1.1 admits a sequence of solutions 1,px Ω such that Iun → ∞. {un } ∈ W0 1,px Proof. First we show that any P Sc sequence is bounded. Let {un } ∈ W0 Ω and c ∈ R, such that Iun → c and I un → 0 in W −1,p x Ω. Denote inf{1/px − 1/αx} a > 0 and lx 1/px − a/2. We have 1/px − lx a/2 and lx − 1/αx ≥ a/2. 18 Boundary Value Problems We can get Iun − I un , lxun 1 |un |αx − Fx, un dx |∇un |px |un |px − λ αx Ω px 1 |un |αx − lx |∇un |px |un |px − λ − Fx, un dx αx Ω px lxλ|un |αx dx lxfx, un un − |∇un |px−2 ∇un ∇lx · un dx Ω ≥ 3.44 Ω |∇un |px−2 ∇un ∇lx · un dx |∇un |px |un |px dx − a 2 Ω Ω Ω lx − 1 λ|un |αx lxfx, un un − Fx, un dx. αx By Young’s inequality, for ε1 , ε2 ∈ 0, 1, we get Ω |∇un |px−2 ∇un ∇lx · un dx ≤ Cε1 Ω |∇un | px dx 1−p Cε1 ε2 Ω |un | αx dx 1−p −p /α−p− Cε1 ε2 |Ω|. 1−p Take ε1 and ε2 sufficiently small so that Cε1 < a/2 and Cε1 3.45 ε2 ≤ a/2. Then ∞ > Iun − I un , lxun a − Cε1 |∇un |px |un |px − lxfx, un un |Fx, un | dx − C1 ≥ 2 Ω a px px dx − C2 − Cε1 |∇un | ≥ |un | bx|un |qx dx − C3 . Ω 2 Ω 3.46 As in the proof of Theorem 3.5, Iun − I un , lxun → ∞, when un 1,p → ∞. Thus, 1,px we conclude that {un } is bounded in W0 Ω. Then as in the proof of Theorem 3.1{un } possesses a convergent subsequence {un } still denoted by {un }. By Theorem 3.2, we can also get inf u∈Zk ,u1,p rk Iu −→ ∞ as k −→ ∞. 3.47 Q As in the proof of Theorem 3.1 we can find hypercubes {Qi }i1 which mutually have Q no common points such that Ω ⊆ i1 Qi and pi supy∈Ωi py < infy∈Ωi αy α−i , where Boundary Value Problems 19 Ωi Qi ∩ Ω. Since the dimension of Yk is finite, any two norms on Yk are equivalent. Then we need only to consider the cases |u|1,p,Ωi > 1|u|α,Ωi > 1 and |u|q,b,Ωi > 1 for every i. We have Iu ≤ Ω |∇u|px |u|px px px |u|αx −λ dx C4 αx Ω bx|un |qx dx C5 Q pi qi α−i c0 u1,p,Ω − cu C6 . ≤ u 1,p,Ω 1,p,Ω i i i 3.48 i1 Hence we obtain Iu → −∞ as u1,p → ∞. As in the proof of Theorem 3.2, we complete the proof. Acknowledgments This work is supported by the Science Research Foundation in Harbin Institute of Technology HITC200702, The Natural Science Foundation of Heilongjiang Province A2007-04, and the Program of Excellent Team in Harbin Institute of Technology. References 1 O. Kováčik and J. 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