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We obtain multiple positive solutions of singular p-Laplacian problems using variational methods. The techniques are applicable to other types of singular problems as well. 1. Introduction We consider the singular quasilinear elliptic boundary value problem −∆ p u = a(x)u−γ + λ f (x,u)

in Ω,

u > 0 in Ω,

(1.1)

u = 0 on ∂Ω, where Ω is a bounded C 2 domain in Rn ,n ≥ 1, ∆ p u = div(|∇u| p−2 ∇u) is the p-Laplacian, 1 < p < ∞, a ≥ 0 is a nontrivial measurable function, γ > 0 is a constant, λ > 0 is a parameter, and f is a Carath´eodory function on Ω × [0, ∞) satisfying sup

(x,t)∈Ω×[0,T]




f (x,t)
< ∞

∀T > 0.

(1.2)

The semilinear case p = 2 with γ < 1 and f = 0 has been studied extensively in both bounded and unbounded domains (see [5, 6, 7, 10, 11, 12, 14, 20] and their references). In particular, Lair and Shaker [11] showed the existence of a unique (weak) solution when Ω is bounded and a ∈ L2 (Ω). Their result was extended to the sublinear case f (t) = tβ , 0 < β ≤ 1 by Shi and Yao [15] and Wiegner [18]. In the superlinear case 1 < β < 2∗ − 1 and for small λ, Coclite and Palmieri [4] obtained a solution when a = 1 and Sun et al. [16] obtained two solutions using the Ekeland’s variational principle for more general a’s. Zhang [19] extended their multiplicity result to more general superlinear terms f (t) ≥ 0 using critical point theory on closed convex sets. The ODE case n = 1 was studied by Agarwal and O’Regan [1] using fixed point theory and by Agarwal et al. [2] using variational methods. The purpose of the present paper is to treat the general quasilinear case p ∈ (1, ∞), γ ∈ (0, ∞), and f is allowed to change sign. We use a simple cutoff argument and only the basic critical point theory. Our results seem to be new even for p = 2. Copyright © 2006 Hindawi Publishing Corporation Boundary Value Problems 2005:3 (2005) 377–382 DOI: 10.1155/BVP.2005.377

378

Positive solutions of singular p-Laplacian problems

First we assume (H1 ) ∃ϕ ≥ 0 in C01 (Ω) and q > n such that aϕ−γ ∈ Lq (Ω). This does not require γ < 1 as usually assumed in the literature. For example, when Ω is the unit ball, a(x) = (1 − |x|2 )σ , σ ≥ 0, and γ < σ + 1/n, we can take ϕ(x) = 1 − |x|2 and q < 1/(γ − σ) (resp., q with no additional restrictions) if γ > σ (resp., γ ≤ σ). Theorem 1.1. If (H1 ) and (1.2) hold and f ≥ 0, then ∃λ0 > 0 such that problem (1.1) has a solution ∀λ ∈ (0,λ0 ). Corollary 1.2. Problem (1.1) with f = 0 has a solution if (H1 ) holds. Next we allow f to change sign, but strengthen (H1 ) to (H2 ) a ∈ L∞ (Ω) with a0 := inf Ω a > 0 and γ < 1/n. This implies that aϕ−γ ∈ Lq (Ω) for any ϕ whose interior normal derivative ∂ϕ/∂ν > 0 on ∂Ω and q < 1/γ. Theorem 1.3. If (H2 ) and (1.2) hold, then ∃λ0 > 0 such that problem (1.1) has a solution ∀λ ∈ (0,λ0 ).

Finally we assume that f is C 1 in t, satisfies

 
ft (x,t)
≤ C t r −2 + 1

(1.3)

for some 2 ≤ r < p∗ , and p-superlinear: 0 < θF(x,t) ≤ t f (x,t),

t large

(1.4)

for some θ > p. Here p∗ = np/(n − p) (resp., ∞) if p < n (resp., p ≥ n) is the critical Sobolev expo