On a Class of Weighted p-Laplace Equation with Singular Nonlinearity
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On a Class of Weighted p-Laplace Equation with Singular Nonlinearity Prashanta Garain and Tuhina Mukherjee Abstract. This article deals with the existence of the following quasilinear degenerate singular elliptic equation: −div(w(x)|∇u|p−2 ∇u) = gλ (u), u > 0 in Ω, (Pλ ) u = 0 on ∂Ω, where Ω ⊂ Rn is a smooth bounded domain, n ≥ 3, λ > 0, p > 1, and w is a Muckenhoupt weight. Using variational techniques, for gλ (u) = λf (u)u−q and certain assumptions on f , we show existence of a solution to (Pλ ) for each λ > 0. Moreover, when gλ (u) = λu−q + ur , we establish existence of at least two solutions to (Pλ ) in a suitable range of the parameter λ. Here, we assume q ∈ (0, 1) and r ∈ (p − 1, p∗s − 1). Mathematics Subject Classification. 35R11, 35R09, 35A15. Keywords. Weighted p-Laplacian, Singular nonlinearity, Multiple weak solutions, Variational method.
1. Introduction In this article, we are interested in the question of existence of weak solutions to the following singular weighted p-Laplace equation: −Δp,w u = gλ (u), u > 0 in Ω, (Pλ ) u = 0 on ∂Ω, where Ω ⊂ Rn is a smooth bounded domain, n ≥ 3, λ > 0 and p > 1. We consider the nonlinearity gλ of the following two types: Case (I) gλ (u) = λf (u)u−q where q ∈ (0, 1) and f : [0, ∞) → R satisfies (f1) f (0) > 0, such that f is non-decreasing and satisfies the following hypothesis: lim
f (t)
t→∞ tq+p−1
= 0 and lim
t→0
f (t) = ∞. tq
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P. Garain and T. Mukherjee
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Case (II) gλ (u) = λu−q + ur where q ∈ (0, 1), r ∈ (p − 1, p∗s − 1). Here, p∗s = nps ps 1 n n−ps for 1 ≤ ps < n where ps = s+1 and s ∈ [ p−1 , ∞) ∩ ( p , ∞). We observe that in both the cases, Case (I) and Case (II), gλ is singular in the sense that: lim gλ (t) = +∞.
t→0+
Here: Δp,w u := div(w(x)|∇u|p−2 ∇u) is the weighted p-Laplace operator for some weight function w. When w ≡ 1, Δp,w = Δp u (which is the usual p-Laplace operator) which further reduces to the classical Laplace operator ’Δ’ for p = 2. In this article, we discuss the existence of weak solutions to the problem (Pλ ) depending on the range of λ. The study of singular elliptic problems has been a topic of considerable attention throughout the last 3 decades and there is a colossal amount of work done in this direction. Now, we state some known results in this direction which are essential to understand the difficulties and the framework of our problem. The following quasilinear singular problem has been investigated in quite a large number of papers: ⎧ h(x, u) ⎨ + μur , u > 0 in Ω, −Δp u = λ (1.1) uq ⎩ u = 0 on ∂Ω. When p = 2, λ > 0, and μ = 0 (that is the purely singular case), Crandall, Rabinowitz, and Tartar [7] proved the existence of a unique classical solution uλ ∈ C 2 (Ω) ∩ C(Ω) of the problem (1.1) for any q > 0 and h(x, u) = h(x) being nonnegative and bounded in Ω. For the same problem, existence of a weak solution in H01 (Ω) was proved by Lazer-Mckenna [18] when 0 < q < 3. Boccardo-Orsina [5] investigated the following purely singular problem in case of arbitrary q > 0 with a weight f
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