Multiplicity of Solution for a Quasilinear Equation with Singular Nonlinearity
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Multiplicity of Solution for a Quasilinear Equation with Singular Nonlinearity Kaushik Bal and Prashanta Garain Abstract. For an open, bounded domain Ω in RN which is strictly convex with smooth boundary, we show that there exists a Λ > 0 such that for 0 < λ < Λ, the quasilinear singular problem −Δp u = λu−δ + uq in Ω u = 0 on ∂Ω; u > 0 in Ω 1,p admits at least two distinct solutions u and v in Wloc (Ω) ∩ L∞ (Ω) N (p−1) 2N +2 provided δ ≥ 1, N +2 < p < N and p − 1 < q < N −p .
Mathematics Subject Classification. 35B09, 35B44, 35B45, 35D99. Keywords. Quasilinear problem, singular nonlinearity, a priori estimates, topological degree.
1. Introduction In this paper, we study the multiplicity of weak solution to the quasilinear singular problem given by − Δp u = λu−δ + uq in Ω u = 0 on ∂Ω; u > 0 in Ω (1) where Ω(⊂ R ) is a strictly convex bounded domain with smooth boundary. Here N
Δp u := div(|∇u|p−2 ∇u) is the p-Laplacian operator for 1 < p < ∞. We also assume that λ > 0, δ ≥ 1, N (p−1) 2N +2 N +2 < p < N and p − 1 < q < p∗ − 1 where p∗ = N −p is the critical Sobolev exponent. The first author is supported by DST-Inspire Faculty Award MA-2013029. The second author is supported by NBHM Fellowship No: 2/39(2)/2014/NBHM/R&D-II/8020/June 26, 2014.
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K. Bal and P. Garain
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We start with a brief background of the problem (1) which was available in the literature and is critical for a clear understanding of the issues and the framework of our study. About 3 decades of work on the study of singular elliptic equation can be traced back to the pioneering work of of Crandall et al. [1], where the problem −Δu = u−δ in Ω; u = 0 on ∂Ω was shown to admit a unique classical solution for any δ > 0 provided Ω bounded. Following this Lazer–Mckenna [2] elaborating that the unique classical solution u is also in H01 (Ω) iff 0 < δ < 3. They also showed that the ¯ provided 0 < δ < 1. This was followed by the work solution belongs to C 1 (Ω) of Haitao [3] who studied the perturbed singular problem − Δu = λu−δ + uq in Ω u = 0 on ∂Ω; u > 0 in Ω
(2)
and showed the existence of Λ > 0 such that there exists at least two solutions u, v ∈ H01 (Ω) to problem (2) for λ < Λ, no solution for λ > Λ and at least N +2 one solution for λ = Λ provided 0 < δ < 1 < q ≤ N −2 using fibering method on Nehari Manifold. These results were generalised for p-Laplacian by Giacomoni et al. [4] who showed among other results the existence of at least two solutions for 0 < δ < 1 and p−1 < q ≤ p∗ −1. It should be noted that in the above-mentioned works on perturbed problem, the solution so obtained satisfied the equation in the trace sense and the restriction 0 < δ < 1 is due to the use of variational methods which requires the associated functional to be well defined on W01,p (Ω). Boccardo and Orsina [5] took a different approach and showed that the problem f (x) in Ω; u = 0 in ∂Ω (3) uδ 1 (Ω) for any non-negative f ∈ L1 (Ω) in the sense admits a solution u ∈ Hloc that fφ ∇u∇φ = , φ ∈ C01 (Ω) δ u Ω Ω − div(M (x)∇u) =
f
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