Variational methods for a resonant problem with the p-Laplacian in ℝN
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The solvability of the resonant Cauchy problem
-Δpu = λ1m (|x|)|u|p-2 u + ƒ(x) in ℝN; u ∈ D1,p (ℝN),
in the entire Euclidean space ℝN (N ≥ 1) is investigated as a part of the Fredholm alternative at the first (smallest) eigenvalue λ1 of the positive p-Laplacian -Δp on D1,p (ℝN) relative to the weight m(|x|). Here Δp stands for the p-Laplacian, m: ℝ+ → ℝ+ is a weight function assumed to be radially symmetric, m ≢ 0 in ℝ+, and ƒ : ℝN → ℝ is a given function satisfying a suitable integrability condition. The weight m(r) is assumed to be bounded and to decay fast enough as r → +∞. Let φ1 denote the (positive) eigenfunction associated with the (simple) eigenvalue λ1 of -Δp. If ∫ℝN ƒφ1 dx = 0, we show that problem has at least one solution u in the completion D1,p (ℝN) of C1c (ℝN) endowed with the norm (∫ℝN |∇u|p dx)1/p. To establish this existence result, we employ a saddle point method if 1 < p < 2, and in improved Poincaré inequality if 2 ≤ p < N. We use weighted Lebesgue and Sobolov spaces with weights depending on φ1. The asymptotic behavior of φ1(x) = φ1(|x|) as |x| → ∞ plays a crucial role.
CitationAlziary, B., Fleckinger, J., & Takac, P. (2004). Variational methods for a resonant problem with the p-Laplacian in ℝN. Electronic Journal of Differential Equations, 2004(76), pp. 1-32.
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