Variational methods for a resonant problem with the p-Laplacian in ℝN
| dc.contributor.author | Alziary, Benedicte | |
| dc.contributor.author | Fleckinger, Jacqueline | |
| dc.contributor.author | Takac, Peter | |
| dc.date.accessioned | 2021-04-26T13:29:27Z | |
| dc.date.available | 2021-04-26T13:29:27Z | |
| dc.date.issued | 2004-05-26 | |
| dc.description.abstract | 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 λ1of 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. | |
| dc.description.department | Mathematics | |
| dc.format | Text | |
| dc.format.extent | 32 pages | |
| dc.format.medium | 1 file (.pdf) | |
| dc.identifier.citation | Alziary, B., Fleckinger, J., & Takac, P. (2004). Variational methods for a resonant problem with the p-Laplacian in ℝN. <i>Electronic Journal of Differential Equations, 2004</i>(76), pp. 1-32. | |
| dc.identifier.issn | 1072-6691 | |
| dc.identifier.uri | https://hdl.handle.net/10877/13429 | |
| dc.language.iso | en | |
| dc.publisher | Southwest Texas State University, Department of Mathematics | |
| dc.rights | Attribution 4.0 International | |
| dc.rights.uri | https://creativecommons.org/licenses/by/4.0/ | |
| dc.source | Electronic Journal of Differential Equations, 2004, San Marcos, Texas: Southwest Texas State University and University of North Texas. | |
| dc.subject | p-Laplacian | |
| dc.subject | Degenerate quasilinear Cauchy problem | |
| dc.subject | Fredholm alternative | |
| dc.subject | (p-1)-homogeneous problem at resonance | |
| dc.subject | Saddle point geometry | |
| dc.subject | Improved Poincare inequality | |
| dc.subject | Second-order Taylor formula | |
| dc.title | Variational methods for a resonant problem with the p-Laplacian in ℝN | |
| dc.type | Article |