In the present paper we prove some uniqueness results for weak solutions to a class of problems, whose prototype is −div ((ε + |∇u|^2)^{(p−2)/2 ∇uϕ) = f ϕ in Ω u = 0 on ∂Ω, where ε ≥ 0,1 < p < +∞, φ(x) is the density of the N-dimensional Gauss measure, Ω is an open subset of RN (N > 1) with Gauss measure less than one and datum f belongs to the natural dual space. When p ≤ 2 we obtain a uniqueness result for ε = 0. While for p > 2 we have to consider ε > 0 unless the sign of f is constant. Some counterexamples are given too.
Uniqueness results for strongly monotone operators related to Gauss measure
M. F. Betta;F. Feo
;
2019-01-01
Abstract
In the present paper we prove some uniqueness results for weak solutions to a class of problems, whose prototype is −div ((ε + |∇u|^2)^{(p−2)/2 ∇uϕ) = f ϕ in Ω u = 0 on ∂Ω, where ε ≥ 0,1 < p < +∞, φ(x) is the density of the N-dimensional Gauss measure, Ω is an open subset of RN (N > 1) with Gauss measure less than one and datum f belongs to the natural dual space. When p ≤ 2 we obtain a uniqueness result for ε = 0. While for p > 2 we have to consider ε > 0 unless the sign of f is constant. Some counterexamples are given too.File in questo prodotto:
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