Fucik spectrum for operators with rapidly increasing weight and applications
- 1Beijing Technology and Business Univ., Beijing, China
Abstract
In this article, we study the Fucik spectrum for operators with rapidly increasing weight, which is defined as a set Σ comprising those (α, β) ∈ ℝ2 such that displaylines L u:=−Δ u−1/2(x · ∇ u)=α u+−β u−, in ℝN,cr u∈ X, has a non-trivial solution u, where, N≥1, u ±=max ( ± u, 0), u=u+−u−. The existence of a first nontrivial curve 𝒞 of this spectrum, along with some of its properties (e.g., Lipschitz continuity, strict decrease and asymptotic behavior) is investigated in this paper. Our difficulty is that the problem is defined on the whole space ℝN, and therefore certain estimates do not carry over from the Fucik problem on bounded domains. As an application, we establish the multiplicity of solutions to the problem displaylines −Δ u−1/2(x · ∇ u)=f(x,u), in ℝN,cr u∈ X, where, N≥1 and the nonlinearity f is asymptotically linear at zero and at infinity.
Related
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BibTeX
@article{bai2026fucik,
title = {{Fucik spectrum for operators with rapidly increasing weight and applications}},
author = {Jinzi Bai and Fei Fang},
journal = {Electronic Journal of Differential Equations},
year = {2026},
month = jul,
url = {http://ejde.math.txstate.edu/Volumes/2026/58/abstr.html},
note = {CC BY},
}
RIS
TY - JOUR
TI - Fucik spectrum for operators with rapidly increasing weight and applications
AU - Jinzi Bai
AU - Fei Fang
PY - 2026
DA - 2026/07/01
JO - Electronic Journal of Differential Equations
UR - http://ejde.math.txstate.edu/Volumes/2026/58/abstr.html
AB - In this article, we study the Fucik spectrum for operators with rapidly increasing weight, which is defined as a set Σ comprising those (α, β) ∈ ℝ^2 such that displaylines L u:=−Δ u−1/2(x · ∇ u)=α u^+−β u^−, in ℝ^N,cr u∈ X, has a non-trivial solution u, where, N≥1, u^ ±=max ( ± u, 0), u=u^+−u^−. The existence of a first nontrivial curve 𝒞 of this spectrum, along with some of its properties (e.g., Lipschitz continuity, strict decrease and asymptotic behavior) is investigated in this paper. Our difficulty is that the problem is defined on the whole space ℝ^N, and therefore certain estimates do not carry over from the Fucik problem on bounded domains. As an application, we establish the multiplicity of solutions to the problem displaylines −Δ u−1/2(x · ∇ u)=f(x,u), in ℝ^N,cr u∈ X, where, N≥1 and the nonlinearity f is asymptotically linear at zero and at infinity.
SN - 1072-6691
ER -