关键词: 单侧全局区间分歧, 半线性问题, 结点解, p-Laplacian方程

Abstract:

In this paper, we establish a unilateral global interval bifurcation result for the p-Laplacian equation. Furthermore, we shall prove the existence of the principal half-eigenvalues for the half-linear p-Laplacian equation. Moreover, we also investigate the existence of radial nodal solutions for the problems.
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where 1< p < +∞, ψ_p(s)=|s|^p-2s, a(r) ∈ C[0, 1], a(r) ≥ 0,a(r)?0 on any subinterval of[0, 1]; λ is a parameter, u⁺=max{u, 0}, u^-=-min{u,0}, α, β ∈ C[0, 1] are radially symmetric; f ∈ C(R, R), sf(s) > 0 for s∈R⁺, and f₀ ∈[0, ∞) and f_∞ ∈ (0, ∞) or f₀ ∈ (0, ∞] and f_∞=0 or f₀=0 and f_∞=∞, where f₀= f(s)/s, f_∞=f(s)/s. We use unilateral global bifurcation techniques and the approximation of connected components to prove our main results.

Key words: Unilateral interval bifurcation, Half-quasilinear problems, Nodal solutions, p-Laplacian equation