一类拟线性SchrO^¨dinger方程正解的存在性

Existence of Positive Solutions for Quasilinear Schr^¨dinger Equations

摘要: 本文讨论如下一类拟线性SchrO^¨dinger方程其中V(x):ℝ^N→ℝ为位势函数，γ > 0,且N≥3.当γ∈(0,γ₀)时，我们得到了上述问题的正解.此外当位势函数V(x)≡V_∞ > 0,我们在H²(ℝ^N)∩ C²(ℝ^N)上证明了经典径向正解u_γ的存在性,且γ→0⁺时，满足u_γ→u₀，其中u₀是以下半线性问题的基态解:

Abstract: This paper focuses on a class of quasilinear SchrO^¨dinger equations:where V(x):ℝ^N→ℝ is a given potential,γ > 0 and N≥3. Firstly, we obtain a positive solution for the above problem in γ∈(0,γ₀). The potential function V(x) is considered in V(x)≡V_∞ > 0. We prove the existence of a positive classical radial solution u_γ and up to a subsequence,u_γ→u₀ in H²(ℝ^N)∩ C²(ℝ^N) as γ→0⁺, where u₀ is the ground state of the following semilinear problem: