Thresholds for the existence of solutions to inhomogeneous elliptic equations with general exponential nonlinearity
Thresholds for the existence of solutions to inhomogeneous elliptic equations with general exponential nonlinearity
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DOI：
10.1515/anona-2021-0220
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发表时间：
2022-01
影响因子：
4.2
通讯作者：
Kazuhiro Ishige;S. Okabe;Tokushi Sato
中科院分区：
文献类型：
--
作者：
Kazuhiro Ishige;S. Okabe;Tokushi Sato
摘要 本文研究了非齐次非线性椭圆问题 (P) −Δu+u=F(u)+κμ 在 RN 中解的存在与不存在，在 RN 中 u>0，u(x)→0 as |x|→∞, - \Delta u + u = F(u) + \kappa \mu \quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u > 0\quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u(x) \to 0\quad {\kern 1pt} {\rm as}{\kern 1pt} \quad |x| \to \infty ，其中 F = F(t) 随着 t → ∞ 呈指数增长（至少）。这里 N ≥ 2，κ > 0，并且 μεLc1(RN)\{0} \mu \in L_{\rm{c}}^1({{\bf R}^N})\backslash \{ 0\} 是非负的。那么，在μ上合适的可积性条件下，存在阈值参数κ* > 0，使得如果κ*为0则问题(P)有解。此外，在2≤N≤9的情况下，如果κ=κ*，问题(P)具有唯一解。
Abstract In this paper we study the existence and the nonexistence of solutions to an inhomogeneous non-linear elliptic problem (P) −Δu+u=F(u)+κμ in RN, u>0 in RN, u(x)→0 as |x|→∞, - \Delta u + u = F(u) + \kappa \mu \quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u > 0\quad {\kern 1pt} {\rm in}{\kern 1pt} \quad {{\bf R}^N},\quad u(x) \to 0\quad {\kern 1pt} {\rm as}{\kern 1pt} \quad |x| \to \infty , where F = F(t) grows up (at least) exponentially as t → ∞. Here N ≥ 2, κ > 0, and μ∈Lc1(RN)\{0} \mu \in L_{\rm{c}}^1({{\bf R}^N})\backslash \{ 0\} is nonnegative. Then, under a suitable integrability condition on μ, there exists a threshold parameter κ* > 0 such that problem (P) possesses a solution if 0 κ*. Furthermore, in the case of 2 ≤ N ≤ 9, problem (P) possesses a unique solution if κ = κ*.