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 < κ * and it does not possess no solutions if κ > κ * . Furthermore, in the case of 2 ≤ N ≤ 9, problem (P) possesses a unique solution if κ = κ * .

中文翻译：

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具有一般指数非线性的非齐次椭圆方程解存在的阈值

在本文中，我们研究了非齐次非线性椭圆问题（P）-Δu+u=F(u)+κμ 在 RN 中的存在和不存在解，在 RN 中 u>0，u(x)→0 为|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 → ∞ （至少）呈指数增长。这里 N ≥ 2, κ > 0, μ∈Lc1(RN)\{0} \mu \in L_{\rm{c}}^1({{\bf R}^N})\反斜杠 \{ 0 \} 是非负数。然后，在 μ 上合适的可积性条件下，存在一个阈值参数 κ * > 0 使得问题 (P) 如果 0 < κ < κ * 有解，如果 κ > κ * 则不无解。此外，在 2 ≤ N ≤ 9 的情况下，