Abstract Let Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 3 {N\geq 3} ) be a C 2 {C^{2}} bounded domain, and let δ be the distance to ∂ ⁡ Ω {\partial\Omega} . We study equations ( E ± ) {(E_{\pm})} , - L μ ⁢ u ± g ⁢ ( u , | ∇ ⁡ u | ) = 0 {-L_{\mu}u\pm g(u,\lvert\nabla u\rvert)=0} in Ω, where L μ = Δ + μ δ 2 {L_{\mu}=\Delta+\frac{\mu}{\delta^{2}}} , μ ∈ ( 0 , 1 4 ] {\mu\in(0,\frac{1}{4}]} and g : ℝ × ℝ + → ℝ + {g\colon\mathbb{R}\times\mathbb{R}_{+}\to\mathbb{R}_{+}} is nondecreasing and locally Lipschitz in its two variables with g ⁢ ( 0 , 0 ) = 0 {g(0,0)=0} . We prove that, under some subcritical growth assumption on g, equation ( E + ) {(E_{+})} with boundary condition u = ν {u=\nu} admits a solution for any nonnegative bounded measure on ∂ ⁡ Ω {\partial\Omega} , while equation ( E - ) {(E_{-})} with boundary condition u = ν {u=\nu} admits a solution provided that the total mass of ν is small. Then we analyze the model case g ⁢ ( s , t ) = | s | p ⁢ t q {g(s,t)=\lvert s\rvert^{p}t^{q}} and obtain a uniqueness result, which is even new with μ = 0 {\mu=0} . We also describe isolated singularities of positive solutions to ( E + ) {(E_{+})} and establish a removability result in terms of Bessel capacities. Various existence results are obtained for ( E - ) {(E_{-})} . Finally, we discuss existence, uniqueness and removability results for ( E ± ) {(E_{\pm})} in the case g ⁢ ( s , t ) = | s | p + t q {g(s,t)=\lvert s\rvert^{p}+t^{q}} .

中文翻译：

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具有哈代势和梯度相关非线性的椭圆方程

摘要 令Ω ⊂ ℝ N {\Omega\subset\mathbb{R}^{N}} ( N ≥ 3 {N\geq 3} ) 是一个C 2 {C^{2}} 有界域，令δ 为到∂ ⁡ Ω {\partial\Omega} 的距离。我们研究方程 ( E ± ) {(E_{\pm})} , - L μ ⁢ u ± g ⁢ ( u , | ∇ ⁡ u | ) = 0 {-L_{\mu}u\pm g(u, \lvert\nabla u\rvert)=0} in Ω, 其中 L μ = Δ + μ δ 2 {L_{\mu}=\Delta+\frac{\mu}{\delta^{2}}} , μ ∈ ( 0 , 1 4 ] {\mu\in(0,\frac{1}{4}]} 和 g : ℝ × ℝ + → ℝ + {g\colon\mathbb{R}\times\mathbb{R} _{+}\to\mathbb{R}_{+}} 在它的两个变量中是非递减的和局部 Lipschitz，g ⁢ ( 0 , 0 ) = 0 {g(0,0)=0} 。我们证明，在 g 的一些亚临界增长假设下，方程 ( E + ) {(E_{+})} 与边界条件 u = ν {u=\nu} 承认 ∂ ⁡ Ω {\partial\Omega 上任何非负有界测度的解} , 而边界条件为 u = ν {u=\nu} 的方程 ( E - ) {(E_{-})} 承认一个解，条件是 ν 的总质量很小。然后我们分析模型案例 g ⁢ ( s , t ) = | | p ⁢ tq {g(s,t)=\lvert s\rvert^{p}t^{q}} 并获得唯一性结果，这对于 μ = 0 {\mu=0} 甚至是新的。我们还描述了 ( E + ) {(E_{+})} 的正解的孤立奇点，并根据贝塞尔容量建立了可移除性结果。对于 ( E - ) {(E_{-})} 获得各种存在结果。最后，我们讨论在 g ⁢ ( s , t ) = | 情况下 ( E ± ) {(E_{\pm})} 的存在性、唯一性和可移除性结果。| p + tq {g(s,t)=\lvert s\rvert^{p}+t^{q}} 。这甚至是新的 μ = 0 {\mu=0} 。我们还描述了 ( E + ) {(E_{+})} 的正解的孤立奇点，并根据贝塞尔容量建立了可移除性结果。对于 ( E - ) {(E_{-})} 获得各种存在结果。最后，我们讨论在 g ⁢ ( s , t ) = | 情况下 ( E ± ) {(E_{\pm})} 的存在性、唯一性和可移除性结果。| p + tq {g(s,t)=\lvert s\rvert^{p}+t^{q}} 。这甚至是新的 μ = 0 {\mu=0} 。我们还描述了 ( E + ) {(E_{+})} 的正解的孤立奇点，并根据贝塞尔容量建立了可移除性结果。对于 ( E - ) {(E_{-})} 获得各种存在结果。最后，我们讨论在 g ⁢ ( s , t ) = | 情况下 ( E ± ) {(E_{\pm})} 的存在性、唯一性和可移除性结果。| p + tq {g(s,t)=\lvert s\rvert^{p}+t^{q}} 。