Positive radial solutions to classes of singular problems on the exterior domain of a ball

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• 作者：Eun Kyoung Lee^a ; ^{eunkyoung165@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; R. Shivaji^b ; ^{shivaji@uncg.edu" class="auth_mail" title="E-mail the corresponding author} ; Byungjae Son^b ; ^{b_son@uncg.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Nonlinear boundary conditions ; Singular problems ; Exterior domains ; Positive radial solutions
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 February 2016
• 年：2016
• 卷：434
• 期：2
• 页码：1597-1611
• 全文大小：417 K

文摘

We study positive radial solutions to singular boundary value problems of the form: where 836979aef09b9fed0c00d5775bf" title="Click to view the MathML source">Δu:=div(∇u)$\mathrm{\Delta }u:=\mathrm{div}(\mathrm{\nabla }u)$ is the Laplacian operator of u  , Ω={x∈R^N‖x|>r₀>0,N>2}$\mathrm{\Omega }=\{x\in {\mathbb{R}}^N\Vert x|>r_0>0,N>2\}$, λ>0$\lambda >0$, K∈C([r₀,∞),(0,∞))$K\in C([r_0,\infty ),(0,\infty ))$ is such that $K(s)\le \frac{1}{s^{N+\hat{\beta }}}$ for s≫1$s\gg 1$ for some $\hat{\beta }>1$, $\alpha <\min \{1,\frac{\hat{\beta }}{N-2}\}$ and $\frac{\partial u}{\partial \eta }$ is the outward normal derivative of u   on |x|=r₀$|x|=r_0$. Here, 83f" title="Click to view the MathML source">f∈C¹([0,∞),R)$f\in C^1([0,\infty ),\mathbb{R})$ is such that $\frac{f(s)}{s^{1+\alpha }}\to 0$ as s→∞$s\to \infty$, and $\tilde{c}\in C([0,\infty ),(0,\infty ))$. We analyse the cases when (a  ) f(0)>0$f(0)>0$ and (b  ) 3c75896d5c7fa450f02def81d17" title="Click to view the MathML source">f(0)<0$f(0)<0$. We discuss existence, non-existence, multiplicity and uniqueness results. We prove our existence results by the method of sub and supersolutions.