Existence of positive radial solutions for superlinear, semipositone problems on the exterior of a ball

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• 作者：R. Dhanya^a ; ^{dhanya.tr@gmail.com" class="auth_mail" title="E-mail the corresponding author} ; Q. Morris^b ; ^{qamorris@uncg.edu" class="auth_mail" title="E-mail the corresponding author} ; R. Shivaji^b ; ¹ ; ^{r_shivaji@uncg.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Existence ; Superlinear semipositone problems ; Nonlinear and Dirichlet boundary conditions ; Exterior domain
• 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：1533-1548
• 全文大小：382 K

文摘

We study positive radial solutions to −Δu=λK(|x|)f(u)$-\mathrm{\Delta }u=\lambda K(|x|)f(u)$; x∈Ω_e$x\in {\mathrm{\Omega }}_e$ where λ>0$\lambda >0$ is a parameter, Ω_e={x∈R^N‖x|>r₀,r₀>0,N>2}${\mathrm{\Omega }}_e=\{x\in {\mathbb{R}}^N\Vert x|>r_0,r_0>0,N>2\}$, Δ is the Laplacian operator, K∈C([r₀,∞),(0,∞))$K\in C([r_0,\infty ),(0,\infty ))$ satisfies $K(r)\le \frac{1}{r^{N+\mu }};\mu >0$ for r>>1$r>>1$, and f∈C¹([0,∞),R)$f\in C^1([0,\infty ),\mathbb{R})$ is a class of non-decreasing functions satisfying ${\lim }_{s\to \infty }\frac{f(s)}{s}=\infty$ (superlinear) and f(0)<0$f(0)<0$ (semipositone). We consider solutions, u  , such that u→0$u\to 0$ as |x|→∞$|x|\to \infty$, and which also satisfy the nonlinear boundary condition $\frac{\partial u}{\partial \eta }+\tilde{c}(u)u=0$ when |x|=r₀$|x|=r_0$, where $\frac{\partial }{\partial \eta }$ is the outward normal derivative, and $\tilde{c}\in C([0,\infty ),(0,\infty ))$. We will establish the existence of a positive radial solution for small values of the parameter λ. We also establish a similar result for the case when u   satisfies the Dirichlet boundary condition (u=0)$(u=0)$ for |x|=r₀$|x|=r_0$. We establish our results via variational methods, namely using the Mountain Pass Lemma.