A uniqueness result for a semipositone p-Laplacian problem on the exterior of a ball

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• 作者：R. Shivaji^a ; ^{shivaji@uncg.edu" class="auth_mail" title="E-mail the corresponding author} ; Inbo Sim^b ; ^{ibsim@ulsan.ac.kr" class="auth_mail" title="E-mail the corresponding author} ; Byungjae Son^a ; ^{b_son@uncg.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Uniqueness results ; Semipositone problems ; p-Laplacian ; Exterior domains ; Positive radial solutions
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2017
• 出版时间：1 January 2017
• 年：2017
• 卷：445
• 期：1
• 页码：459-475
• 全文大小：390 K

文摘

We consider steady state reaction diffusion equations on the exterior of a ball, namely, boundary value problems of the form: where Δ_pz:=div(|∇z|^p−2∇z)${\mathrm{\Delta }}_{p}z:=\text{div}(|\mathrm{\nabla }z{|}^{p-2}\mathrm{\nabla }z)$, e5a4a8f8f619e47d760eb" title="Click to view the MathML source">1<p<n$1<p<n$, λ &#xA0; is a positive parameter, r₀>0$r_0>0$ and bcd8320f8" title="Click to view the MathML source">Ω_E:={x∈Rⁿ | |x|>r₀}${\mathrm{\Omega }}_E:=\{x\in {\mathbb{R}}^n||x|>r_0\}$. Here the weight function a0bfa1b2c54103e" title="Click to view the MathML source">K∈C¹[r₀,∞)$K\in C^1[r_0,\infty )$ satisfies K(r)>0$K(r)>0$ for bc477a26e4e7bb76170b3b326cb2a7" title="Click to view the MathML source">r≥r₀$r\ge r_0$, lim_r→∞⁡K(r)=0${\lim }_{r\to \infty }K(r)=0$, and the reaction term f∈C[0,∞)∩C¹(0,∞)$f\in C[0,\infty )\cap C^1(0,\infty )$ is strictly increasing and satisfies e9c4dae6a3f3497ab947229" title="Click to view the MathML source">f(0)<0$f(0)<0$ (semipositone), bc77500934873d946bd58fe18">${\lim \sup }_{s\to 0+}s{f}^{\prime }(s)<\infty$, lim_s→∞⁡f(s)=∞${\lim }_{s\to \infty }f(s)=\infty$, e95f6cd2">${\lim }_{s\to \infty }\frac{f(s)}{s^{p-1}}=0$ and bc7ad01fc07568a7">$\frac{f(s)}{s^q}$ is nonincreasing on [a,∞)$[a,\infty )$ for some e9b" title="Click to view the MathML source">a>0$a>0$ and q∈(0,p−1)$q\in (0,p-1)$. For a class of such steady state equations it turns out that every nonnegative radial solution is strictly positive in the exterior of a ball, and exists for λ≫1$\lambda \gg 1$. We establish the uniqueness of this positive radial solution for λ≫1$\lambda \gg 1$.