On the existence of positive solutions for an ecological model with indefinite weight

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• 作者：Saleh Shakeri^a ; ^{s.shakeri@iauamol.ac.ir" class="auth_mail" title="E-mail the corresponding author} ; Ghasem A. Afrouzi^b ; ^{afrouzi@umz.ac.ir" class="auth_mail" title="E-mail the corresponding author} ; Armin Hadjian^c ; ^{hadjian83@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Ecological systems ; Indefinite weight ; Grazing and constant yield harvesting ; Sub&ndash ; super solution method
• 刊名：Arab Journal of Mathematical Sciences
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：22
• 期：1
• 页码：132-137
• 全文大小：203 K

文摘

This study concerns the existence of positive solutions for the following nonlinear boundary value problem: where class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1319516615000067&_mathId=si2.gif&_user=111111111&_pii=S1319516615000067&_rdoc=1&_issn=13195166&md5=07d01382c1a49944bf3b7641a69fbbb9" title="Click to view the MathML source">Δu=div(∇u)class="mathContainer hidden">class="mathCode">$\Delta u=\text{div}\left(\nabla u\right)$ is the Laplacian of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1319516615000067&_mathId=si3.gif&_user=111111111&_pii=S1319516615000067&_rdoc=1&_issn=13195166&md5=9d0887ac25fca237ad15c1e7c6f682f6" title="Click to view the MathML source">uclass="mathContainer hidden">class="mathCode">$u$, while class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1319516615000067&_mathId=si4.gif&_user=111111111&_pii=S1319516615000067&_rdoc=1&_issn=13195166&md5=e4c69c81e49473d0197ce06167ef805c" title="Click to view the MathML source">a,b,c,p,Kclass="mathContainer hidden">class="mathCode">$a,b,c,p,K$ are positive constants with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1319516615000067&_mathId=si5.gif&_user=111111111&_pii=S1319516615000067&_rdoc=1&_issn=13195166&md5=d9d6ef04d65ab736f10bd3077aadca5c" title="Click to view the MathML source">p≥2class="mathContainer hidden">class="mathCode">$p\ge 2$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1319516615000067&_mathId=si6.gif&_user=111111111&_pii=S1319516615000067&_rdoc=1&_issn=13195166&md5=cf45253c6df3fd753cf2b7c7581ccf36" title="Click to view the MathML source">Ωclass="mathContainer hidden">class="mathCode">$\Omega$ is a bounded smooth domain of class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1319516615000067&_mathId=si7.gif&_user=111111111&_pii=S1319516615000067&_rdoc=1&_issn=13195166&md5=ef54d19f949bc377fad229403ba8ed6e" title="Click to view the MathML source">R^Nclass="mathContainer hidden">class="mathCode">${\mathbb{R}}^N$ with class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1319516615000067&_mathId=si8.gif&_user=111111111&_pii=S1319516615000067&_rdoc=1&_issn=13195166&md5=b936b4ebe5ddde765e81a53e6053bd7b" title="Click to view the MathML source">∂Ωclass="mathContainer hidden">class="mathCode">$\partial \Omega$ in class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1319516615000067&_mathId=si9.gif&_user=111111111&_pii=S1319516615000067&_rdoc=1&_issn=13195166&md5=2211eb123a56b495ccf68b2fd58370ba" title="Click to view the MathML source">C²class="mathContainer hidden">class="mathCode">$C^2$. The weight function class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1319516615000067&_mathId=si10.gif&_user=111111111&_pii=S1319516615000067&_rdoc=1&_issn=13195166&md5=4c715c1e33e4af5f97c80b8317311e3a" title="Click to view the MathML source">mclass="mathContainer hidden">class="mathCode">$m$ satisfies class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1319516615000067&_mathId=si11.gif&_user=111111111&_pii=S1319516615000067&_rdoc=1&_issn=13195166&md5=814237a10f3e8c6e3f931877dfbd535b" title="Click to view the MathML source">m∈C(Ω)class="mathContainer hidden">class="mathCode">$m\in C\left(\Omega \right)$ and class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1319516615000067&_mathId=si12.gif&_user=111111111&_pii=S1319516615000067&_rdoc=1&_issn=13195166&md5=b6c42dc161d1e29ea52e28fed7f5708a" title="Click to view the MathML source">m(x)≥m₀>0class="mathContainer hidden">class="mathCode">$m\left(x\right)\ge m_0>0$ for class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1319516615000067&_mathId=si13.gif&_user=111111111&_pii=S1319516615000067&_rdoc=1&_issn=13195166&md5=b8ab32c4323e29234b56941a7b548e41" title="Click to view the MathML source">x∈Ωclass="mathContainer hidden">class="mathCode">$x\in \Omega$, also class="mathmlsrc">class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S1319516615000067&_mathId=si14.gif&_user=111111111&_pii=S1319516615000067&_rdoc=1&_issn=13195166&md5=eed5f1af79eb4587093162de8fa0baec" title="Click to view the MathML source">‖m‖_∞=l<∞class="mathContainer hidden">class="mathCode">${\Vert m\Vert }_{\infty }=l<\infty$. We prove the existence of positive solutions under certain conditions.