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 thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si2.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=1ea29e86e6da6c268a9789e4eb4e64b8" title="Click to view the MathML source">Δ_pz:=div(|∇z|^p−2∇z)thContainer hidden">thCode">th altimg="si2.gif" overflow="scroll">thvariant="normal">Δpz:=div(|thvariant="normal">∇z|p−2thvariant="normal">∇z)th>, thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si3.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=92874edfe89e5a4a8f8f619e47d760eb" title="Click to view the MathML source">1<p<nthContainer hidden">thCode">th altimg="si3.gif" overflow="scroll">1<p<nth>, λ   is a positive parameter, thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si4.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=68ed7b76aad25e0a2c735f8841b469a4" title="Click to view the MathML source">r₀>0thContainer hidden">thCode">th altimg="si4.gif" overflow="scroll">r0>0th> and thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si32.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=a92e08d603c87c379d56a30bcd8320f8" title="Click to view the MathML source">Ω_E:={x∈Rⁿ | |x|>r₀}thContainer hidden">thCode">th altimg="si32.gif" overflow="scroll">thvariant="normal">ΩE:={x∈thvariant="double-struck">Rn | |x|>r0}th>. Here the weight function thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si34.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=b30dd4c71b5b45379a0bfa1b2c54103e" title="Click to view the MathML source">K∈C¹[r₀,∞)thContainer hidden">thCode">th altimg="si34.gif" overflow="scroll">K∈C1[r0,∞)th> satisfies thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si7.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=139f937c8e741ece054603ce048d523b" title="Click to view the MathML source">K(r)>0thContainer hidden">thCode">th altimg="si7.gif" overflow="scroll">K(r)>0th> for thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si8.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=9fbc477a26e4e7bb76170b3b326cb2a7" title="Click to view the MathML source">r≥r₀thContainer hidden">thCode">th altimg="si8.gif" overflow="scroll">r≥r0th>, thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si9.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=2284f79017f1903d7a3f873dce73529f" title="Click to view the MathML source">lim_r→∞⁡K(r)=0thContainer hidden">thCode">th altimg="si9.gif" overflow="scroll">thvariant="normal">limr→∞⁡K(r)=0th>, and the reaction term thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si10.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=46cf1ad9b16ec0f7ac0b7dd83dfae450" title="Click to view the MathML source">f∈C[0,∞)∩C¹(0,∞)thContainer hidden">thCode">th altimg="si10.gif" overflow="scroll">f∈C[0,∞)∩C1(0,∞)th> is strictly increasing and satisfies thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si11.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=6f41e8cf1e9c4dae6a3f3497ab947229" title="Click to view the MathML source">f(0)<0thContainer hidden">thCode">th altimg="si11.gif" overflow="scroll">f(0)<0th> (semipositone), thmlsrc">the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si12.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=1bae8bdbc77500934873d946bd58fe18">th="170" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303535-si12.gif">thContainer hidden">thCode">th altimg="si12.gif" overflow="scroll">thvariant="normal">limth="0.2em">thvariant="normal">sups→0+th="0.2em">sf′(s)<∞th>, thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si13.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=2ceda53bd48c510d1b7389e722600904" title="Click to view the MathML source">lim_s→∞⁡f(s)=∞thContainer hidden">thCode">th altimg="si13.gif" overflow="scroll">thvariant="normal">lims→∞⁡f(s)=∞th>, thmlsrc">the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si14.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=d5f4bfc3f672426254e7940ee95f6cd2">th="120" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303535-si14.gif">thContainer hidden">thCode">th altimg="si14.gif" overflow="scroll">thvariant="normal">lims→∞⁡f(s)sp−1=0th> and thmlsrc">the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si15.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=f7ee135fde423808bc7ad01fc07568a7">th="27" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303535-si15.gif">thContainer hidden">thCode">th altimg="si15.gif" overflow="scroll">f(s)sqth> is nonincreasing on thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si16.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=9913f49afa626400eb66cbdec95e43bf" title="Click to view the MathML source">[a,∞)thContainer hidden">thCode">th altimg="si16.gif" overflow="scroll">[a,∞)th> for some thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si17.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=bf366bb1c45589078abf9ed957f85e9b" title="Click to view the MathML source">a>0thContainer hidden">thCode">th altimg="si17.gif" overflow="scroll">a>0th> and thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si18.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=2159a9e3cd7553221b843992323ff62a" title="Click to view the MathML source">q∈(0,p−1)thContainer hidden">thCode">th altimg="si18.gif" overflow="scroll">q∈(0,p−1)th>. 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 thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si19.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=b42884d51fa4f47191b7807ed63df861" title="Click to view the MathML source">λ≫1thContainer hidden">thCode">th altimg="si19.gif" overflow="scroll">λ≫1th>. We establish the uniqueness of this positive radial solution for thmlsrc">thImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022247X16303535&_mathId=si19.gif&_user=111111111&_pii=S0022247X16303535&_rdoc=1&_issn=0022247X&md5=b42884d51fa4f47191b7807ed63df861" title="Click to view the MathML source">λ≫1thContainer hidden">thCode">th altimg="si19.gif" overflow="scroll">λ≫1th>.