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 an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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(|&nabla;z|^p−2&nabla;z)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">row>athvariant="normal">Δrow>row>prow>z:=divalse">(alse">|athvariant="normal">&nabla;zrow>alse">|row>row>p−2row>athvariant="normal">&nabla;zalse">)ath>an>an>an>, an id="mmlsi3" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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<nan>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">1<p<nath>an>an>an>, λ   is a positive parameter, an id="mmlsi4" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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>0an>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">row>rrow>row>0row>>0ath>an>an>an> and an id="mmlsi32" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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}an>an class="mathContainer hidden">an class="mathCode">ath altimg="si32.gif" overflow="scroll">row>athvariant="normal">Ωrow>row>Erow>:=alse">{x∈row>athvariant="double-struck">Rrow>row>nrow> alse">| alse">|xalse">|>row>rrow>row>0row>alse">}ath>an>an>an>. Here the weight function an id="mmlsi34" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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,∞)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si34.gif" overflow="scroll">K∈row>Crow>row>1row>alse">[row>rrow>row>0row>,∞alse">)ath>an>an>an> satisfies an id="mmlsi7" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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)>0an>an class="mathContainer hidden">an class="mathCode">ath altimg="si7.gif" overflow="scroll">Kalse">(ralse">)>0ath>an>an>an> for an id="mmlsi8" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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≥r₀an>an class="mathContainer hidden">an class="mathCode">ath altimg="si8.gif" overflow="scroll">r≥row>rrow>row>0row>ath>an>an>an>, an id="mmlsi9" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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&rarr;∞⁡K(r)=0an>an class="mathContainer hidden">an class="mathCode">ath altimg="si9.gif" overflow="scroll">row>athvariant="normal">limrow>row>ralse">&rarr;∞row>⁡Kalse">(ralse">)=0ath>an>an>an>, and the reaction term an id="mmlsi10" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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,∞)&cap;C¹(0,∞)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si10.gif" overflow="scroll">f∈Calse">[0,∞alse">)&cap;row>Crow>row>1row>alse">(0,∞alse">)ath>an>an>an> is strictly increasing and satisfies an id="mmlsi11" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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)<0an>an class="mathContainer hidden">an class="mathCode">ath altimg="si11.gif" overflow="scroll">false">(0alse">)<0ath>an>an>an> (semipositone), an id="mmlsi12" class="mathmlsrc"><a title="View 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">ass="imgLazyJSB inlineImage" height="18" width="170" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303535-si12.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si12.gif" overflow="scroll">row>row>athvariant="normal">limrow>ace width="0.2em">ace>row>athvariant="normal">suprow>row>row>salse">&rarr;0+row>ace width="0.2em">ace>srow>frow>row>′row>alse">(salse">)<∞ath>an>an>an>, an id="mmlsi13" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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&rarr;∞⁡f(s)=∞an>an class="mathContainer hidden">an class="mathCode">ath altimg="si13.gif" overflow="scroll">row>athvariant="normal">limrow>row>salse">&rarr;∞row>⁡false">(salse">)=∞ath>an>an>an>, an id="mmlsi14" class="mathmlsrc"><a title="View 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">ass="imgLazyJSB inlineImage" height="22" width="120" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303535-si14.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si14.gif" overflow="scroll">row>athvariant="normal">limrow>row>salse">&rarr;∞row>⁡ac>row>false">(salse">)row>row>srow>row>p−1row>ac>=0ath>an>an>an> and an id="mmlsi15" class="mathmlsrc"><a title="View 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">ass="imgLazyJSB inlineImage" height="22" width="27" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0022247X16303535-si15.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si15.gif" overflow="scroll">ac>row>false">(salse">)row>row>srow>row>qrow>ac>ath>an>an>an> is nonincreasing on an id="mmlsi16" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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,∞)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si16.gif" overflow="scroll">alse">[a,∞alse">)ath>an>an>an> for some an id="mmlsi17" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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>0an>an class="mathContainer hidden">an class="mathCode">ath altimg="si17.gif" overflow="scroll">a>0ath>an>an>an> and an id="mmlsi18" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si18.gif" overflow="scroll">q∈alse">(0,p−1alse">)ath>an>an>an>. 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 an id="mmlsi19" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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≫1an>an class="mathContainer hidden">an class="mathCode">ath altimg="si19.gif" overflow="scroll">λ≫1ath>an>an>an>. We establish the uniqueness of this positive radial solution for an id="mmlsi19" class="mathmlsrc">an class="formulatext stixSupport mathImg" 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≫1an>an class="mathContainer hidden">an class="mathCode">ath altimg="si19.gif" overflow="scroll">λ≫1ath>an>an>an>.