Pointwise estimates of Brezis-Kamin type for solutions of sublinear elliptic equations

详细信息    查看全文

• 作者：Dat T. Cao^a ; ^{dcao4@utk.edu" class="auth_mail" title="E-mail the corresponding author} ; Igor E. Verbitsky^b ; ^{verbitskyi@missouri.edu" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35B05 ; 35J92
• 刊名：Nonlinear Analysis
• 出版年：2016
• 出版时间：November 2016
• 年：2016
• 卷：146
• 期：Complete
• 页码：1-19
• 全文大小：732 K

文摘

We study quasilinear elliptic equations of the type where an id="mmlsi2" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301791&_mathId=si2.gif&_user=111111111&_pii=S0362546X16301791&_rdoc=1&_issn=0362546X&md5=45907af3af7f9dbef4480d303f3cc499" title="Click to view the MathML source">Δ_pu=&nabla;⋅(&nabla;u|&nabla;u|^p−2)an>an class="mathContainer hidden">an class="mathCode">ath altimg="si2.gif" overflow="scroll">Δpu=&nabla;⋅(&nabla;u|&nabla;u|p−2)ath>an>an>an> is the an id="mmlsi3" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301791&_mathId=si3.gif&_user=111111111&_pii=S0362546X16301791&_rdoc=1&_issn=0362546X&md5=bc55ce8e740a2a02cbf7e6490e7b0ce6" title="Click to view the MathML source">pan>an class="mathContainer hidden">an class="mathCode">ath altimg="si3.gif" overflow="scroll">path>an>an>an>-Laplacian (or a more general an id="mmlsi4" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301791&_mathId=si4.gif&_user=111111111&_pii=S0362546X16301791&_rdoc=1&_issn=0362546X&md5=0a6cd3a2b6d7b38b0370596c8e403819" title="Click to view the MathML source">Aan>an class="mathContainer hidden">an class="mathCode">ath altimg="si4.gif" overflow="scroll">athvariant="script">Aath>an>an>an>-Laplace operator an id="mmlsi5" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301791&_mathId=si5.gif&_user=111111111&_pii=S0362546X16301791&_rdoc=1&_issn=0362546X&md5=f026e42ec39c787f0aa0ca969e3ad9ec">ass="imgLazyJSB inlineImage" height="13" width="75" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X16301791-si5.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si5.gif" overflow="scroll">divace width="0.16667em">ace>athvariant="script">A(x,&nabla;u)ath>an>an>an>), an id="mmlsi6" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301791&_mathId=si6.gif&_user=111111111&_pii=S0362546X16301791&_rdoc=1&_issn=0362546X&md5=40894a616201244c2b1eb07dec0d533b" title="Click to view the MathML source">0<q<p−1an>an class="mathContainer hidden">an class="mathCode">ath altimg="si6.gif" overflow="scroll">0<q<p−1ath>an>an>an>, and an id="mmlsi7" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301791&_mathId=si7.gif&_user=111111111&_pii=S0362546X16301791&_rdoc=1&_issn=0362546X&md5=acca5d56fcf517c7ce0c10bcf372c529" title="Click to view the MathML source">σ≥0an>an class="mathContainer hidden">an class="mathCode">ath altimg="si7.gif" overflow="scroll">σ≥0ath>an>an>an> is an arbitrary locally integrable function or measure on an id="mmlsi8" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301791&_mathId=si8.gif&_user=111111111&_pii=S0362546X16301791&_rdoc=1&_issn=0362546X&md5=8a9dd6eed7e7399299163ceff14161ae" title="Click to view the MathML source">Rⁿan>an class="mathContainer hidden">an class="mathCode">ath altimg="si8.gif" overflow="scroll">athvariant="double-struck">Rnath>an>an>an>.

We obtain necessary and sufficient conditions for the existence of positive solutions (not necessarily bounded) which satisfy global pointwise estimates of Brezis&ndash;Kamin type given in terms of Wolff potentials. Similar problems with the fractional Laplacian an id="mmlsi9" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301791&_mathId=si9.gif&_user=111111111&_pii=S0362546X16301791&_rdoc=1&_issn=0362546X&md5=d120aaa5d614b30f49e7574214fe3fde" title="Click to view the MathML source">(−Δ)^αan>an class="mathContainer hidden">an class="mathCode">ath altimg="si9.gif" overflow="scroll">(−Δ)αath>an>an>an> for an id="mmlsi10" class="mathmlsrc"><a title="View the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301791&_mathId=si10.gif&_user=111111111&_pii=S0362546X16301791&_rdoc=1&_issn=0362546X&md5=5fda0416747c0f343dfb37985a5bad12">ass="imgLazyJSB inlineImage" height="15" width="62" alt="View the MathML source" style="margin-top: -5px; vertical-align: middle" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X16301791-si10.gif">a>an class="mathContainer hidden">an class="mathCode">ath altimg="si10.gif" overflow="scroll">0<α<ac>n2ac>ath>an>an>an> are treated as well, including explicit estimates for radially symmetric an id="mmlsi11" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301791&_mathId=si11.gif&_user=111111111&_pii=S0362546X16301791&_rdoc=1&_issn=0362546X&md5=1d7ee35d1aba12f5a8ae29886dd9c72b" title="Click to view the MathML source">σan>an class="mathContainer hidden">an class="mathCode">ath altimg="si11.gif" overflow="scroll">σath>an>an>an>. Our results are new even in the classical case an id="mmlsi12" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301791&_mathId=si12.gif&_user=111111111&_pii=S0362546X16301791&_rdoc=1&_issn=0362546X&md5=81146ea6baad7f3b03feca6c4a818806" title="Click to view the MathML source">p=2an>an class="mathContainer hidden">an class="mathCode">ath altimg="si12.gif" overflow="scroll">p=2ath>an>an>an> and an id="mmlsi13" class="mathmlsrc">an class="formulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16301791&_mathId=si13.gif&_user=111111111&_pii=S0362546X16301791&_rdoc=1&_issn=0362546X&md5=034fd6aa819647671220ddaf8e9a1edb" title="Click to view the MathML source">α=1an>an class="mathContainer hidden">an class="mathCode">ath altimg="si13.gif" overflow="scroll">α=1ath>an>an>an>.