Some results for a class of quasilinear elliptic equations with singular nonlinearity

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• 作者：Jo&atilde ; o Marcos do Ó ; ^a ; ^{jmbo@pq.cnpq.br" class="auth_mail" title="E-mail the corresponding author} ; Esteban da Silva^b ; ^{esteban@dmat.ufpe.br" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35J62 ; 35J70 ; 34B15 ; 35B35
• 刊名：Nonlinear Analysis
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：148
• 期：Complete
• 页码：1-29
• 全文大小：750 K

文摘

In this paper, motivated by recent works on the study of the equations which model the electrostatic MEMS devices, we study the quasilinear elliptic equation involving a singular nonlinearity <div class="formula" id="fd000005"><div class="mathml">d="mmlsi1" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302231&_mathId=si1.gif&_user=111111111&_pii=S0362546X16302231&_rdoc=1&_issn=0362546X&md5=534bc74ad2e6424d0a10c0a38dcf3e03">dth="282" alt="View the MathML source" title="View the MathML source" src="/sd/grey_pxl.gif" data-inlimgeid="1-s2.0-S0362546X16302231-si1.gif">dden">de">mi>rmi><mi>αmi>|<mi>umi>′(<mi>rmi>)|<mi>βmi><mi>umi>′(<mi>rmi>))′=<mi>λmi><mi>rmi><mi>γmi><mi>fmi>(<mi>rmi>)(1&minus;<mi>umi>(<mi>rmi>))2,d>d><mi>rmi>∈(0,1),d>d>0≤<mi>umi>(<mi>rmi>)<1,d>d><mi>rmi>∈(0,1),d>d><mi>umi>′(0)=<mi>umi>(1)=0.d>d>d>d/blank.gif">div>div> According to the choice of the parameters d="mmlsi2" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302231&_mathId=si2.gif&_user=111111111&_pii=S0362546X16302231&_rdoc=1&_issn=0362546X&md5=2a647da511b31f17216ab4b8527a76e5">dth="28" 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-S0362546X16302231-si2.gif">dden">de">mi>αmi>,<mi>βmi> and d="mmlsi16" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302231&_mathId=si16.gif&_user=111111111&_pii=S0362546X16302231&_rdoc=1&_issn=0362546X&md5=52453dd584fc57c6c0e2e1acdbf4c12f" title="Click to view the MathML source">γdden">de">mi>γmi>, the differential operator which we are dealing with corresponds to the radial form of the Laplacian, the d="mmlsi4" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302231&_mathId=si4.gif&_user=111111111&_pii=S0362546X16302231&_rdoc=1&_issn=0362546X&md5=c09f5b29c4e6f78c33813412438553dc" title="Click to view the MathML source">pdden">de">mi>pmi>-Laplacian and the d="mmlsi5" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302231&_mathId=si5.gif&_user=111111111&_pii=S0362546X16302231&_rdoc=1&_issn=0362546X&md5=add9ec4d8ea092b2e8f4a43876351d08" title="Click to view the MathML source">kdden">de">mi>kmi>-Hessian. In this work we present conditions over which we can assert regularity for solutions, including the case d="mmlsi6" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302231&_mathId=si6.gif&_user=111111111&_pii=S0362546X16302231&_rdoc=1&_issn=0362546X&md5=7a01a57538cae36b4114b685530c9046" title="Click to view the MathML source">λ=λ^∗dden">de">mi>λmi>=<mi>λmi>∗, where d="mmlsi7" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302231&_mathId=si7.gif&_user=111111111&_pii=S0362546X16302231&_rdoc=1&_issn=0362546X&md5=282c3631c26e9063ad91473470115426" title="Click to view the MathML source">λ^∗dden">de">mi>λmi>∗ is a critical value for the existence of solutions. Moreover, we prove that whenever the critical solution is regular, there exists another solution of mountain pass type for d="mmlsi8" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302231&_mathId=si8.gif&_user=111111111&_pii=S0362546X16302231&_rdoc=1&_issn=0362546X&md5=972b82e00a011dd1e0c765ec6c94e6ae" title="Click to view the MathML source">λdden">de">mi>λmi> close to the critical one. In addition, we use the Shooting Method to prove uniqueness of solutions for d="mmlsi8" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302231&_mathId=si8.gif&_user=111111111&_pii=S0362546X16302231&_rdoc=1&_issn=0362546X&md5=972b82e00a011dd1e0c765ec6c94e6ae" title="Click to view the MathML source">λdden">de">mi>λmi> in a neighborhood of d="mmlsi10" class="mathmlsrc">data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0362546X16302231&_mathId=si10.gif&_user=111111111&_pii=S0362546X16302231&_rdoc=1&_issn=0362546X&md5=36c85ea3c459bb31244bdbb14b1c6060" title="Click to view the MathML source">0dden">de">$0$.