Existence of solutions for a bi-nonlocal fractional -Kirchhoff type problem

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• 作者：Mingqi Xiang^a ; ^{xiangmingqi_hit@163.com" class="auth_mail" title="E-mail the corresponding author} ; Binlin Zhang^b ; ^{zhangbinlin2012@163.com" class="auth_mail" title="E-mail the corresponding author} ; Vicenţiu D. Rădulescu^c ; ^d ; ^{vicentiu.radulescu@imar.ro" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Fractional pp-Kirchhoff equation ; Integro-differential operator ; Mountain Pass Theorem ; Iterative method
• 刊名：Computers & Mathematics with Applications
• 出版年：2016
• 出版时间：January 2016
• 年：2016
• 卷：71
• 期：1
• 页码：255-266
• 全文大小：405 K

文摘

In this paper, we are concerned with the existence of nonnegative solutions for a pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089812211500560X&_mathId=si1.gif&_user=111111111&_pii=S089812211500560X&_rdoc=1&_issn=08981221&md5=2a41752479b1b9e1707560a10df81bb7" title="Click to view the MathML source">p$p$-Kirchhoff type problem driven by a non-local integro-differential operator with homogeneous Dirichlet boundary data. As a particular case, we study the following problem where ${(-\Delta )}_p^s$ is a fractional pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089812211500560X&_mathId=si1.gif&_user=111111111&_pii=S089812211500560X&_rdoc=1&_issn=08981221&md5=2a41752479b1b9e1707560a10df81bb7" title="Click to view the MathML source">p$p$-Laplace operator, pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089812211500560X&_mathId=si6.gif&_user=111111111&_pii=S089812211500560X&_rdoc=1&_issn=08981221&md5=bded4382214a4f41eafc0f8fb4b31f3a" title="Click to view the MathML source">Ω$\Omega$ is an open bounded subset of pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089812211500560X&_mathId=si7.gif&_user=111111111&_pii=S089812211500560X&_rdoc=1&_issn=08981221&md5=ddb6804023df14bbefac2dbf63ab3c8f" title="Click to view the MathML source">R^N${\mathbb{R}}^N$ with Lipschitz boundary, $M:\Omega \times {\mathbb{R}}_0^{+}\to {\mathbb{R}}^{+}$ is a continuous function and $f:\Omega \times \mathbb{R}\times {\mathbb{R}}_0^{+}\to \mathbb{R}$ is a continuous function satisfying the Ambrosetti–Rabinowitz type condition. The existence of nonnegative solutions is obtained by using the Mountain Pass Theorem and an iterative scheme. The main feature of this paper lies in the fact that the Kirchhoff function pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089812211500560X&_mathId=si10.gif&_user=111111111&_pii=S089812211500560X&_rdoc=1&_issn=08981221&md5=d1a6b4fd82315c66a7df026e7951dd62" title="Click to view the MathML source">M$M$ depends on pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089812211500560X&_mathId=si11.gif&_user=111111111&_pii=S089812211500560X&_rdoc=1&_issn=08981221&md5=eae27f853e81016a498ff365d440ef23" title="Click to view the MathML source">x∈Ω$x\in \Omega$ and the nonlinearity pport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S089812211500560X&_mathId=si12.gif&_user=111111111&_pii=S089812211500560X&_rdoc=1&_issn=08981221&md5=b61d1151d2128a78005ec6aadfcb785c" title="Click to view the MathML source">f$f$ depends on the energy of solutions.