Ground state sign-changing solutions for Kirchhoff type problems in bounded domains

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• 作者：X.H. Tang^a ; ^{tangxh@csu.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Bitao Cheng^a ; ^b ; ^{chengbitao2006@126.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：35J20 ; 35J65
• 刊名：Journal of Differential Equations
• 出版年：2016
• 出版时间：15 August 2016
• 年：2016
• 卷：261
• 期：4
• 页码：2384-2402
• 全文大小：338 K

文摘

In the present paper, we consider the existence of ground state sign-changing solutions for a class of Kirchhoff-type problems where 616300663&_mathId=si2.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=a8874bb6ebb7d130d65668c5c4bd35e3" title="Click to view the MathML source">Ω⊂R^N$\mathrm{\Omega }\subset {\mathbb{R}}^N$ is a bounded domain with a smooth boundary ∂Ω, 616300663&_mathId=si3.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=739e27528faf3518b9cf38b3314324ae">616300663-si3.gif">$N=1,2,3$, 616300663&_mathId=si4.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=9349eba5f3194466bcbe493ed6d96e31" title="Click to view the MathML source">a>0$a>0$, 616300663&_mathId=si5.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=2014a505c067ae5cef583274dde04fed" title="Click to view the MathML source">b>0$b>0$ and 616300663&_mathId=si6.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=aeac26397355b17c3b28c0747196c409">616300663-si6.gif">$f\in C(\mathbb{R},\mathbb{R})$. Under some weak assumptions on f, with the aid of some new analytical skills and Non-Nehari manifold method, we prove that (0.1) possesses one ground state sign-changing solution 616300663&_mathId=si7.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=906d47e197e86b6da131b0b83a07970a" title="Click to view the MathML source">u_b$u_b$, and its energy is strictly larger than twice that of the ground state solutions of Nehari-type. Furthermore, we establish the convergence property of 616300663&_mathId=si7.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=906d47e197e86b6da131b0b83a07970a" title="Click to view the MathML source">u_b$u_b$ as the parameter 616300663&_mathId=si75.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=db7941552617b41841f5aeb65556fdd8" title="Click to view the MathML source">b↘0$b\searrow 0$. Our results improve and generalize some results obtained by W. Shuai (2015) [34].