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 mmlsi2" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616300663&_mathId=si2.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=a8874bb6ebb7d130d65668c5c4bd35e3" title="Click to view the MathML source">Ω⊂R^NmathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll"><mi mathvariant="normal">Ωmi><mo>⊂mo><msup><mrow><mi mathvariant="double-struck">Rmi>mrow><mrow><mi>Nmi>mrow>msup>math> is a bounded domain with a smooth boundary ∂Ω, mmlsi3" class="mathmlsrc">w the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616300663&_mathId=si3.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=739e27528faf3518b9cf38b3314324ae">mg class="imgLazyJSB inlineImage" height="14" width="77" 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-S0022039616300663-si3.gif">mathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll"><mi>Nmi><mo>=mo><mn>1mn><mo>,mo><mspace width="0.2em">mspace><mn>2mn><mo>,mo><mspace width="0.2em">mspace><mn>3mn>math>, mmlsi4" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616300663&_mathId=si4.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=9349eba5f3194466bcbe493ed6d96e31" title="Click to view the MathML source">a>0mathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll"><mi>ami><mo>>mo><mn>0mn>math>, mmlsi5" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616300663&_mathId=si5.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=2014a505c067ae5cef583274dde04fed" title="Click to view the MathML source">b>0mathContainer hidden">mathCode"><math altimg="si5.gif" overflow="scroll"><mi>bmi><mo>>mo><mn>0mn>math> and mmlsi6" class="mathmlsrc">w the MathML source" class="mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616300663&_mathId=si6.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=aeac26397355b17c3b28c0747196c409">mg class="imgLazyJSB inlineImage" height="15" width="85" 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-S0022039616300663-si6.gif">mathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll"><mi>fmi><mo>∈mo><mi>Cmi><mo stretchy="false">(mo><mi mathvariant="double-struck">Rmi><mo>,mo><mspace width="0.2em">mspace><mi mathvariant="double-struck">Rmi><mo stretchy="false">)mo>math>. Under some weak assumptions on m>fm>, with the aid of some new analytical skills and Non-Nehari manifold method, we prove that m0010">(0.1) possesses one ground state sign-changing solution mmlsi7" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616300663&_mathId=si7.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=906d47e197e86b6da131b0b83a07970a" title="Click to view the MathML source">u_bmathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll"><msub><mrow><mi>umi>mrow><mrow><mi>bmi>mrow>msub>math>, and its energy is strictly larger than twice that of the ground state solutions of Nehari-type. Furthermore, we establish the convergence property of mmlsi7" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616300663&_mathId=si7.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=906d47e197e86b6da131b0b83a07970a" title="Click to view the MathML source">u_bmathContainer hidden">mathCode"><math altimg="si7.gif" overflow="scroll"><msub><mrow><mi>umi>mrow><mrow><mi>bmi>mrow>msub>math> as the parameter mmlsi75" class="mathmlsrc">mulatext stixSupport mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0022039616300663&_mathId=si75.gif&_user=111111111&_pii=S0022039616300663&_rdoc=1&_issn=00220396&md5=db7941552617b41841f5aeb65556fdd8" title="Click to view the MathML source">b↘0mathContainer hidden">mathCode"><math altimg="si75.gif" overflow="scroll"><mi>bmi><mo stretchy="false">↘mo><mn>0mn>math>. Our results improve and generalize some results obtained by W. Shuai (2015) [34].