Standing waves with a critical frequency for nonlinear Schrödinger equations involving critical growth

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• 作者：Jianjun Zhang ; ^{jianjun.zhang@uninsubria.it" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Critical frequency ; Truncation approach ; Critical growth ; Ambrosetti open problem
• 刊名：Applied Mathematics Letters
• 出版年：2017
• 出版时间：January 2017
• 年：2017
• 卷：63
• 期：Complete
• 页码：53-58
• 全文大小：635 K

文摘

We consider the following singularly perturbed Schrödinger equation where mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302063&_mathId=si2.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=a9acfab3bd89e367be224e0de1709a5c" title="Click to view the MathML source">N≥3mathContainer hidden">mathCode"><math altimg="si2.gif" overflow="scroll">N≥3math>, mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302063&_mathId=si3.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=834294a5c6f55cce50b4927a3be4f7d3" title="Click to view the MathML source">VmathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll">Vmath> is a nonnegative continuous potential and the nonlinear term mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302063&_mathId=si4.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=820b20afa69e05ef7d7bffdfa48bb3b0" title="Click to view the MathML source">fmathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll">fmath> is of critical growth. In this paper, with the help of a truncation approach, we prove that if mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302063&_mathId=si3.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=834294a5c6f55cce50b4927a3be4f7d3" title="Click to view the MathML source">VmathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll">Vmath> has a positive local minimum, then for small mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302063&_mathId=si6.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=62c9239737b4e084fdfe6076479ecdb7" title="Click to view the MathML source">εmathContainer hidden">mathCode"><math altimg="si6.gif" overflow="scroll">εmath> the problem admits positive solutions which concentrate at an isolated component of positive local minimum points of mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302063&_mathId=si3.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=834294a5c6f55cce50b4927a3be4f7d3" title="Click to view the MathML source">VmathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll">Vmath> as mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302063&_mathId=si8.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=3915f0fbfb7ed26169b9cc7325d86bb9" title="Click to view the MathML source">ε→0mathContainer hidden">mathCode"><math altimg="si8.gif" overflow="scroll">ε→0math>. In particular, the potential mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302063&_mathId=si3.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=834294a5c6f55cce50b4927a3be4f7d3" title="Click to view the MathML source">VmathContainer hidden">mathCode"><math altimg="si3.gif" overflow="scroll">Vmath> is allowed to be either compactly supported   or decay faster than mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302063&_mathId=si10.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=6d8bc0aa5724428b90ccd5ed9e272027" title="Click to view the MathML source">∣x∣⁻²mathContainer hidden">mathCode"><math altimg="si10.gif" overflow="scroll">∣x∣−2math> at infinity. Moreover, a general nonlinearity mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302063&_mathId=si4.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=820b20afa69e05ef7d7bffdfa48bb3b0" title="Click to view the MathML source">fmathContainer hidden">mathCode"><math altimg="si4.gif" overflow="scroll">fmath> is involved, i.e., the monotonicity   of mathmlsrc">mathImg" data-mathURL="/science?_ob=MathURL&_method=retrieve&_eid=1-s2.0-S0893965916302063&_mathId=si12.gif&_user=111111111&_pii=S0893965916302063&_rdoc=1&_issn=08939659&md5=263be07dc382d8710e221ee5f9c9677d" title="Click to view the MathML source">f(s)/smathContainer hidden">mathCode"><math altimg="si12.gif" overflow="scroll">f(s)/smath> and the Ambrosetti–Rabinowitz condition are not required.