{"title":"Soliton solutions for a class of critical Schrödinger equations with Stein–Weiss convolution parts in $$\\mathbb {R}^2$$","authors":"Claudianor Oliveira Alves, Liejun Shen","doi":"10.1007/s00605-024-01980-0","DOIUrl":null,"url":null,"abstract":"<p>We consider the following class of quasilinear Schrödinger equations introduced in plasma physics and nonlinear optics with Stein–Weiss convolution parts </p><span>$$\\begin{aligned} -\\Delta u+V(x) u+\\frac{\\kappa }{2} u\\Delta (u^2)=\\frac{1}{|x|^\\beta }\\Bigg (\\int _{\\mathbb {R}^2}\\frac{H(u)}{|x-y|^\\mu |y|^\\beta }dy\\Bigg ) h(u),~x\\in \\mathbb {R}^2, \\end{aligned}$$</span><p>where <span>\\(\\kappa \\in \\mathbb {R}\\backslash \\{0\\}\\)</span> is a parameter, <span>\\(\\beta >0\\)</span>, <span>\\(0<\\mu <2\\)</span> with <span>\\(0<2\\beta +\\mu <2\\)</span> and <i>H</i> is the primitive of <i>h</i> that fulfills the critical exponential growth in the Trudinger–Moser sense. For <span>\\(\\kappa <0\\)</span>: (i) via using a change of variable argument and the mountain-pass theorem, we investigate the existence of ground state solutions only assuming that <span>\\(V\\in C^0(\\mathbb {R}^2,\\mathbb {R}^+)\\)</span> and <span>\\(\\inf _{x \\in \\mathbb {R}^2}V(x)>0\\)</span>, which complements and generalizes the problems proposed in our recent work in Alves and Shen (J Differ Equ 344:352–404, 2023); (ii) by developing a new type of Trudinger–Moser inequality, we establish a Pohoz̆aev type ground solution by the constraint minimization approach when <span>\\(V\\equiv 1\\)</span>. Moreover, if <span>\\(\\kappa >0\\)</span> is small, combining the mountain-pass theorem and Nash–Moser iteration procedure, we obtain the existence of nontrivial solutions, where the asymptotical behavior is also considered when <span>\\(\\kappa \\rightarrow 0^+\\)</span>. It seems that the results presented above are even new for the case <span>\\(\\kappa =0\\)</span>.</p>","PeriodicalId":18913,"journal":{"name":"Monatshefte für Mathematik","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-05-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Monatshefte für Mathematik","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s00605-024-01980-0","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
We consider the following class of quasilinear Schrödinger equations introduced in plasma physics and nonlinear optics with Stein–Weiss convolution parts
where \(\kappa \in \mathbb {R}\backslash \{0\}\) is a parameter, \(\beta >0\), \(0<\mu <2\) with \(0<2\beta +\mu <2\) and H is the primitive of h that fulfills the critical exponential growth in the Trudinger–Moser sense. For \(\kappa <0\): (i) via using a change of variable argument and the mountain-pass theorem, we investigate the existence of ground state solutions only assuming that \(V\in C^0(\mathbb {R}^2,\mathbb {R}^+)\) and \(\inf _{x \in \mathbb {R}^2}V(x)>0\), which complements and generalizes the problems proposed in our recent work in Alves and Shen (J Differ Equ 344:352–404, 2023); (ii) by developing a new type of Trudinger–Moser inequality, we establish a Pohoz̆aev type ground solution by the constraint minimization approach when \(V\equiv 1\). Moreover, if \(\kappa >0\) is small, combining the mountain-pass theorem and Nash–Moser iteration procedure, we obtain the existence of nontrivial solutions, where the asymptotical behavior is also considered when \(\kappa \rightarrow 0^+\). It seems that the results presented above are even new for the case \(\kappa =0\).