{"title":"A Multiplicity Result for a Fractional Kirchhoff Equation with a General Nonlinearity","authors":"V. Ambrosio","doi":"10.1007/978-3-030-60220-8_10","DOIUrl":"https://doi.org/10.1007/978-3-030-60220-8_10","url":null,"abstract":"","PeriodicalId":311016,"journal":{"name":"Nonlinear Fractional Schrödinger Equations in R^N","volume":"8 4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2020-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116814500","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Concentrating Solutions for a Fractional Kirchhoff Equation with Critical Growth","authors":"V. Ambrosio","doi":"10.3233/ASY-191543","DOIUrl":"https://doi.org/10.3233/ASY-191543","url":null,"abstract":"In this paper we consider the following class of fractional Kirchhoff equations with critical growth: begin{equation*} left{ begin{array}{ll} left(varepsilon^{2s}a+varepsilon^{4s-3}bint_{mathbb{R}^{3}}|(-Delta)^{frac{s}{2}}u|^{2}dxright)(-Delta)^{s}u+V(x)u=f(u)+|u|^{2^{*}_{s}-2}u quad &mbox{ in } mathbb{R}^{3}, uin H^{s}(mathbb{R}^{3}), quad u>0 &mbox{ in } mathbb{R}^{3}, end{array} right. end{equation*} where $varepsilon>0$ is a small parameter, $a, b>0$ are constants, $sin (frac{3}{4}, 1)$, $2^{*}_{s}=frac{6}{3-2s}$ is the fractional critical exponent, $(-Delta)^{s}$ is the fractional Laplacian operator, $V$ is a positive continuous potential and $f$ is a superlinear continuous function with subcritical growth. Using penalization techniques and variational methods, we prove the existence of a family of positive solutions $u_{varepsilon}$ which concentrates around a local minimum of $V$ as $varepsilonrightarrow 0$.","PeriodicalId":311016,"journal":{"name":"Nonlinear Fractional Schrödinger Equations in R^N","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2018-10-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121902742","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}