{"title":"无渐近条件的Kirchhoff型问题正解的多重性","authors":"X. Qian","doi":"10.12775/tmna.2022.031","DOIUrl":null,"url":null,"abstract":"In this paper, we are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem\n\\[\n\\begin{cases}\n-\\bigg({\\varepsilon}^2a+{\\varepsilon}b\\int_{\\mathbb{R}^3} |\\n u|^2dx\\bigg)\\Delta u+u=Q(x)|u|^{p-2}u, & x\\in\\mathbb{R}^3,\\\\\nu\\in H^1\\big(\\mathbb{R}^3\\big), \\quad u> 0, & x\\in\\mathbb{R}^3,\n\\end{cases}\n\\]\nwhere $\\varepsilon> 0$ is a small parameter, $a,b> 0$ are constants, $4< p< 6$, $Q$\n is a nonnegative continuous potential and does not satisfy any asymptotic condition.\n Combining Nehari manifold and concentration compactness principle, we study how the shape of the graph of $Q(x)$ affects the number of positive solutions.","PeriodicalId":23130,"journal":{"name":"Topological Methods in Nonlinear Analysis","volume":" ","pages":""},"PeriodicalIF":0.7000,"publicationDate":"2023-06-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Multiplicity of positive solutions for a Kirchhoff type problem without asymptotic conditions\",\"authors\":\"X. Qian\",\"doi\":\"10.12775/tmna.2022.031\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"In this paper, we are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem\\n\\\\[\\n\\\\begin{cases}\\n-\\\\bigg({\\\\varepsilon}^2a+{\\\\varepsilon}b\\\\int_{\\\\mathbb{R}^3} |\\\\n u|^2dx\\\\bigg)\\\\Delta u+u=Q(x)|u|^{p-2}u, & x\\\\in\\\\mathbb{R}^3,\\\\\\\\\\nu\\\\in H^1\\\\big(\\\\mathbb{R}^3\\\\big), \\\\quad u> 0, & x\\\\in\\\\mathbb{R}^3,\\n\\\\end{cases}\\n\\\\]\\nwhere $\\\\varepsilon> 0$ is a small parameter, $a,b> 0$ are constants, $4< p< 6$, $Q$\\n is a nonnegative continuous potential and does not satisfy any asymptotic condition.\\n Combining Nehari manifold and concentration compactness principle, we study how the shape of the graph of $Q(x)$ affects the number of positive solutions.\",\"PeriodicalId\":23130,\"journal\":{\"name\":\"Topological Methods in Nonlinear Analysis\",\"volume\":\" \",\"pages\":\"\"},\"PeriodicalIF\":0.7000,\"publicationDate\":\"2023-06-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Topological Methods in Nonlinear Analysis\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.12775/tmna.2022.031\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Topological Methods in Nonlinear Analysis","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.12775/tmna.2022.031","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
Multiplicity of positive solutions for a Kirchhoff type problem without asymptotic conditions
In this paper, we are concerned with the multiplicity of positive solutions for the following Kirchhoff type problem
\[
\begin{cases}
-\bigg({\varepsilon}^2a+{\varepsilon}b\int_{\mathbb{R}^3} |\n u|^2dx\bigg)\Delta u+u=Q(x)|u|^{p-2}u, & x\in\mathbb{R}^3,\\
u\in H^1\big(\mathbb{R}^3\big), \quad u> 0, & x\in\mathbb{R}^3,
\end{cases}
\]
where $\varepsilon> 0$ is a small parameter, $a,b> 0$ are constants, $4< p< 6$, $Q$
is a nonnegative continuous potential and does not satisfy any asymptotic condition.
Combining Nehari manifold and concentration compactness principle, we study how the shape of the graph of $Q(x)$ affects the number of positive solutions.
期刊介绍:
Topological Methods in Nonlinear Analysis (TMNA) publishes research and survey papers on a wide range of nonlinear analysis, giving preference to those that employ topological methods. Papers in topology that are of interest in the treatment of nonlinear problems may also be included.