{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,5,9]],"date-time":"2025-05-09T11:26:06Z","timestamp":1746789966977},"reference-count":28,"publisher":"American Institute of Mathematical Sciences (AIMS)","issue":"1","content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["MFC"],"published-print":{"date-parts":[[2022]]},"abstract":"<p style='text-indent:20px;'>In this work, we investigate a class of fractional Schr\u00f6dinger - Poisson systems<\/p><p style='text-indent:20px;'><disp-formula> <label\/> <tex-math id=\"FE1\"> \\begin{document}$ \\begin{equation*} \\left\\{\\begin{array}{ll}(-\\triangle)^s u +V(x)u+\\lambda\\phi u = \\mu u+|u|^{p-1}u, &amp; x\\in\\ \\mathbb{R}^3, \\\\(-\\triangle)^s \\phi = u^2, &amp; x\\in\\ \\mathbb{R}^3, \\end{array}\\right. \\end{equation*} $\\end{document} <\/tex-math><\/disp-formula><\/p><p style='text-indent:20px;'>where <inline-formula><tex-math id=\"M1\">\\begin{document}$ s\\in(\\frac{3}{4}, 1) $\\end{document}<\/tex-math><\/inline-formula>, <inline-formula><tex-math id=\"M2\">\\begin{document}$ p\\in(3, 5) $\\end{document}<\/tex-math><\/inline-formula>, <inline-formula><tex-math id=\"M3\">\\begin{document}$ \\lambda $\\end{document}<\/tex-math><\/inline-formula> is a positive parameter. By the variational method, we show that there exists <inline-formula><tex-math id=\"M4\">\\begin{document}$ \\delta(\\lambda)&gt;0 $\\end{document}<\/tex-math><\/inline-formula> such that for all <inline-formula><tex-math id=\"M5\">\\begin{document}$ \\mu\\in[\\mu_1, \\mu_1+\\delta(\\lambda)) $\\end{document}<\/tex-math><\/inline-formula>, the above fractional Schr\u00f6dinger -Poisson systems possess a nonnegative bound state solutions with positive energy. Here <inline-formula><tex-math id=\"M6\">\\begin{document}$ \\mu_1 $\\end{document}<\/tex-math><\/inline-formula> is the first eigenvalue of <inline-formula><tex-math id=\"M7\">\\begin{document}$ (-\\triangle)^s +V(x) $\\end{document}<\/tex-math><\/inline-formula>.<\/p><\/jats:p>","DOI":"10.3934\/mfc.2021023","type":"journal-article","created":{"date-parts":[[2021,10,20]],"date-time":"2021-10-20T04:01:49Z","timestamp":1634702509000},"page":"57","source":"Crossref","is-referenced-by-count":3,"title":["Bound state solutions for fractional Schr\u00f6dinger-Poisson systems"],"prefix":"10.3934","volume":"5","author":[{"given":"Xinsheng","family":"Du","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Qi","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Zengqin","family":"Zhao","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gen","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"2321","reference":[{"key":"key-10.3934\/mfc.2021023-1","doi-asserted-by":"publisher","unstructured":"A. Ambrosetti, P. H. Rabinowitz.Dual variational methods in critical point theory and applications, J. Funct. Anal.<\/i>, 14<\/b> (1973), 349-381.","DOI":"10.1016\/0022-1236(73)90051-7"},{"key":"key-10.3934\/mfc.2021023-2","unstructured":"J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis<\/i>, Wiley, New York, 1984."},{"key":"key-10.3934\/mfc.2021023-3","doi-asserted-by":"publisher","unstructured":"V. Benci, D. Fortunato.An eigenvalue problem for the Schr\u00f6dinger-Maxwell equations, Topol. Methods Nonlinear Anal.<\/i>, 11<\/b> (1998), 283-293.","DOI":"10.12775\/TMNA.1998.019"},{"key":"key-10.3934\/mfc.2021023-4","doi-asserted-by":"publisher","unstructured":"V. Benci, D. Fortunato.Solitary waves of nonlinear Klein-Gordon equation coupled with Maxwell equations, Rev. Math. Phys.<\/i>, 14<\/b> (2002), 409-420.","DOI":"10.1142\/S0129055X02001168"},{"key":"key-10.3934\/mfc.2021023-5","doi-asserted-by":"publisher","unstructured":"L. Caffarelli, L. 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