{ "id": "2004.09451", "version": "v1", "published": "2020-04-20T17:06:58.000Z", "updated": "2020-04-20T17:06:58.000Z", "title": "Nehari manifold for fractional p(.)-Laplacian system involving concave-convex nonlinearities", "authors": [ "Reshmi Biswas", "Sweta Tiwari" ], "categories": [ "math.AP" ], "abstract": "In this article using Nehari manifold method we study the multiplicity of solutions of the following nonlocal elliptic system involving variable exponents and concave-convex nonlinearities: \\begin{equation*} \\;\\;\\; \\begin{array}{rl} (-\\Delta)_{p(\\cdot)}^{s} u&=\\lambda~ a(x)| u|^{q(x)-2}u+\\frac{\\alpha(x)}{\\alpha(x)+\\beta(x)}c(x)| u|^{\\alpha(x)-2}u| v| ^{\\beta(x)},\\hspace{2mm} x\\in \\Omega; \\\\ (-\\Delta)_{p(\\cdot)}^{s} v&=\\mu~ b(x)| v|^{q(x)-2}v+\\frac{\\alpha(x)}{\\alpha(x)+\\beta(x)}c(x)| v|^{\\alpha(x)-2}v| u| ^{\\beta(x)},\\hspace{2.5mm} x\\in \\Omega; \\\\ u=v&=0 ,\\hspace{1cm} x\\in \\Omega^c:=\\mathbb R^N\\setminus\\Omega, \\end{array} \\end{equation*} where $\\Omega\\subset\\mathbb R^N,~N\\geq2$ is a smooth bounded domain, $\\lambda,\\mu>0$ are the parameters, $s\\in(0,1),$ $p\\in C(\\mathbb R^N\\times \\mathbb R^N,(1,\\infty))$ and $q,\\alpha,\\beta\\in C(\\overline{\\Omega},(1,\\infty))$ are the variable exponents and $a,b,c\\in C(\\overline{\\Omega},[0,\\infty))$ are the non-negative weight functions. We show that there exists $\\Lambda>0$ such that for all $\\lambda+\\mu<\\Lambda$, there exist two non-trivial and non-negative solutions of the above problem under some assumptions on $q,\\alpha,\\beta$.", "revisions": [ { "version": "v1", "updated": "2020-04-20T17:06:58.000Z" } ], "analyses": { "keywords": [ "concave-convex nonlinearities", "fractional", "nehari manifold method", "nonlocal elliptic system", "variable exponents" ], "note": { "typesetting": "TeX", "pages": 0, "language": "en", "license": "arXiv", "status": "editable" } } }