A nonlocal type problem involving a mixed local and nonlocal operator

Abstract

In this paper, we consider the nonlocal elliptic problem involving a mixed local and nonlocal operator, (P)∫Ωf(x,u)dxβLp,s(u)=fα(x,u)inΩ,u>0inΩ,u=0inRN\\Ω,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} (P)\\left\\{ \\begin{array}{rcll} \\left( \\displaystyle \\int \\limits _\\Omega f(x,u)dx\\right) ^{\\beta }\\mathfrak {L_{p,s}}(u)&{}= &{} f^\\alpha (x,u) &{} \ ext { in }\\Omega , \\\\ u &{}> &{} 0 &{} \ ext {in }\\Omega , \\\\ u &{} = &{} 0 &{} \ ext {in }{\\mathbb {R}}^N \\setminus \\Omega , \\end{array} \\right. \\end{aligned}$$\\end{document}where Ω⊂RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Omega \\subset {\\mathbb {R}}^N$$\\end{document} is a bounded regular domain, Lp,s≡-Δp+(-Δ)ps\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathfrak {L_{p,s}}\\equiv -\\Delta _p+(-\\Delta )^s_p$$\\end{document}, 0<s<1<p<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$0<s<1<p<N$$\\end{document}, α,β∈R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha ,\\,\\beta \\in {\\mathbb {R}}$$\\end{document} and f:Ω×R→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f: \\Omega \ imes {\\mathbb {R}}\\rightarrow {\\mathbb {R}}$$\\end{document} be a nonnegative function which is defined almost everywhere with respect to the variable x. Using Schauder and Tychonoff fixed point theorems, we get two existence theorems of weak positive solutions under some hypothesis on α,β\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\alpha , \\beta $$\\end{document} and f.