An Exact Multiplicity Result for Singular Subcritical Elliptic Problems

Authors : Tomas Godoy

Pages : 87-107

Doi:10.33401/fujma.1376919

View : 35 | Download : 50

Publication Date : 2024-06-30

Article Type : Research

Abstract :For a bounded and smooth enough domain $\\Omega$ in $\\mathbb{R}^{n}$, with $n\\geq2,$ we consider the problem $-\\Delta u=au^{-\\beta}+\\lambda h\\left( .,u\\right) $ in $\\Omega,$ $u=0$ on $\\partial\\Omega,$ $u>0$ in $\\Omega,$ where $\\lambda>0,$ $0<\\beta<3,$ $a\\in L^{\\infty}\\left( \\Omega\\right) ,$ $ess\\,inf\\,\\,(a)>0,$ and with $h=h\\left( x,s\\right) \\in C\\left( \\overline{\\Omega}\\times\\left[ 0,\\infty\\right) \\right) $ positive on $\\Omega\\times\\left( 0,\\infty\\right) $ and such that, for any $x\\in\\Omega,$ $h\\left( x,.\\right) $ is strictly convex on $\\left( 0,\\infty\\right) $, nondecreasing, belongs to $C^{2}\\left( 0,\\infty\\right) ,$ and satisfies, for some $p\\in\\left( 1,\\frac{n+2}{n-2}\\right) ,$ that $\\lim_{s\\rightarrow\\infty }\\frac{h_{s}\\left( x,s\\right) }{s^{p}}=0$ and $\\lim_{s\\rightarrow\\infty }\\frac{h\\left( x,s\\right) }{s^{p}}=k\\left( x\\right) ,$ in both limits uniformly respect to $x\\in\\overline{\\Omega}$, and with $k\\in C\\left( \\overline{\\Omega}\\right)$ such that $\\min_{\\overline{\\Omega}}k>0.$ Under these assumptions it is known the existence of $\\Sigma > 0 $ such that for $ \\lambda =0 $ and $ \\lambda = \\Sigma $ the above problem has exactly a weak solution, whereas for $ \\lambda \\in \\left( 0, \\Sigma \\right) $ it has at least two weak solutions, and no weak solutions exist if $ \\lambda > \\Sigma $. For such a $ \\Sigma $ we prove that for $ \\lambda \\in \\left( 0, \\Sigma \\right) $ the considered problem has it has exactly two weak solutions.
Keywords : Bifurcation problems, Implicit function theorem, Positive solutions, Singular elliptic problem, Sub and supersolutions method