Bifurcation analysis for a modified quasilinear equation with negative exponent

Abstract

AbstractIn this paper, we consider the following modified quasilinear problem:−Δu−κuΔu2=λa(x)u−α+b(x)uβinΩ,u>0inΩ,u=0on∂Ω,$$\begin{array}{} \left\{\begin{array}{c}\, -{\it\Delta} u-\kappa u{\it\Delta} u^2 = \lambda a(x)u^{-\alpha}+b(x)u^\beta \, \, in\, {\it\Omega}, \\\!\! u \gt 0 \, \, in\, {\it\Omega}, \, \, \, \, \, \, \, u = 0 \, \, on \, \partial{\it\Omega} , \\ \end{array}\right. \end{array} $$whereΩ⊂ ℝNis a smooth bounded domain,N≥ 3,a,bare two bounded continuous functions,α> 0, 1 <β≤ 22*− 1 andλ> 0 is a bifurcation parameter. We use the framework of analytic bifurcation theory to obtain an analytic global unbounded path of solutions to the problem. Moreover, we get the direction of solution curve at the asmptotic point.