On asymptotics of solutions to semilinear elliptic equations near the first eigenvalue of the nonperturbed problem

The following elliptic equations withp-Laplacian\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$ - \Delta _p u = \lambda g(x)\left| u \right|^{p - 2} u + f(x)\left| u \right|^{\gamma - 2} u$$ \end{document} are considered in the entire space ℝN and in the bounded domain with the Dirichlet boundary conditions. By the fibering method for the basic positive solutions of these equations, we derive the following asymptotic formula\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$u^\lambda = (\lambda _1 - \lambda )^{1/(\gamma - p)} u_1 + o((\lambda _1 - \lambda )^{1/(\gamma - p)} )$$ \end{document} for λ↑λ1, where λ1 is the first eigenvalue and u1 is the corresponding eigenfunction of nonperturbed problem (ƒ=0).