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

The following elliptic equations with p-Laplacian -Delta(p)u = lambda g(x)/u/(P-2)u + f(x)/u/(gamma-2)u are considered in the entire space R-N 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 u(lambda) = (lambda(1) - lambda)(1/(gamma-p))u(1) + o((lambda(1) - lambda)(1/(gamma-p))) for lambda dagger lambda(1), where lambda(1) is the first eigenvalue and u(1) is the corresponding eigenfunction of nonperturbed problem (f = 0).