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

Ya. Sh. Il'yasov

Matematičeskie zametki, Tome 64 (1998) no. 4, pp. 543-548

Résumé

The following elliptic equations with $p$-Laplacian $$ -\Delta_pu=\lambda g(x)|u|^{p-2}u+f(x)|u|^{\gamma-2}u $$ are considered in the entire space $\mathbb R^N$ 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 $$ u^\lambda=(\lambda_1-\lambda)^{1/(\gamma-p)}u_1 +o\bigl((\lambda_1-\lambda)^{1/(\gamma-p)}\bigr) $$ for $\lambda\uparrow\lambda_1$, where $\lambda_1$ is the first eigenvalue and $u_1$ is the corresponding eigenfunction of nonperturbed problem ($f=0$).