Geodesic
On asymptotics of solutions to semilinear elliptic equations near the first eigenvalue of the nonperturbed problem
Matematičeskie zametki, Tome 64 (1998) no. 4, pp. 543-548.

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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 ($f=0$). 
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     author = {Ya. Sh. Il'yasov},
     title = {On asymptotics of solutions to semilinear elliptic equations near the first eigenvalue of the nonperturbed problem},
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Ya. Sh. Il'yasov. On asymptotics of solutions to semilinear elliptic equations near the first eigenvalue of the nonperturbed problem. Matematičeskie zametki, Tome 64 (1998) no. 4, pp. 543-548. http://geodesic.mathdoc.fr/item/MZM_1998_64_4_a5/

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