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We investigate the multiplicity of the weak solutions for the nonlinear elliptic boundary value problem. We get a theorem which shows the existence of at least four weak solutions for the asymptotically linear elliptic problem with Dirichlet boundary condition. We obtain this result by using the Leray-Schauder degree theory, the variational reduction method and critical point theory.

Existence of four solutions for the elliptic problem with nonlinearity crossing one eigenvalue