A Class Of Degenerate Elliptic Equations Existence And Multiplicity

This paper mainly study the existence and multiplicity for degenerate elliptic equation and systems with main eigenvalues by variational methods such as Mountain Pass Lemma and Ekeland variational principle.Firstly we study Semilinear Elliptic equation:In addition, Ω∈Rn is bounded domain, with smooth boundary (?)Ω, f satisfy the sublinear conditions, we obtained three solutions about the above equations near the main eigenvalues.Then we study the degenerate elliptic equation: where Ω∈Rn is bounded domain, with smooth boundary (?)Ω, f satisfy sublinear conditions, we also adopt Ekeland variational principle and Mountain Pass Lemma proved the existence and multiplicity of its solution near the main eigenvalue with four steps.Finally, we consider the degenerate elliptic equations: where Ω∈Rn is bounded domain, with smooth boundary (?)Ω, F satisfy coercive conditions, we prove the existence and multiplicity of the system equations near the main eigenvalues.