Abstract
We get a theorem which shows the existence of at least three solutions for some elliptic system with Dirichlet boundary condition. We obtain this result by using the finite dimensional reduction method for the dimension of the system which reduces the infinite dimensional problem to the finite dimensional one. We also use critical point theory on the reduced finite dimensional subspace.
Highlights
1 Introduction In this paper we are concerned with multiple solutions for a class of systems of elliptic equations with Dirichlet boundary condition
Let λ < λ ≤ · · · ≤ λk ≤ · · · be eigenvalues of the eigenvalue problem – u = λu in, u = on ∂, and φk be an eigenfunction belonging to the eigenvalue λk, k ≥
The authors obtained some results for those problems by approaching the variational method, critical point theory and the topological method
Summary
Let λ < λ ≤ · · · ≤ λk ≤ · · · be eigenvalues of the eigenvalue problem – u = λu in , u = on ∂ , and φk be an eigenfunction belonging to the eigenvalue λk, k ≥. (F ) There exist eigenvalues λh+ , . (F ) There exist γ and C such that λh+m < γ < β and. Some papers of Chang [ ] and Choi and Jung [ ] considered the existence and multiplicity of weak solutions for nonlinear boundary value problems with asymptotically linear term. The authors obtained some results for those problems by approaching the variational method, critical point theory and the topological method. Let E be a cartesian product of the Sobolev spaces W , ( , R), i.e.,.
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