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In this paper I present the most important results of my research studying the solutions of nonlinear partial differential equations of the type.
In this study, we focus on nonlinear partial differential equations of the form ∆u + λf(u) = 0 in Ω, u = 0 on ∂Ω, where λ ∈ R, Ω is a smooth bounded domain in RN, ∆ = Σ ∂²/∂x²ᵢ is the Laplacian operator, and f : R → R is a nonlinear function. Our theorems were obtained using bifurcation theory, variational methods, and a minmax principle developed by the author in collaboration with A. Castro and J. M. Neuberger ([Cas-Cos-Nu1], 1997). We also present some theorems related to algorithms for approximating solutions to nonlinear problems of type (1), along with a set of open questions.
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