Existence and multiplicity of solutions to nonlinear differential equations at resonance
The problem \(Lu+g(u)=f\) in \(\Omega\), \(B_ ku=0\), \(k=1,...,m-1\), where L is a uniformly elliptic differential operator of order 2m, \(B_ k\) are boundary operators, \(\Omega\) is a smooth domain, and g together with its derivative g' are continuous and bounded, is considered in this paper. It is assumed that the kernel of L is a one-dimensional subspace spanned by a function \(\theta\) (x) satisfying \(0<\alpha \leq \theta(x)\leq \beta\), with \(\alpha\) and \(\beta\) constants, a very strong assumption. Existence and multiplicity of solutions for small f is studied under additional assumptions, concerning mainly the zeroes of g. The alternative method of Cesari is the main tool used by the authors. The results are, in some sense, complementary of the previous ones where Landesman-Lazer type assumptions are made by several authors (Fučik, Hess, Ambrosetti-Mancini, etc.). Section 2 includes some fundamental estimates on the solutions. Existence results are proved in Section 3 for g with a finite (or countable) number of separated zeroes, and multiplicity results are given in Section 4.
scientific article; zbMATH DE number 3842366
scientific article