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A topological version of the Ambrosetti-Prodi theorem

Bogdan Przeradzki (1996)

Annales Polonici Mathematici

The existence of at least two solutions for nonlinear equations close to semilinear equations at resonance is obtained by the degree theory methods. The same equations have no solutions if one slightly changes the right-hand side. The abstract result is applied to boundary value problems with specific nonlinearities.

A transmission problem

Irena Rachůnková (1992)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

An elliptic equation with no monotonicity condition on the nonlinearity

Gregory S. Spradlin (2006)

ESAIM: Control, Optimisation and Calculus of Variations

An elliptic PDE is studied which is a perturbation of an autonomous equation. The existence of a nontrivial solution is proven via variational methods. The domain of the equation is unbounded, which imposes a lack of compactness on the variational problem. In addition, a popular monotonicity condition on the nonlinearity is not assumed. In an earlier paper with this assumption, a solution was obtained using a simple application of topological (Brouwer) degree. Here, a more subtle degree...

Boundary value problems for nonlinear perturbations of some ϕ-Laplacians

J. Mawhin (2007)

Banach Center Publications

This paper surveys a number of recent results obtained by C. Bereanu and the author in existence results for second order differential equations of the form (ϕ(u'))' = f(t,u,u') submitted to various boundary conditions. In the equation, ϕ: ℝ → ≤ ]-a,a[ is a homeomorphism such that ϕ(0) = 0. An important motivation is the case of the curvature operator, where ϕ(s) = s/√(1+s²). The problems are reduced to fixed point problems in suitable function space, to which Leray-Schauder...

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