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The Collatz–Wielandt formula obtained by Lothar Collatz (1942) and Helmut Wielandt (1950) provides a simple variational characterization of the Perron–Frobenius eigenvalue of certain types of matrices. We introduce a nonlinear generalization of the Collatz–Wielandt formula and show that it enables finding the so-called maximal saddle-node bifurcations of systems of nonlinear equations. As a consequence, we obtain an easily verifiable criterion for the detection of such saddle-node bifurcations. We illustrate our approach by examples of finite-difference approximations of nonlinear partial differential equations and a system of power flow. [ABSTRACT FROM AUTHOR]

This paper is concerned with variational continuation of branches of solutions for nonlinear boundary value problems, which involve the p-Laplacian, an indefinite nonlinearity, and depend on a real parameter λ. A special focus is given to the extreme value λ∗ of the Nehari manifold that determines the threshold of applicability of the Nehari manifold method. In the main result the existence of two branches of positive solutions for the cases where the parameter λ lies above the threshold λ∗ is obtained. [ABSTRACT FROM AUTHOR]

This paper is concerned with the existence of global, blow-up and bifurcation solutions for parametrized families of elliptic and parabolic equations with p-Laplacian and concave-convex nonlinearity. The main results are obtained by means of a generalised Collatz-Wielandt formula. [ABSTRACT FROM AUTHOR]

A new class of inverse problems is considered. In the context of classical theory, inverse problems are concerned with finding a model that has the observed measurements. It is well known that such problems usually are ill-posed. At the same time, it is often the case when there is some a priori information about the system. This naturally leads to the following inverse optimal problem : find data F ˆ of a model which is the nearest to a priori given data F 0 and sufficient to ensure the model has the observed measurements S. In this note, an approach to a complete solution to such a problem is developed. Within the framework of this approach, we consider a model problem of recovering the potential field V ˆ from the m observed eigenvalues of the Schrödinger operator, provided that such potential field is at the minimum distance from a priori given potential V a. In the main result, we establish a new type of relationship between the linear spectral problems and systems of nonlinear differential equations which enables us to find a solution to the inverse optimal spectral problem and obtain novel results on the existence of solutions to nonlinear problems as well. • A new class of inverse optimal spectral problem. • The relationship between the spectral problems and the nonlinear problem. • Existence result for the system of NLS equations. [ABSTRACT FROM AUTHOR]

The anti-maximum principle for the homogeneous Dirichlet problem to − Δ p u = λ | u | p − 2 u + f (x) with positive f ∈ L ∞ (Ω) states the existence of a critical value λ f > λ 1 such that any solution of this problem with λ ∈ (λ 1 , λ f) is strictly negative. In this paper, we give a variational upper bound for λ f and study its properties. As an important supplementary result, we investigate the branch of ground state solutions of the considered boundary value problem in (λ 1 , λ 2). [ABSTRACT FROM AUTHOR]

We study the solvability of the Zakharov equation Δ 2 u + (κ − ω 2) Δ u − κ div e − | ∇ u | 2 ∇ u = 0 in a bounded domain under homogeneous Dirichlet or Navier boundary conditions. This problem is a consequence of the system of equations derived by Zakharov to model the Langmuir collapse in plasma physics. Assumptions for the existence and nonexistence of a ground state solution as well as the multiplicity of solutions are discussed. Moreover, we consider formal approximations of the Zakharov equation obtained by the Taylor expansion of the exponential term. We illustrate that the existence and nonexistence results are substantially different from the corresponding results for the original problem. [ABSTRACT FROM AUTHOR]