We present existence results for a Neumann problem involving critical Sobolev nonlinearities both on the right hand side of the equation and at the boundary condition. Positive solutions are obtained through constrained minimization on the Nehari manifold. Our approach is based on the concentration compactness principle of P. L. Lions and M. Struwe.

We present an abstract approach to locate multiple solutions of some superlinear variational problems in a Hilbert space $H$. The approach has many points in common with existing methods, but we add a new tool by using a decomposition technique related to dual cones in $H$ which goes back to Moreau. As an application we deduce new existence results for sign changing solutions for some superlinear biharmonic boundary value problems.

Using recent results of M. Tang [Uniqueness of positive radial solutions for $\Delta u − u + u^p = 0$ on an annulus, J. Differential Equations 189 (2003), no. 1, 148–160], we provide a simple approach to proving positivity for the linearized problem of semilinear equations, which is crucial for establishment of exact multiplicity results, and for symmetry breaking.

We give a sufficient condition in order that a sequence of cap or Steiner symmetrizations or of polarizations approximates some fixed cap or Steiner symmetrization. This condition is used to obtain the almost sure convergence for random sequences of symmetrization taken in an appropriate set. The results are applicable to the symmetrization of sets. An application is given to the study of the symmetry of critical points obtained by minimax methods based on the Krasnosel'skiĭ genus.

We propose general and unified concepts of sub- supersolutions for boundary value problems that encompass several types of boundary conditions for nonlinear elliptic equations and variational inequalities. Various, by now classical, sub- and supersolution existence and comparison results are covered by the general theory presented here.

We study a class of $p$-laplacian-type problems with various unbounded weights and a forcing term on open subsets of $\mathbb R^N$ or on the positive real axis. To prove the existence of solution, we use variational methods involving concentration-compactness technique and Hardy-type inequalities.

In this paper we prove the existence of two positive solutions for a second order nonlinear functional nonlocal boundary value problem. The results are obtained by using a fixed point theorem on a Banach space, ordered by an appropriate cone, due to Avery and Henderson [Two positive fixed points of nonlinear operators on ordered Banach spaces, Comm. Appl. Nonlinear Anal. 8 (2001), 27–36]. Using this theorem we have the advantage that the obtained two solutions have their values at three points of their domain upper and lower bounded by a-priori given constants.

Necessary and sufficient conditions for single-valued extensions of multivalued maps are discussed. Moreover, a quantitative version of a generalization of Dugundji's extension theorem for multivalued maps is obtained. Finally, the extension problem for compact maps is studied. Many of the results are new even for single-valued maps.

The Michelson system [D. Michelson, Steady solutions of the Kuramoto–Sivashinsky equation, Physica D 19 (1986), 89–111] $x'''+x'+0.5x^2=c^2$ for the parameter value $c=1$ is investigated. It was proven in [W. C. Troy, The existence of steady solutions of the Kuramoto–Sivashinsky equation, J. Differential Equations 82 (1989), 269–313] that the system possesses two odd periodic solutions. We shall show that there exist infinitely many homoclinic and heteroclinic connections between them. Moreover, we shall show that the family of homoclinic solutions contains a countable set of odd homoclinic solutions.

We consider periodic perturbations of an isochronous hamiltonian system in the plane, depending on a parameter, which generalize the classical asymmetric oscillator. We compute the associated topological degree, and consider situations where large-amplitude periodic solutions can arise.

Motivated by the Emden differential equation we discuss in this paper the existence of positive solutions to the integral equation $$ y(t)=\int^t_0 k(t,s)f(y(s))ds \quad\text{for } t\in [0,T). $$