For a class of non-conservative hyperbolic systems of partial differential equations endowed with a strictly convex mathematical entropy, we formulate the initial-value problem by supplementing the equations with a kinetic relation prescribing the rate of entropy dissipation across shock waves. Our condition can be regarded as a generalization to non-conservative systems of a similar concept introduced by Abeyaratne, Knowles and Truskinovsky for subsonic phase transitions and by LeFloch for non-classical undercompressive shocks to nonlinear hyperbolic systems. The proposed kinetic relation for non-conservative systems turns out to be equivalent, for the class of systems under consideration at least, to Dal Maso, LeFloch and Murat's definition based on a prescribed family of Lipschitz continuous paths. In agreement with previous theories, the kinetic relation should be derived from a phase-plane analysis of travelling-wave solutions associated with an augmented version of the non-conservative system. We illustrate with several examples that non-conservative systems arising in the applications fit in our framework, and for a typical model of turbulent fluid dynamics we provide a detailed analysis of the existence and properties of travelling waves which yields the corresponding kinetic function.

We use the method of upper and lower solutions combined with degree-theoretic techniques to prove the existence of multiple positive solutions to some superlinear elliptic systems of the form

on a smooth, bounded domain Ω⊂ℝn with Dirichlet boundary conditions, under suitable conditions on g1 and g2. Our techniques apply generally to subcritical, superlinear problems with a certain concave–convex shape to their nonlinearity.

We give sharp conditions under which the composition of two homeomorphisms of finite distortion is of finite distortion and has integrable distortion. As an application, we obtain a generalization of the classical uniqueness theorem of homeomorphic solution to the measurable Riemann mapping problem.

Let (M, g) be a compact smooth N-dimensional Riemannian manifold without boundary. We consider the multiple existence of positive solutions of the problem

where Δg stands for the Laplacian in M and f ε C2(M).

We study the existence and multiplicity of positive solutions for the Dirichlet problem

where λ > 0, 1 < q < 2, p = 2* = 2N/(N − 2), 0 ε Ω ⊂ ℝN, N ≥ 3, is a bounded domain with smooth boundary ∂Ω and f is a non-negative continuous function on . Assuming that f satisfies some hypothesis, we prove that the equation admits at least three positive solutions for sufficiently small λ.

We are concerned with the solvability of nonlinear second-order elliptic partial differential equations with nonlinear boundary conditions. We study the generalized Steklov–Robin eigenproblem (with possibly singular weights) in which the spectral parameter is both in the differential equation and on the boundary. We prove the existence of solutions for nonlinear problems when both nonlinearities in the differential equation and on the boundary interact, in some sense, with the generalized spectrum. The proofs are based on variational methods and a priori estimates.

The abstract version of Struwe's monotonicity trick developed by Jeanjean and Toland for functionals depending on a real parameter is strengthened in the sense that it provides, for almost every value of the parameter, the existence of a bounded almost symmetric Palais–Smale sequence at the mountain-pass level whenever a mild symmetry assumption is set on the energy functional. In addition, the whole theory is extended to the case of continuous functionals on Banach spaces, in the framework of non-smooth critical point theory.

The Green function of singular limit-circle problems is constructed directly for the problem, not as a limit of sequences of regular Green's functions. This construction is used to obtain adjointness and self-adjointness conditions which are entirely analogous to the regular case. As an application, a new and explicit formula for the Green function of the classical Legendre problem is found.

We consider the elliptic system

where N ≥ 4, λ > 0, α1, α2, β ε ℝ, p, p > 1, p + q = 2* = 2N/(N − 2) and α1(x), a2(x) ≥ 0 have potential well. By using variational methods and the category theory, we establish the existence of least energy and multiplicity of solutions.

We discuss a new class of Banach spaces which are wider than the strongly convex spaces introduced by Congxin Wu and Yongjin Li. We prove that the new class of Banach spaces lies strictly between either the class of uniformly convex spaces and strongly convex spaces or the class of fully k-convex spaces and strongly convex spaces. The new class of Banach spaces has inclusive relations with neither the class of locally uniformly convex spaces nor the class of nearly uniformly convex spaces. We obtain in addition some characterizations of this new class of Banach spaces.