In this paper we consider the nonlinear elliptic problem −Δu + αu = g(∣∇u∣) + λh(x) in Ω, u = 0 on ∂Ω, where Ω is a smooth bounded domain of ℝN, α ≥ 0, g is an arbitrary C1 increasing function and h ∈ C1() is non-negative. We completely analyse the existence and non-existence of (positive) classical solutions in terms of the parameter λ. We show that there exist solutions for every λ when α = 0 and the integral 1/g(s)ds = ∞, or when α > 0 and the integral s/g(s)ds = ∞. Conversely, when the respective integrals converge and h is non-trivial on ∂Ω, existence depends on the size of λ. Moreover, non-existence holds for large λ. Our proofs mainly rely on comparison arguments, and on the construction of suitable supersolutions in annuli. Our results include some cases where the function g is superquadratic and existence still holds without assuming any smallness condition on λ.

We present an extension of the Allen-Cahn/Cahn-Hilliard system that incorporates a geometrically linear ansatz for the elastic energy of the precipitates. The model contains both the elastic Allen-Cahn system and the elastic Cahn-Hilliard system as special cases, and accounts for the microstructures on the microscopic scale. We prove the existence of weak solutions to the new model for a general class of energy functionals. We then give several examples of functionals that belong to this class. This includes the energy of geometrically linear elastic materials for dimensions D < 3. Moreover, we show this for D = 3 in the setting of scalar-valued deformations, which corresponds to the case of anti-plane shear. All this is based on explicit formulae for relaxed energy functionals newly derived in this article for D = 1 and D = 3. In these cases we can also prove the uniqueness of the weak solutions.

We use Melnikov function techniques together with geometric methods of bifurcation theory to study the interactions of forcing, damping and detuning on resonant periodic orbits for single and coupled forced Van der Pol oscillators. For a coupled pair the local bifurcation geometry is almost everywhere described in terms of the singularities of a line congruence in three dimensions.

A saddle-point property of the self-similar solutions in the curve shortening flow was conjectured by Abresch and Langer and confirmed by Au. An improvement on Au's solution is presented.

We study the excision property for Hochschild and cyclic homologies in the category of simplicial algebras. We extend Wodzicki's notion of H-unital algebras to simplicial algebras and then show that a simplicial algebra I* satisfies excision in Hochschild and cyclic homologies if and only if it is H-unital. We use this result in the category of crossed modules of algebras and provide an answer to the question posed in the recent paper by Donadze et al. We also give (based on work by Guccione and Guccione) the excision theorem in Hochschild homology with coefficients.

In this paper, we deal with the nonlinear Schrödinger–Poisson system

where λ > 0, V and Q are radial functions, which can be vanishing or coercive at ∞. With assumptions on f just in a neighbourhood of the origin, existence and multiplicity of non-trivial radial solutions are obtained via variational methods. In particular, if f is sublinear and odd near the origin, we obtain infinitely many solutions of (SP)λ for any λ < 0.

In this paper, we study the problem of the existence of solutions for a class of fractional semilinear evolution inclusions with non-convex right-hand side. We mainly focus on the existence of the extreme solution and the relationship of the solution sets between the original problem and the convexified problem. An example is provided to illustrate the application of the obtained theory.

In this paper, we consider the Cauchy problem for an inviscid two-phase gas—liquid model with external force, in order to demonstrate the smoothing effect on the damping mechanism. It is shown that the Cauchy problem admits a unique global smooth solution provided that the initial data are smooth and the C0-norm of the derivative of the initial data are small.

This paper is devoted to the study of a typical (p, q)-gradient elliptic system. We discuss the existence of multiple non-trivial solutions. More precisely, we establish the existence of three non-trivial solutions, obtained by applying the fibering method introduced by Pohozaev.

We study the weighted p-Laplacian with an isolated singular point, and discuss some properties of solutions near such a point, including removable singularities, boundedness and growth rate estimates. In particular, some relevant critical exponents are exactly determined.

In this paper, we revisit a reaction—diffusion autocatalytic chemical reaction model with decay. For higher-order reactions, we prove that the system possesses at least two positive steady-state solutions; hence, it has bistable dynamics similar to the system without decay. For the linear reaction, we determine the necessary and sufficient condition to ensure the existence of a solution. Moreover, in the one-dimensional case, we prove that the positive steady-state solution is unique. Our results demonstrate the drastic difference in dynamics caused by the order of chemical reactions.