Consider the equation $-\Delta u = u_{+}^{p-1}-u_{-}^{q-1}$ in the unit ball $B$ with a homogeneous Dirichlet boundary condition. We assume $2< p,q< 2^{*}$. Let $\varphi(u)=(1/2)\int_{B} |\nabla u|^{2} dx-(1/p)\int_{B}u_{+}^{p}dx -(1/q)\int_{B}u_{-}^{q}dx$ be the functional associated to this equation. The nodal Nehari set is defined by $\mathcal M=\{u\in H^{1}_{0}(B): u_{+}\neq 0,\ u_{-}\neq 0,\ \langle\varphi'(u_{+}),u_{+}\rangle= \langle\varphi'(u_{-}),u_{-}\rangle=0\}$. Now let $\mathcal M_{\rm rad}$ denote the subset of $\mathcal M$ consisting of radial functions and let $\beta_{\\rm rad}$ be the infimum of $\varphi$ restricted to $\mathcal M_{\\rm rad}$. Furthermore fix two disjoint half balls $B^{+}$ and $B^{-}$ and denote by $\mathcal M_{h}$ the subset of $\mathcal M$ consisting of functions which are positive in $B^{+}$ and negative in $B^{-}$. We denote by $\beta_{h}$ the infimum of $\varphi$ restricted to $\mathcal M_{h}$. In this note we are interested in obtaining inequalities between $\beta_{\\rm rad}$ and $\beta_{h}$. This problem is related to the study of symmetry properties of least energy nodal solutions of the equation under consideration. We also consider the case of the homogeneous Neumann boundary condition.

The paper is concerned with the existence and uniqueness of solutions to the Allen-Cahn equation coupled with elasticity. The system represents a particular, simple version of the Fried-Gurtin model for solid-solid transitions with phase characterized by an order parameter.

The system is studied with the help of the Leray-Schauder fixed point theorem. The main tool applied in the existence proof is the solvability theory of parabolic problems in anisotropic Sobolev spaces with mixed time-space norms.

Using global variational methods and coordinate free assumptions, we obtain existence and multiplicity results on stationary Lorentzian manifolds for solutions to the Lorentz force equation joining two spacelike submanifolds. Some examples and applications are provided.

We study a semilinear nonautonomous second order periodic system with a nonsmooth potential function which exhibits a quadratic or superquadratic growth. We establish the existence of a solution, using minimax methods of the nonsmooth critical point theory.

Let $X$ be a Banach space, $X^*$ its dual and let $T\colon X\to L^p(\Omega ,\mathbb {R}^k)$ be a linear, continuous operator, where $p, k\ge 1$, $\Omega $ being a bounded open set in $\mathbb {R}^N$. Let $K$ be a subset of $X$, ${\mathcal A}\colon K\rightsquigarrow X^*$, $G\colon K\times X\rightsquigarrow \mathbb {R}$ and $F\colon \Omega \times \mathbb {R}^k\times \mathbb {R}^k\rightsquigarrow \mathbb {R}$ set-valued maps with nonempty values. Using mainly set-valued analysis, under suitable conditions on the involved maps, we shall guarantee solutions to the following inclusion problem:

Find $u\in K$ such that, for every $v\in K$ $$ \sigma ({\mathcal A}(u),v-u)+G(u,v-u)+ \int_\Omega F(x,T{u}(x),T{v}(x)-T{u}(x))dx \subseteq \mathbb {R}_+. $$ In particular, well-known variational and hemivariational inequalities can be derived.

In this paper we use the topological degree to estimate the minimal number of solutions of the sections (defined by fixing a parameter) of the semi-bounded components of a general class of one-parameter abstract nonlinear equations by means of the signature of the semi-bounded component. A semi-bounded component is, roughly speaking, a component that is bounded along one direction of the parameter. The signature consists of the set of bifurcation values from the trivial state of the component together with their associated parity indices. The parity is a local invariant measuring the change of the local index of the trivial state.

The paper presents new approximation and fixed point results for $\mathcal U_{c}^{\kappa}$ maps in Hausdorff locally convex spaces.

The existence of continuous approximate selections is proved for a class of upper semicontinuous multifunctions taking closed $\alpha$-convex values in a metric space equipped with an appropriate notion of $\alpha$-convexity. The approach is based on the definition of pseudo-barycenter of an ordered $n$-tuple of points. As an application, a notion of topological degree for a class of $\alpha$-convex multifunctions is developed.

The present paper studies the local versions of a fixed point theorem of Dhage (1987) in Banach algebras. An application of the newly developed fixed point theorem is also discussed for proving the existence results to a nonlinear functional integral equation of mixed type.

We describe a class of functions $f\colon \mathcal B/\mathbb Z \to \mathcal B$ such that for each irrational rotation $Tx=x+\alpha$, where $\alpha$ has the property that the sequence of aritmethical means of its partial quotients is bounded, the corresponding weighted unitary operators $L^2({\mathbb{R} / \mathbb{Z})\ni g \mapsto e^{2 \pi icf} \cdot g \circ T$ have a Lebesgue spectrum for each $c\in \mathbb R\setminus\{0\}$. We show that for such $f$ and $T$ and for an arbitrary ergodic $\mathcal B$-action $\mathcal S=(S_t)_{t\in\mathcal B}$ on $(Y,\mathcal C,\nu)$ the corresponding Rokhlin cocycle extension $T_{f,\mathcal S}(x,y)=(Tx,S_{f(x)}y)$ acting on $(\mathcal B/\mathbb Z\times Y,\mu \otimes \nu)$ has also a Lebesgue spectrum in the orthogonal complement of $L^2(\mathcal B/\mathbb Z,\mu)$ and moreover the weak closure of powers of $T_{f,\mathcal S}$ in the space of self-joinings consists of ergodic elements.