We consider a two-component competition-diffusion system with equal diffusion coefficients and inhomogeneous Dirichlet boundary conditions. When the interspecific competition parameter tends to infinity, the system solution converges to that of a free-boundary problem. If all stationary solutions of this limit problem are non-degenerate and if a certain linear combination of the boundary data does not identically vanish, then for sufficiently large interspecific competition, all non-negative solutions of the competition-diffusion system converge to stationary states as time tends to infinity. Such dynamics are much simpler than those found for the corresponding system with either homogeneous Neumann or homogeneous Dirichlet boundary conditions.

We study existence and multiplicity results for semilinear elliptic equations of the type $-\Delta u=g(x,u)-te_1+\mu$ with homogeneous Dirichlet boundary conditions. Here $g(x,u)$ is a jumping nonlinearity, $\mu$ is a Radon measure, $t$ is a positive constant and $e_1> 0$ is the first eigenfunction of $-\Delta$. Existence results strictly depend on the asymptotic behavior of $g(x,u)$ as $u\rightarrow\pm \infty$. Depending on this asymptotic behavior, we prove existence of two and three solutions for $t> 0$ large enough. In order to find solutions of the equation, we introduce a suitable action functional $I_t$ by means of an appropriate iterative scheme. Then we apply to $I_t$ standard results from the critical point theory and we prove existence of critical points for this functional.

In this paper we investigate existence and multiplicity of solutions, with prescribed nodal properties, to a two-point boundary value problem of asymptotically linear $n$-th order equations. The proof follows a shooting approach and it is based on the weighted eigenvalue theory for linear $n$-th order boundary value problems.

We study bifurcation phenomena in control flows and the bifurcation of control sets. A Mel'nikov method and the Conley index together with exponential dichotomy theory and integral manifold theory are used.

Given a Hilbert space $( {\mathcal H}, \langle \cdot,\cdot\rangle)$, $\Lambda$ an interval of $\mathbb R$ and $K \in C^{1,1} ({\mathcal H}, {\mathbb R})$ whose gradient is a compact mapping, we consider a family of functionals of the type: $$ I(\lambda, u) = \frac{1}{2} \langle u , u\rangle - \lambda K(u), \quad (\lambda,u) \in \Lambda \times {\mathcal H}. $$ Though the Palais-Smale condition may fail under just these assumptions, we present a deformation lemma to detect critical points. As a corollary, if $I(\overline \lambda,\cdot)$ has a "mountain pass geometry"' for some $\overline \lambda \in \Lambda$, we deduce the existence of a sequence $\lambda_n \to \overline \lambda$ for which each $I(\lambda_n,\cdot)$ has a critical point. To illustrate such results, we consider the problem: $$ - \Delta u = \lambda \bigg( \frac{e^u}{\int_{\Omega} e^u } - \frac{T}{|\Omega|} \bigg), \quad u \in H_0^1 (\Omega), $$ where $\Omega \subset \subset {\mathbb R}^2$ and $T$ belongs to the dual $H^{-1}$ of $H^1_0 (\Omega)$. It is known that the associated energy functional does not satisfy the Palais-Smale condition. Nevertheless, we can prove existence of multiple solutions under some smallness condition on $\| T-1 \|_{H^{-1}}$, where $1$ denotes the constant function identically equal to $1$ in the domain.

In this article we deal with pairs of polynomial planar foliations. The main results concern global and local structural stability as well as the finite determinacy for these pairs. These results can be applied to study a special class of quadratic differential forms in the plane.

In this paper the compression-expansion fixed point theorems are extended to operators which are compositions of two multi-valued nonlinear maps and satisfy compactness conditions of Mönch type with respect to the weak or the strong topology. As an application, the existence of positive solutions for $p$-Laplace inclusions is studied.

In this paper we present some new existence results for singular boundary value problems for second order impulsive differential equations. Our nonlinearity may be singular in its dependent variable.

Given a compact dynamical system $(X,T,m)$ and a pair $(G,\sigma)$ consisting of a compact group $G$ and a continuous group automorphism $\sigma$ of $G$, we consider the twisted skew-product transformation on $G\times X$ given by $$ T_\varphi (g,x) = (\sigma [(\varphi (x)g],Tx), $$ where $\varphi \colon X\rightarrow G$ is a continuous map. If $(X,T,m)$ is ergodic and aperiodic, we develop a new technique to show that for a large class of groups $G$, the set of $\varphi$'s for which the map $T_\varphi$ is ergodic (with respect to the product measure $\nu\times m$, where $\nu$ is the normalized Haar measure on $G$) is residual in the space of continuous maps from $X$ to $G$. The class of groups for which the result holds contains the class of all connected abelian and the class of all connected Lie groups. For the class of non-abelian fiber groups, this result is the only one of its kind.