We consider a family of quasilinear elliptic problems on bounded domains with perturbed critical growth term; we prove the existence of nontrivial solutions when the perturbation depends on a parameter which forces the solutions to have a suitable behavior in order to interact with it. We also study the behavior of the solutions for varying parameter, and we show that concentration phenomena appear as the parameter tends to infinity.

It is well known that, for $\varepsilon =0$, all solutions of the equations \[ \begin{cases} x''(t)+p(t)x(t-\varepsilon )=0,\quad t\in [1,+\infty ), \\ {x''(t)+p(t)x(t)+\frac{1}{t^{\alpha }} x\left( t-\frac{\varepsilon }{t^{\beta }}\right) =0,\quad t\in [1,+\infty ),} \end{cases} p(t)\ge c>0, \] are bounded on $[1,+\infty )$ and even tend to zero, as $p(t { \mathrel{\mathop{\longrightarrow}_{t\to +\infty}}}+\infty $. Here we obtain the following results: 1) for each positive $\varepsilon $ there exist unbounded solutions of the first equation; 2) for each positive $\varepsilon $ there exist unbounded solutions of the second equation in the case when $\alpha \ge 0$, $\beta \ge 0$, $\alpha +\beta \le 1$ and $p(t)$ is bounded; 3) all solutions of the equation \[ x^{\prime\prime}(t)+p(t)x(t-\tau(t))=0,\quad t\in [1,+\infty), \] with positive nondecreasing and bounded on $[1,+\infty)$ coefficient $p(t)$ are bounded if and only if $ \int _1^{+\infty} \tau(t)\,dt <\infty. $

The local exact controllability of a reaction--diffusion system with homogeneous Neumann boundary conditions is studied. The methods we use combine the Schauder--Kakutani fixed point theorem, Carleman estimates for the backward adjoint linearized system and some estimates in the theory of parabolic boundary value problems in $L^k.$

In this paper we study the degenerate parabolic problem \[ \begin{cases} u_t+a\,x\cdot\nabla u-|x|^2\Delta u=f(u), & x\in \mathbb R^N,\quad t>0,\\ u(x,0)=u_0(x)\ge0, & x\in \mathbb R^N, \end{cases} \] where $a\in \mathbb R$ and $f: \mathbb R\to \mathbb v$ is a $C^1$ function. We obtain local existence results and then focus on the blow-up behavior when $f$ is such that \[ f(u)>0\text{ and }\int_u^\infty\frac{ds}{f(s)} <\infty\quad\forall u>0. \] In particular, we describe the blow-up time and rate of the nonlocal solutions under quite general conditions. Differences with the corresponding problem with uniform diffusivity (the heat equation) are stressed.

Weakly-coupled elliptic system describing models of simple three-species food webs such as the one-predator, two-prey model is discussed. We show that there is no nonconstant solution if diffusions or interspecific competitions are strong, or if the intrinsic growths of the prey are slow and the intrinsic drop rates of predator are fast.

This paper is concerned with degenerate Volterra equations $Mu(t)+\int_0^tk(t-s)Lu(s)\, ds=f(t)$ in Banach spaces, both in the hyperbolic case and the parabolic one. The key assumption is played by the representation of the underlying space $X$ as a direct sum $X=N(T)\oplus\overline{R(T)}$, where $T$ is the bounded linear operator $T=ML^{-1}$. Hyperbolicity means that the part $\tilde{T}$ of $T$ in $\overline{R(T)}$ is an abstract potential operator, i.e., $-\tilde{T}^{-1}$ generates a $C_0$-semigroup, and parabolicity means that $-\tilde{T}^{-1}$ generates an analytic semigroup. A maximal regularity result is obtained for parabolic equations. We will also investigate the cases where the kernel $k(\cdot)$ is degenerate or singular at $t=0$ using the results of Prüss [8] on analytic resolvents. Finally we consider the case where $\lambda$ is a pole for $(\lambda L+M)^{-1}$.