We consider nonlinear perturbations of the $p$-Laplacian depending on a real parameter and subject to zero Dirichlet boundary data; we establish results which guarantee the existence of at least one and at least two solutions for certain parameter ranges.

Let $(A_p)_{1 < p < \infty}$ be a consistent family of sectorial operators on $L^p(\Omega; X)$, where $\Omega$ is a homogeneous space with doubling property and $X$ is a Banach space having the Radon-Nykodým property. If $A_{p_0}$ has a bounded {$H^\infty$ calculus}{} for some $1 < p_0 < \infty$ and the resolvent or the semigroup generated by $A_{p_0}$ fulfills a Poisson estimate, then it is proved that $A_p$ has a bounded {$H^\infty$ calculus}{} for all $1 < p \le p_0$ and even for $1 < p < \infty$ if $X$ is reflexive. In order to do so, the Calderón-Zygmund decomposition is generalized to the vector-valued setting.

This note is concerned with a stability estimate for parabolic evolution operators and its application to abstract quasilinear Cauchy problems.

We consider the Dirichlet problem associated to a $2\times 2$ semilinear elliptic system of equations in a bounded domain of $\mathbb R^n$. This system depends on two parameters $(\eta,\mu)$ and the nonlinear terms are of power type. The existence of solutions for such a problem in the plane $(\eta,\mu)$ is discussed. Our results are natural extensions of those obtained for the corresponding nonlinear eigenvalue problem for elliptic scalar equations.

This paper studies the limit, as a small parameter $\varepsilon\rightarrow0$, of a simplified Magneto-Hydro-Dynamic system in a rotating frame. After proving existence and uniqueness results, we show that, in the strong rotation limit and for a finite time $T$, the solution tends to that of a limit ("averaged") system.

In this paper, the asymptotic behavior in time of solutions for the nonlinear Schrödinger equation with the quadratic nonlinearity $\lambda |u|^2$ with $\lambda \in {\mathbb C} \setminus \{ 0 \}$ in two space dimensions is studied. The nonexistence of asymptotically free solutions for this equation is proved.

The present paper deals with special solutions for linear functional differential equations with small delays. The importance of special solutions relies on the fact that, under some conditions, they can be used to describe the asymptotic behaviour of all solutions. Sufficient conditions for the existence of special solutions for nonautonomous equations in $\mathbb R^n$ are given. For autonomous equations in $\mathbb R^n$, results relating special solutions and characteristic values are presented. We also consider linear autonomous functional differential equations in Banach spaces, and use special solutions to study the asymptotic behaviour of their solutions. Several applications are given.