In this paper we study the structure of all solutions to the boundary value problem, which is the open problem D of Ambrosetti, Brezis and Cerami [1] $$ -u'' = \lambda |u|^{q-1}u+|u|^{p-1}u, \quad t\in [a, b], \ \ u(a)=u(b)=0, $$ where $0 <q <1 <p,$ $ \lambda >0 .$ We obtain a complete characterization of its solutions and the bifurcation graph. By perturbation, we show also instablity of the structure of the solutions for the above problem (see Figure 3).

Let $L$ be a selfadjoint operator with compact resolvent and $\lambda $ an eigenvalue of $L$. When $\lambda $ is simple, it is well known that the Fucik spectrum $\Sigma $ near $\lambda $ consists of two nonincreasing curves. In this paper, we show that when $\lambda $ is not simple, $\Sigma $ contains two nonincreasing curves such that all points above or under both curves are not in $\Sigma $. After that, we give some existence results of solutions of the equation $Lu=\alpha u^{+}-\beta u^{-}+g(.,u)$ where $u^{\pm }=\max (0,\pm u)$.

We prove a result on the existence and uniqueness of a local solution to a generalized Boussinesq equation for initial data of low regularity. We also discuss the existence of global solutions and the occurrence of blow-up phenomena. Our results are applicable to several physically relevant equations that are obtained as special cases of our model equation.

In this paper, we shall prove that the scattering operator for nonlinear Schr\"{o}dinger equations $ iu_t+(-\Delta)^m u=\lambda_1|u|^{p_1}u+ \lambda_2|u|^{p_2}u$ carries a band $\dot B(p_1,p_2,\delta)$ in $H^s$ into $H^s$ for some $\delta>0$, where $\dot B(p_1,p_2,\delta)=\{\varphi\in H^s: \|\varphi\|_{\dot H^{s(p_1)}\cap\dot H^{s(p_2)}}\leq \delta\}$, $s(p_i)=n/2-2m/p_i$, $s(p_2)\leq s\le s(p_1)+1$, $4m/n\leq p_1\leq p_2\leq 4m/(n-2s)$.

In this paper we study the generation of analytic semigroup in the space $L^2(\mathbb R^d)$, and the characterization of the domain, for a family of degenerate elliptic operators with unbounded coefficients, which includes some well-known operators arising in Mathematical Finance. To prove the generation of analytic semigroups, the operators of the family are assumed to satisfy suitable growth and compensation conditions. Under stronger assumptions, we obtain also the characterization of the domain. Finally, various consequences of the obtained results are considered in connection with some applications (see, e.g., [6] for financial applications). In a forthcoming paper, part II of this work, we shall examine the problem of the generation of analytic semigroup in $L^p((\mathbb R^d)$, where $p\in(2,+\infty]$, and the characterization of the domain, for the same family of operators.

In this paper we give a simple proof of the regularity of a class of solutions to the 3D Navier-Stokes equations for a fluid filling {any} smooth three-dimensional domain. The regularity in the same class was proved by Beir\~ao da Veiga in reference [3], for the Cauchy problem in $\mathbb R^n.$

We study the behavior of large amplitude solutions for forced nonlinear oscillators at resonance, the nonlinearity being a bounded perturbation of a function deriving from an isochronous potential, i.e., a potential leading to free oscillations that all have the same period.