The aim of this paper is to prove multiplicity of solutions for nonlocal fractional equations modeled by $$ \left\{ \begin{array}{ll} (-\Delta)^s u-\lambda u=f(x,u) & {\mbox{ in }} \Omega\\ u=0 & {\mbox{ in }} \mathbb R^n\setminus \Omega\,, \end{array} \right. $$ where $s\in (0,1)$ is fixed, $(-\Delta)^s$ is the fractional Laplace operator, $\lambda$ is a real parameter, $\Omega\subset \mathbb R^n$, $n>2s$, is an open bounded set with continuous boundary and nonlinearity $f$ satisfies natural superlinear and subcritical growth assumptions. Precisely, along the paper, we prove the existence of at least three non-trivial solutions for this problem in a suitable left neighborhood of any eigenvalue of $(-\Delta)^s$. For this purpose, we employ a variational theorem of mixed type (one of the so-called $\nabla$-theorems).

The existence and uniqueness of almost automorphic solutions for linear and semilinear nonconvolution Volterra equations on time scales is studied. The existence of asymptotically almost automorphic solutions is proved. Examples that illustrate our results are given.

The well-posedness and the flow invariance are studied for a semilinear functional differential equation governed by a family of non-densely defined operators in a general Banach space. The notion of mild solutions is introduced through a new type of variation of constants formula and the well-posedness is established under a semilinear stability condition with respect to a metric-like functional and a subtangential condition. The abstract result is applied to a size-structured model with birth delay.

In this paper, we consider the logarithmic Schrödinger equation on a star graph. By using a compactness method, we construct a unique global solution of the associated Cauchy problem in a suitable functional framework. Then we show the existence of several families of standing waves. We also prove the existence of ground states as minimizers of the action on the Nehari manifold. Finally, we show that the ground states are orbitally stable via a variational approach.

We study the Cauchy problem for the Zakharov system in spatial dimension $d\ge 4$ with initial datum $ (u(0), n(0), \partial_t n(0) )\in H^k(\mathbb R^d)\times \dot{H}^l(\mathbb R^d)\times \dot{H}^{l-1}(\mathbb R^d)$. According to Ginibre, Tsutsumi and Velo ([9]), the critical exponent of $(k,l)$ is $ ((d-3)/2,(d-4)/2 ). $ We prove the small data global well-posedness and the scattering at the critical space. It seems difficult to get the crucial bilinear estimate only by applying the $U^2,\ V^2$ type spaces introduced by Koch and Tataru ([23], [24]). To avoid the difficulty, we use an intersection space of $V^2$ type space and the space-time Lebesgue space $E:=L^2_tL_x^{2d/(d-2)}$, which is related to the endpoint Strichartz estimate.

We study the Evolutionary $p$-Laplace Equation in the singular case $1 < p < 2$. We prove that a weak solution has a time derivative (in Sobolev's sense) which is a function belonging (locally) to a $L^q$-space.