By means of nonsmooth critical-point theory, we prove existence of infinitely many solutions $(u^m)\subseteq H^1_0(\Omega,\mathbb R^N)$ for a class of perturbed $\mathbb Z_2-$symmetric elliptic systems.

Nonlinear partial differential equations with $p$--structure for $1 <p <\infty$ are investigated under mixed boundary value conditions on $n$-dimensional domains with a piecewise-smooth boundary. $L^q(\Omega)$--properties of the first derivatives of weak solutions in Sobolev and in Morrey spaces are proven.

We prove that the periodic initial value problem for a completely integrable shallow-water equation is not locally well-posed for initial data in the Sobolev space $H^s(\mathbb{T})$ whenever $s <3/2$. Since on the other hand this problem is locally well-posed in the sense of Hadamard for $s>3/2$ our result suggests that $s=3/2$ is the critical Sobolev index for well-posedness. We also show that the nonperiodic initial value problem is not locally well-posed in $H^s(\mathbb{R})$ for $s <3/2$.

The objective of this paper is to initiate the study of second-order nonlinear functional evolutions of the type $$ \begin{cases} & u''(t) \in A(t)u(t) + G(t,u_t),\quad t>0,\\ & u(0)=x, \ \ u_0 = \phi, \ \ \sup_{t \ge 0} \lbrace \|u(t)\| \rbrace < +\infty, \end{cases} \tag*{($P$)} $$ in a real, uniformly smooth Banach space $X$ with strongly monotone duality mapping. The operators $A(t)$ are $m$-accretive and the operators $G$ are Lipschitzian. The problem is lifted into the space $L_2([-r,\infty);X),$ in which it becomes an elliptic-type problem of the type $$\mathcal Au+\mathcal Bu+\mathcal G(\cdot,u_\cdot) \ni 0$$ with $\mathcal A$ and $\mathcal B$ $m$-accretive. Unperturbed results of Xue, Song and Ma are extended to the present case. The main difficulty in the solvability of these problems is due to the presence of a delay and the fact that certain monotonicity properties of some real-valued functions (defined via the duality mapping) which are present in the homogeneous case do not continue to hold in the perturbed case.

We prove that the set $\big\{(\alpha,\beta)\in\mathbb R^2 : $ the problem $-\Delta_pu=\alpha (u^+)^{p-1}-\beta (u^-)^{p-1}$ in $ \Omega,$ $ u=0$ on $ \partial\Omega$ has a nontrivial solution $\,\,\, \} $ contains infinitely many curves which exist locally in the neighbourhood of suitable eigenvalues of the p-Laplacian operator.

The differential problem $$ A(t)u'(t)+B(t)u(t)+M(t)u'(t-\omega)+ N(t)u(t-\omega)=f(t) , \ \ \ t \geq 0, $$ $u(t)=g(t)$, $-\omega \leq t \leq 0,$ where $g$ is a given strongly continuous, $X$-valued function on $[-\omega,0]$, $f$ is strongly continuous from $[0,\infty)$ into $Y$, and $A(t)$, $B(t)$, $M(t)$, $N(t)$ are closed linear operators from the complex Banach space $X$ into the complex Banach space $Y$, is studied. Solvability on $[0,T]$ with $T < \infty$ is considered, too. Moreover, the case where $A(t)$, $B(t)$ are independent of $t$ and $\lambda = 0$ is a multiple pole for $(\lambda B+A)^{-1}$ is investigated.