This paper deals with the existence of one-bump and multibump solutions for the following nonlinear field equation: $$-\Delta u+V(h x)u-\Delta_{p}u+ W'(u)=0$$ where $u:\mathbb R^{N}\rightarrow\mathbb R^{N+1},$ $N\geq 2,$ $p>N,$ $h>0,$ the potential $V$ is positive and $W$ is an appropriate singular function. Existence results are established provided that $h$ is sufficiently small, and we find solutions exhibiting a concentration behaviour in the semiclassical limit (i.e., as $h\rightarrow 0^{+}$) at any prescribed finite set of local minima, possibly degenerate, of the potential. Such solutions are obtained as local minima for the associated energy functional. No restriction on the global behaviour of $V$ is required except that it is bounded below away from zero. In the proofs of these results we use a variational approach, and the method relies on the study of the behaviour of sequences with bounded energy, in the spirit of the concentration-compactness principle.

We consider elliptic boundary value problems of the form $u''+p(x)|u|^{-\gamma}{{\rm sign}\:} u=0$ and $\Delta u + p(x)|u|^{-\gamma}{{\rm sign}\:} u=0$ for $\gamma>1$ and $p>0$ on intervals in ${\mathbb {R}}$ and bounded domains in ${\mathbb {R}}^n$. Prescribed vanishing Dirichlet boundary data and the concept of sign-changing solutions make the problem singular both on the boundary and in the interior of the underlying domain. We introduce a solution concept for sign-changing solutions based on the distributional principal value, and we show existence of such solutions. A principal part of our analysis is a (relatively) accurate description of the boundary behavior of positive solutions. Sign-changing principal-value solutions are constructed by gluing together solutions of one sign.

In this paper, we study type II Streater's models. These models describe the coupled evolution of the density of a cloud of particles in an external potential and a temperature, preserving the energy, with eventually a nonlocal Poisson coupling. We introduce an entropy and consider in a bounded domain, or in an unbounded domain with a~confining external potential, the stationary solutions (with given mass and energy), for which we have existence and uniqueness results. The entropy is reinterpreted as a relative entropy which controls the convergence to the stationary solutions. We consider also the whole $\mathbb R^d$ space problems without exterior potential using time-dependent rescalings and show the existence of intermediate asymptotics in special cases.

Let $A$ be the generator of an analytic semigroup on a Banach space $X$. We study the Sobolev regularity of the solutions $u$ of the problem $u'(t)=Au(t)+f(t)$, for almost every $t\in (0,T)$, $u(0)=u_0$. Using extrapolation theory we can investigate optimal conditions for the data $f$ and $u_0$ guaranteeing $u\in W^{\alpha,p}(0,T;X)$ and obtain new existence results for generalized as well as differentiable solutions for parabolic equations.