We present here some results concerning the existence, uniqueness and regularity of the solutions of second order integro-differential Heisenberg equations in a bounded region with homogeneous Dirichlet boundary conditions.

In this paper we study essentially the questions of uniqueness and stability of solutions of boundary value problems associated with equations of the type: $$ \text{div}(\hat a(x,u,\nabla u))+b(x)|u|^{\gamma-1}u=\mu\in L^1(\Omega) $$ on an arbitrary open subset $\Omega$ of $\mathbb{R}^N$ with $\hat a(x,u,\nabla u)$ a Carathéodory nonlinear function satisfying the general conditions of Leray-Lions where the coerciveness condition is weakened to allow degeneracies and becomes $$ \hat a(x,u,\xi)\cdot\xi\ge a(x)|\xi|^p,\ \forall u\in\mathbb{R},\ \forall\xi\in\mathbb{R}^N, \ \mbox{and $x$ a.e. in } \Omega, $$ with $p>1$ an arbitrary real number and $a$ an $L^1$-weight which might vanish or go to infinity on $S$, a closed subset of $\overline\Omega$ whose measure is zero. Here $b$ is an $L^1$-nonnegative function with properties similar to those of the weight $a$, $\gamma$ a positive number belonging to suitable intervals. For $p>1$ arbitrary and general weight $a$, we need new functional sets called "local T-sets" which are extensions of local Sobolev spaces. The "localization" is to handle the degeneracy. We get uniqueness and stability for $S$ satisfying a geometrical condition or $S$ and $a$ an analytic-geometrical one.

In this paper we study a class of parabolic inverse problems where unknown coefficients are assumed to be time-dependent only and the extra condition is given at a fixed interior or boundary point. Several local and global existence results are established.

This paper is concerned with a class of nonlinear elliptic Dirichlet problems approximating degenerate equations. If the degeneration set consists of $k$ connected components, by using variational methods, it is proved the existence of $k^{2}$ distinct nodal solutions, having exactly two nodal regions, whose positive and negative parts concentrate near subsets of the degeneration set.

We consider an elliptic-parabolic equation in divergence form $b(v)_t = \text{div}\alpha(v, Dv)+ f$ with Dirichlet boundary conditions and initial condition. Under rather general assumptions, we prove existence of mild solutions satisfying an $L^1$-comparison principle; under some additional conditions, these solutions are shown to be weak solutions. Moreover, under the general assumptions, uniqueness of integral solutions is established; under certain conditions, we show that weak solutions are integral solutions. The notions of mild and integral solutions are derived from nonlinear semigroup theory; by this approach, we extend and make precise former results on existence and uniqueness of weak solutions.

Let $\Omega$ be a piecewise-regular domain in $\Bbb R^N$ and O an irregular point on its boundary $\partial \Omega$. We study under what conditions on $q$ any solution $u$ of (E) $-\Delta u + g(x,u)=0$ where $g$ has a $q$-power-like growth at infinity ($q>1$) which coincides on $\partial \Omega \setminus {\{\text{O}}\}$ with a continuous function defined on whole $\partial \Omega$, can be extended as a continuous function in $\bar \Omega$.

This paper is concerned with the existence of positive solutions for boundary value problems of nonlinear elliptic systems which arise in the study of the Lotka-Volterra prey-predator model with cross-diffusion. Making use of the theory of the fixed point index we can derive sufficient conditions for the coexistence of positive steady states. Moreover, when cross-diffusion effects are comparatively small, we can get a necessary and sufficient condition for the coexistence. The uniqueness result is also given in the special case when the spatial dimension is one.