We prove existence and multiplicity of symmetric solutions for the Schrödinger--Newton system in three-dimensional space using equivariant Morse theory.

We study a hierarchy of electrical conduction problems in biological tissues. These problems are set in a finely mixed periodic medium, and the unknown electric potentials solve standard elliptic equations set in different conductive regions (the intracellular and extracellular spaces), separated by an interface (the cell membranes), which exhibits a capacitive and a nonlinear conductive behavior, due to its biochemical structure. Different scalings in the interface condition correspond to different problems in the hierarchy. As the spatial period of the medium goes to zero, the problems approach a homogenization limit depending on the initial scaling. The macroscopic models are obtained by using the technique of two-scale convergence.

In this paper we show approximation procedures for studying singular elliptic problems whose model is $$ \begin{cases} - \Delta u= b(u)|\nabla u|^2+f(x), & \text{in } \; \Omega;\\ u = 0, & \text{on } \; \partial \Omega; \end{cases} $$ where $b(u)$ is singular in the $u$-variable at $u=0$, and $f \in L^m (\Omega)$, with $m>\frac N2$, is a function that does not have a constant sign. We will give an overview of the landscape that occurs when different problems (classified according to the sign of $b(s)$) are considered. So, in each case and using different methods, we will obtain a priori estimates, prove the convergence of the approximate solutions, and show some regularity properties of the limit.

The Darcy flow problem in a porous medium in an infinite cylinder is looked at as a two-parameter limit problem, in terms of the characteristic pore size and the cylinder length. As the characteristic pore size tends to zero, the Stokes problem on the finite cylinder converges to a Darcy problem, and the Darcy problem in the infinite cylinder is obtained as its limit when the length of the cylinder goes to infinity. But one could do this in the opposite order: first consider the limit of the Stokes problem in an infinite cylinder and then consider the homogenized limit to obtain Darcy in an infinite cylinder. Would these two procedures yield the same result? In other words do the limits commute? The answer is shown to be affirmative.

The paper deals with the existence of bounded positive solutions for a quasilinear singular problem with nonlinear Robin conditions. The nonlinear term is singular with respect to the solution and has a quadratic growth with respect to its gradient. The result is obtained by approximation. We define an approximate problem which does not present the singularity and is bounded with respect to the gradient. In a first step we prove the existence of a bounded solution of this problem, using the Schauder fixed-point theorem. Then, we prove uniform a priori estimates for these solutions. Finally, by some equintegrability arguments, we pass to the limit in the approximate problem and show that the approximate solution converges to a solution of our problem. We also prove a strong-maximum property for the problem. In the last part of the paper, we consider the case of a nonsingular but more general quadratic growth with nonlinear Robin conditions.

The existence of infinitely many radially symmetric weak solutions for non-autonomous elliptic problems involving the $p$-Laplacian in the Euclidan space $\mathbb{R}^N$ is investigated. The approach is based on variational method. A main ingredient of proof is the famous symmetric critically principle of Palais. A concrete example of an application is pointed out.

A parabolic transmission problem with Ventcel'-type boundary conditions on a prefractal mixture Koch-type interface is studied. Regularity results for the solution are proved. The numerical approximation of the problem is considered, and optimal a priori error estimates of the order of convergence are obtained.

Homogenization results for an insulating fractal surface $S$ of Koch type are proved.

An application of dimensional reduction results for gradient constrained problems is provided for 3D-2D dimension reduction for supremal functionals, in the case when the domain is convex.

The Dirichlet problem for a class of quasilinear elliptic systems of equations with small-BMO coefficients in a Reifenberg-flat domain $\Omega$ is considered. The lower-order terms are supposed to satisfy controlled growth conditions in ${\mathbf u}$ and $D\mathbf {u}$. $L^p$-integrability with $p>2$ of $D{\mathbf u}$ is obtained, where $p$ depends explicitly on the data. An analogous result is obtained also for the Cauchy--Dirichlet problem for quasilinear parabolic systems.

Let $\Omega$ be a domain of ${\mathbb R}^N$, $N>2.$ We establish higher integrability for solutions $u \in W^{1,p}_{\text{loc}}(\Omega)$ of nonlinear PDEs whose prototype~is \begin{equation*} \text{div\,}[|\nabla u|^{p-2}\nabla u +B(x)|u|^{p-2}u]=\text{div\,}(|F|^{p-2}F) \end{equation*} with $1 < p < N$. We assume that the vector field $B\colon\Omega\to{\mathbb R}^N$ belongs to the Marcinkiewicz space $L^{\frac N{p-1},\infty}$. We prove that $F\in L^r\text{loc}(\Omega,{\mathbb R}^N)$, $p < r < N$, implies $u\in L^{r^*} \text{loc}(\Omega).$

The existence and uniqueness of solution to a one-dimensional hyperbolic problem arising in viscoelasticity is here considered. Specifically, the case of a singular kernel is analyzed. This choice is suggested by applications according to the literature to model a wider variety of materials.

The central theme of this paper is the study of $G$-convergence of elliptic operators in the plane. We consider the operator $$ \mathcal{M}[u]=\text{Tr}(A(z) D^2u)=a_{11}(z)u_{xx}+2a_{12}(z)u_{xy}+a_{22}(z)u_{yy} $$ and its formal adjoint $$ \mathcal{N}[v]=D^2(A(w)v)= (a_{11}(w)v)_{xx} + 2(a_{12}(w)v)_{xy}+ (a_{22}(w)v)_{yy}, $$ where $u\in W^{2,p}$ and $v\in L^p$, with $p>1$, and $A$ is a symmetric uniformly bounded elliptic matrix such that $\text{det}A=1$ almost everywhere. We generalize a theorem due to Sirazhudinov--Zhikov, which is a counterpart of the Div-Curl lemma for elliptic operators in non-divergence form. As an application, under suitable assumptions, we characterize the $G$-limit of a sequence of elliptic operators. In the last section we consider elliptic matrices whose coefficients are also in $VMO$; this leads us to extend our result to any exponent $p\in (1,2)$.

The present note deals with the continuity property for the Jacobian determinant of orientation preserving maps belonging to the Grand Sobolev space $W^{1,n)}(\Omega, {\mathbb{R}}^n)$.

An existence result for a critical point of mountain-pass type, where the classical Palais--Smale condition is not required, is presented. A multiple-critical-point result is then obtained. As an application, the existence of two positive classical solutions for two-point boundary-value problems, without assuming any asymptotic condition on the nonlinearity, is obtained.

The existence of infinitely many solutions for differential inclusions depending on two positive parameters and involving the $p$-biharmonic operator is established via variational methods.

The problem $$ \hskip 60pt u_{rr}+f(u)=0,\quad u_r(0)=0,\quad u(b)=0 \hskip 60pt {(P)} $$ arises when considering radially symmetric solutions of the problem $$ \Delta u+\varphi(u)=0,\quad u=0, \quad x\in\partial\Omega . $$ The problem (P) is solvable if the upper and lower functions $\alpha$ and $\beta$ exist. Then there exist solutions $u^*$ and $u_*$ (the maximal and minimal solutions), which can be approximated by monotone sequences of solutions of equation (P). Mostly solutions of (P) are of oscillatory types, and they cannot be approximated by monotone sequences. In this article we provide results on non-monotone approximations of solutions of the problem (P).

In Part I we construct an upper bound, in the spirit of $\Gamma$- $\limsup$, achieved by multidimensional profiles, for some general classes of singular perturbation problems, with or without the prescribed differential constraint, taking the form \begin{equation*} \begin{split} E_{\varepsilon}(v):=\int_\Omega \frac{1}{{\varepsilon}}F\big({\varepsilon}^n\nabla^n v,\dots, {\varepsilon}\nabla v,v\big)\,dx\; \forall v:\Omega\subset \mathbb R ^N\to \mathbb R ^k\; \text{s.t.}\; A\cdot\nabla v=0, \end{split} \end{equation*} where the function $F\geq 0$ and $A: \mathbb R ^{k\times N}\to \mathbb R ^m$ is a prescribed linear operator (for example, $A:\equiv 0$, $A\cdot\nabla v:=\operatorname{curl} v$, and $A\cdot\nabla v=\text{div } v$) which includes, in particular, the problems considered in [17]. This bound is in general sharper then the one obtained in [17]. v$) which includes, in particular, the problems considered in [17]. This bound is in general sharper then the one obtained in [17].