In this paper we introduce and analyze some non-overlapping multiplicative Schwarz methods for discontinuous Galerkin (DG) approximations of elliptic problems. The construction of the Schwarz preconditioners is presented in a unified framework for a wide class of DG methods. For symmetric DG approximations we provide optimal convergence bounds for the corresponding error propagation operator, and we show that the resulting methods can be accelerated by using suitable Krylov space solvers. A discussion...

In this paper we are concerned with questions of multiplicity and concentration behavior of positive solutions of the elliptic problem$$\left(P_{}\right)\left\{\begin{array}{c}\hfill {ℒ}_{}u=f\left(u\right)\text{in}{ℝ}^3,\\ \hfill u\>0\text{in}{ℝ}^3,\\ \hfill u\in H^1\left({ℝ}^3\right),\end{array}\right.$$( P ε ) ℒ ε u = f ( u ) in IR 3 , u > 0 in IR 3 , u ∈ H 1 ( IR 3 ) , whereε is a small positive parameter, f : ℝ → ℝ is a continuous function,$${ℒ}_{}$$ℒ ε is a nonlocal operator defined by$${ℒ}_{}u=M\left(\frac{1}{}{\int }_{{ℝ}^3}{\left|\nabla u\right|}^2+\frac{1}{{}^3}{\int }_{{ℝ}^3}V\left(x\right)u^2\right)\left[{-}^2\Delta u+V\left(x\right)u\right],$$ℒ ε u = M 1 ε ∫ IR 3 | ∇ u | 2 + 1 ε 3 ∫ IR 3 V ( x ) u 2 [ − ε 2 Δ u + V ( x ) u ] ,M : IR+ → IR+ and V : IR3 → IR are continuous functions which verify some hypotheses.

We prove two explicit bounds for the multiplicities of Steklov eigenvalues ${\sigma }_k$ on compact surfaces with boundary. One of the bounds depends only on the genus of a surface and the index $k$ of an eigenvalue, while the other depends as well on the number of boundary components. We also show that on any given Riemannian surface with smooth boundary the multiplicities of Steklov eigenvalues ${\sigma }_k$ are uniformly bounded in $k$.

We discuss the existence and multiplicity of positive solutions for a class of second order quasilinear equations. To obtain our results we will use the Ekeland variational principle and the Mountain Pass Theorem.

We show that, under appropriate structure conditions, the quasilinear Dirichlet problem $$\left\{\begin{array}{ccc}-div\left(\right|\nabla u{|}^{p-2}\nabla u)=f(x,u),\hfill & x\in \Omega ,u=0,\hfill & x\in \partial \Omega ,\end{array}\right.$$ where $\Omega$is a bounded domain in ${ℝ}^n$, $1<p<+\infty$, admits two positive solutions $u_0$, $u_1$ in $W_0^{1,p}\left(\Omega \right)$ such that $0<u_0\le u_1$ in $\Omega$, while $u_0$ is a local minimizer of the associated Euler-Lagrange functional.