We study the semilinear problem with the boundary reaction $$-\Delta u+u=0\text{in}\Omega ,\frac{\partial u}{\partial \nu }=\lambda f\left(u\right)\text{on}\partial \Omega ,$$ where $\Omega \subset {ℝ}^N$, $N\ge 2$, is a smooth bounded domain, $f:[0,\infty )\to (0,\infty )$ is a smooth, strictly positive, convex, increasing function which is superlinear at $\infty$, and $\lambda >0$ is a parameter. It is known that there exists an extremal parameter ${\lambda }^{*}>0$ such that a classical minimal solution exists for $\lambda <{\lambda }^{*}$, and there is no solution for $\lambda >{\lambda }^{*}$. Moreover, there is a unique weak solution $u^{*}$ corresponding to the parameter $\lambda ={\lambda }^{*}$. In this paper, we continue to study the spectral properties of $u^{*}$ and show...

In this paper, two algorithms are proposed to solve systems of algebraic equations generated by a discretization procedure of the weak formulation of boundary value problems for systems of nonlinear elliptic equations. The first algorithm, Newton-CG-MG, is suitable for systems with gradient mappings, while the second, Newton-CE-MG, can be applied to more general systems. Convergence theorems are proved and application to the semiconductor device modelling is described.

Si studiano soluzioni positive dell’equazione $-{ϵ}^2\mathrm{\Delta }u+u=u^p$ in $\mathrm{\Omega }$, dove $\mathrm{\Omega }\subseteq {\mathbb{R}}^n$ , $p>1$ ed $ϵ$ è un piccolo parametro positivo. Si impongono in genere condizioni al bordo di Neumann. Quando $ϵ$ tende a zero, dimostriamo esistenza di soluzioni che si concentrano su curve o varietà.