We investigate here the properties of extremal solutions for semilinear elliptic equation $-\Delta u=\lambda f\left(u\right)$ posed on a bounded smooth domain of ${ℝ}^n$ with Dirichlet boundary condition and with $f$ exploding at a finite positive value $a$.

We will be concerned with the existence result of a degenerate elliptic unilateral problem of the form $Au+H\left(x,u,\nabla u\right)=f$, where $A$ is a Leray-Lions operator from $W^{1,p}\left(\Omega ,w\right)$ into its dual. On the nonlinear lower-order term $H\left(x,u,\nabla u\right)$, we assume that it is a Carathéodory function having natural growth with respect to $\left|\nabla u\right|$, but without assuming the sign condition. The right-hand side $f$ belongs to $L^1\left(\Omega \right)$.

We prove the existence and uniqueness of a strong solution for a linear third-order equation with integral boundary conditions. The proof uses energy inequalities and the density of the range of the operator generated.

It is the aim of this paper to show how the classical theory, based on fundamental solutions and explicit representations, via special functions can be combined with the functional analytical approach to partial differential equations, to produce semiclassical representation formulae for the solution of equations in cylinder-like domains.

We prove that the moduli of $U$-convexity, introduced by Gao (1995), of the ultrapower $\tilde{X}$ of a Banach space $X$ and of $X$ itself coincide whenever $X$ is super-reflexive. As a consequence, some known results have been proved and improved. More precisely, we prove that $u_X\left(1\right)>0$ implies that both $X$ and the dual space $X^{\ast }$ of $X$ have uniform normal structure and hence the “worth” property in Corollary 7 of Mazcuñán-Navarro (2003) can be discarded.

We devote this paper to quasiautonomous second-order differential equations in Hilbert spaces governed by maximal monotone operators. Some bilocal boundary conditions are associated. We discuss the continuous dependence of the solution both on the operator and on the boundary values. One uses the methods of nonlinear analysis. Some applications to internal approximate schemes are given.