We consider a semilinear elliptic equation on a smooth bounded domain Ω in R2, assuming that both the domain and the equation are invariant under reflections about one of the coordinate axes, say the y-axis. It is known that nonnegative solutions of the Dirichlet problem for such equations are symmetric about the axis, and, if strictly positive, they are also decreasing in x for x > 0. Our goal...

Abstract : We construct positive solutions of the semilinear elliptic problem ∆u + λu + up = 0 with Dirichet boundary conditions, in a bounded smooth domain Ω ⊂ RN (N ≥ 4), when the exponent p is supercritical and close enough to N+2 N−2 and the parameter λ ∈ R is small enough. As p → N+2 N−2 , the solutions have multiple blow up at finitely many points which are the critical points of a functi...

We consider a control constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of diffusegray conductive-radiative heat transfer. After stating first-order necessary conditions, second-order sufficient conditions are derived that account for strongly active sets. These conditions ensure local op...

This paper deals with the spectrum of a linear, weighted eigenvalue problem associated with a singular, second order, elliptic operator in a bounded domain, with Dirichlet boundary data. In particular, we analyze the existence and uniqueness of principal eigenvalues. As an application, we extend the usual concepts of linearization and Frechet derivability, and the method of sub and supersolutio...

The semilinear reaction-diffusion equation−ε4u+b(x, u) = 0 with Dirichlet boundary conditions is considered in a convex unbounded sector. The diffusion parameter ε is arbitrarily small, and the “reduced equation” b(x, u0(x)) = 0 may have multiple solutions. A formal asymptotic expansion for a possible solution u is constructed that involves boundary and corner layer functions. For this asymptot...

On a bounded smooth domain Ω ⊂ R we study solutions of a semilinear elliptic equation with an exponential nonlinearity and a Hardy potential depending on the distance to ∂Ω. We derive global a priori bounds of the Keller–Osserman type. Using a Phragmen–Lindelöf alternative for generalized sub and super-harmonic functions we discuss existence, nonexistence and uniqueness of so-called large solut...

An optimal control problem for a semilinear elliptic partial differential equation is discussed subject to pointwise control constraints on the control and the state. The main novelty of the paper is the presence of the L1-norm of the control as part of the objective functional that eventually leads to sparsity of the optimal control functions. Second-order sufficient optimality conditions are ...

In this paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are stable or more generally finite Morse index even locally stable.