By a perturbation method and constructing comparison functions, we show the exact asymptotic behaviour of solutions to the semilinear elliptic problem ∆u− |∇u| = b(x)g(u), u > 0 in Ω, u ̨̨ ∂Ω = +∞, where Ω is a bounded domain in RN with smooth boundary, q ∈ (1, 2], g ∈ C[0,∞) ∩ C1(0,∞), g(0) = 0, g is increasing on [0,∞), and b is non-negative non-trivial in Ω, which may be singular or vanishing ...

Semilinear elliptic optimal control problems involving the L norm of the control in the objective are considered. Necessary and sufficient second-order optimality conditions are derived. A priori finite element error estimates for three different discretizations for the control problem are given. These discretizations differ in the use of piecewise constant, piecewise linear and continuous or n...

We show that in dimension two or greater, a certain equivalence class of the scalar coefficient a(x, u) of the semilinear elliptic equation ∆u + a(x, u) = 0 is uniquely determined by the Dirichlet to Neumann map of the equation on a bounded domain with smooth boundary. We also show that the coefficient a(x, u) can be determined by the Dirichlet to Neumann map under some additional hypotheses.

In this paper we prove the existence and uniqueness of the boundary layer solution to a semilinear eigenvalue problem consisting of a coupled system of two elliptic partial differential equations. Although the system is not quasimonotone, there exists a transformation to a quasimonotone system. For the transformed system we may and will use maximum (sweeping) principle arguments to derive point...

Abstract. We study the Cauchy problem for a semilinear stochastic partial differential equation driven by a finite-dimensional Wiener process. In particular, under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives, we consider the equation in the context of power scale generated by a strongly elliptic differential operator. Application of semigroup ar...

Uniqueness of positive radial solutions decaying at infinity is proved for a class of semilinear elliptic equations on R. Complementary results for the same kind of equations were obtained in the early 90’s, on R with N ≥ 3, and in finite balls of R with N ≥ 2. The new result presented here plays a crucial role in the global bifurcation problem, previously studied by the author.

Abstract. Semilinear elliptic optimal control problems involving the L1 norm of the control in the objective are considered. Necessary and sufficient second-order optimality conditions are derived. A priori finite element error estimates for piecewise constant discretizations for the control and piecewise linear discretizations of the state are shown. Error estimates for the variational discret...

In this paper we establish the existence of multiple solutions for the semilinear elliptic problem (1.1) −∆u = g(x, u) in Ω, u = 0 on ∂Ω, where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω, a function g: Ω×R→ R is of class C1 such that g(x, 0) = 0 and which is asymptotically linear at infinity. We considered both cases, resonant and nonresonant. We use critical groups to distinguish the c...

where Ω ⊂ R is a bounded smooth domain, f : Ω × R ® R and M : R ® R are continuous functions. The existence and multiplicity results for Equation (1) are considered in [1-3] by using variational methods and fixed point theorems in cones of ordered Banach space with space dimension is one. On the other hand, The four-order semilinear elliptic problem { 2u + c u = f (x, u), in , u = u = 0, on ∂ ,...

In this paper, we investigate residual-based a posteriori error estimates for the hp finite element approximation of semilinear Neumann boundary elliptic optimal control problems. By using the hp finite element approximation for both the state and the co-state and the hp discontinuous Galerkin finite element approximation for the control, we derive a posteriori error bounds in L2-H1 norms for t...