We develop sufficient optimality conditions for a Moreau-Yosida regularized optimal control problem governed by a semilinear elliptic PDE with pointwise constraints on the state and the control. We make use of the equivalence of a setting of Moreau-Yosida regularization to a special setting of the virtual control concept, for which standard second order sufficient conditions have been shown. Mo...

1.1. Topological derivatives in shape optimization. Topological derivatives are introduced for linear problems in (Sokolowski and Zochowski, 1999) and for variational inequalities in (Sokolowski and Zochowski, 2005). The mathematical theory of asymptotic analysis is applied in (Nazarov and Sokolowski, 2003; 2006) for the derivation of topological derivatives in shape optimization of elliptic bo...

Reaction–diffusion systems are used to model many chemical and biological phenomena in the natural world [25, 26], and systems of coupled partial differential equations are also used in other physical models such as nonlinear Schrödinger systems in multi-component Bose–Einstein condensates and nonlinear optics [19, 23]. The steady-state solutions or standing-wave solutions of such systems of no...

The numerical computation of solitary waves to semilinear elliptic equations in innnite cylinders is investigated. Rather than solving on the innnite cylinder, the equation is approximated by a boundary-value problem on a nite cylinder. Convergence and stability results for this algorithm are given. In addition, it is shown that Galerkin approximations can be used to calculate solitary waves fo...

Abstract: The paper deals with optimal control problems for semilinear elliptic and parabolic PDEs subject to pointwise state constraints. The main issue is that the controls are taken from a restricted control space. In the parabolic case, they are R -vectorvalued functions of time, while they are vectors of R in elliptic problems. Under natural assumptions, firstand second-order sufficient op...

this study concerns the existence and multiplicity of positive weak solutions for a class ofsemilinear elliptic systems with nonlinear boundary conditions. our results is depending onthe local minimization method on the nehari manifold and some variational techniques. alsoby using mountain pass lemma, we establish the existence of at least one solution withpositive energy.

In this paper, we consider some semilinear elliptic equations with asymptotic linear nonlinearity and show the existence, uniqueness, and asymptotic behavior of these solutions.

Abstract We consider the inverse problem of determining some class nonlinear terms appearing in an elliptic equation from boundary measurements. More precisely, we study stability issue for this problems. Under suitable assumptions, prove a Lipschitz and Hölder estimate associated with determination quasilinear semilinear equations measurements restricted to arbitrary part domain. Besides their...

The paper deals with optimal control problems for semilinear elliptic and parabolic PDEs subject to pointwise state constraints. The main issue is that the controls are taken from a restricted control space. In the parabolic case, they are Rm -vector-valued functions of the time, while the are vectors of Rm in elliptic problems. Under natural assumptions, firstand second-order sufficient optima...

Let Ω be a domain in RN , N ≥ 1, and 2∗ = ∞ if N = 1, 2, 2∗ = 2N N−2 if N > 2, 2 < p < 2 ∗. Consider the semilinear elliptic problem −∆u+ u = |u|p−2u in Ω; u ∈ H 0 (Ω). Let H1 0 (Ω) be the Sobolev space in Ω. The existence, the nonexistence, and the multiplicity of positive solutions are affected by the geometry and the topology of the domain Ω. The existence, the nonexistence, and the multipli...