This paper deals with second order necessary conditions for optimal solution of optimal control problems governed by semilinear elliptic partial differential equations with pointwise state-control constraints.

In this paper, we study a semilinear elliptic equation defined on a bounded smooth domain. This type of problem arises from the study of spatial ecology model, and the growth function in the equation has a strong Allee effect and is inhomogeneous. We use variational methods to prove that the equation has at least two positive solutions for a large parameter if it satisfies some appropriate cond...

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...

Via the variational methods, we prove the existence of a nontrivial solution to a singular semilinear elliptic equation with critical Sobolev-Hardy exponent under certain conditions .

We introduce the explosion problem for semilinear elliptic and parabolic PDEs. The explosion problem is a question of existence of nonnegative solutions to −Lφ = λg(x, φ) and nonnegative global solutions to ∂tφ−Lφ = λg(x, t, φ). Under certain conditions there is a λ∗ > 0, called the explosion threshold, for which λ < λ∗ implies a solution exists, and λ > λ∗ implies no solution exists. In this p...

This paper is devoted to a complete classiication of the radial singular and possibly sign changing solutions of a semilinear elliptic equation with critical or subcritical growth.

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

This paper is devoted to a complete classiication of the radial singular and possibly sign changing solutions of a semilinear elliptic equation with critical or subcritical growth.

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...