We consider the semilinear elliptic problem −Δu + u = λK(x)up + f (x) in Ω, u > 0 in Ω, u ∈ H 0 (Ω), where λ ≥ 0, N ≥ 3, 1 < p < (N + 2)/(N − 2), and Ω is an exterior strip domain in RN . Under some suitable conditions on K(x) and f (x), we show that there exists a positive constant λ∗ such that the above semilinear elliptic problem has at least two solutions if λ∈ (0,λ∗), a unique positive sol...

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...

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.

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 .

In this paper, we study the combined effect of concave and convex nonlinearities on the number of solutions for an indefinite semilinear elliptic system ðEk;lÞ involving critical exponents and sign-changing weight functions. Using Nehari manifold, the system is proved to have at least two nontrivial nonnegative solutions when the pair of the parameters ðk;lÞ belongs to a certain subset of R. 20...

We prove existence results for the Dirichlet problem associated with an elliptic semilinear second-order equation of divergence form. Degeneracy in the ellipticity condition is allowed.

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.