Abstract. This paper is devoted to the study of resonant nonlinear boundary value problems with Neumann boundary condition. First we consider the linear situation doing a careful analysis on the existence of nontrivial solutions. This analysis involves Liapunov-type inequalities with the L p − norm of the coefficient function for 1 p ∞ . We carry out a complete treatment of the problem for any constant p 1. Then, this is combined with Schauder fixed point theorem to obtain new results about the existence and uniqueness of solutions for resonant nonlinear problems.Mathematics subject classification (2000): 34B15, 34B05.
This paper is devoted to the study of L p Lyapunov-type inequalities (1 p +∞) for linear partial differential equations. More precisely, we treat the case of Neumann boundary conditions on bounded and regular domains in R N . It is proved that the relation between the quantities p and N/2 plays a crucial role. This fact shows a deep difference with respect to the ordinary case. The linear study is combined with Schauder fixed point theorem to provide new conditions about the existence and uniqueness of solutions for resonant nonlinear problems.
An optimal control problem for a nonlinear elliptic equation of logistic type is considered. Under certain assumptions, the existence of at least an optimal control is shown and an optimality system is derived. Then this system is used for proving the uniqueness of and a constructive approximation to the optimal control.
Abstract. An optimal control problem for a nonlinear elliptic equation of logistic type is considered. Under certain assumptions, the existence of at least an optimal control is shown and an optimality system is derived. Then this system is used for proving the uniqueness of and a constructive approximation to the optimal control.
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