Abstract
Abstract We discuss, by topological methods, the solvability of systems of second-order elliptic differential equations subject to functional boundary conditions under the presence of gradient terms in the nonlinearities. We prove the existence of nonnegative solutions and provide a non-existence result. We present some examples to illustrate the applicability of the existence and non-existence results.
Highlights
We prove the existence of nonnegative solutions and provide a non-existence result
We present some examples to illustrate the applicability of the existence and non-existence results
In this paper we study the solvability of a system of second-order elliptic differential equations subject to functional boundary conditions (BCs for short)
Summary
In this paper we study the solvability of a system of second-order elliptic differential equations subject to functional boundary conditions (BCs for short). A peculiarity of system (1.1) is the dependence on the gradient of the solutions, both in the nonlinearity and in the functionals occurring in the BCs, and this represents the main technical difficulty that we have to deal with in this paper For this purpose, we have to perform a preliminary study of the Green’s function of the partial differential operators which occur in (1.1). Speaking, these estimates yield the a priori bounds needed to compute the fixed point index in suitable cones of nonnegative functions.
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