Study of a nonlinear Kirchhoff equation with non-homogeneous material

Abstract

In this paper we study a non-homogeneous elliptic Kirchhoff equation with nonlinear reaction term. We analyze the existence and uniqueness of positive solution. The main novelty is the inclusion of non-homogeneous term making the problem without a variational structure. We use mainly bifurcation arguments to get the results.

An optimal control problem for a Kirchhoff-type equation

“…Nonlocal problems start out in the papers of Kirchhoff, [8], where he introduced his celebrated equation There are a relevant range of situations in Physics and Technology where problems with nonlocal terms appear, see for instance [1,2,4,10,14] and the survey [13]. In this paper we study an optimal control problem associated to (1.2) in the particular case M (x, r) = a(x)+b(x)r where a(x), b(x) are positive continuous functions.…”

“…Let us stress that since a and b are not constants the problem does not have a variational structure, as a consequence, the knowledge for this problem is very limited. In [15] it is proved that (1.2) has a solution in H 1 0 (Ω), see also [4] for the uniqueness under the restriction f ≥ 0.…”

Existence of least energy nodal solution with two nodal domains for a generalized Kirchhoff problem in an Orlicz–Sobolev space

“…The reader may consult , , , and the references therein, for more physical motivation on Kirchhoff problem.…”

“…is called Kirchhoff type because of the presence of the term M |∇u| 2 dx . Indeed, this operator appears in the The reader may consult [2], [3], [25], [39] and the references therein, for more physical motivation on Kirchhoff problem.…”

A sub-supersolution approach for a quasilinear Kirchhoff equation

“…Figueiredo, Morales-Rodrigo, Santos Junior & Suárez [7] consider a problem whose equation is of the form…”