Existence Of Solution Of Nonlocal-Kirchhoff Partial Differential Equation

In recent years, nonlocal equations, which widely apply to optimal control, physics, financial mathematics, oceanography, climatology and and so on, become a hot issue in the field of mathematics. As an important part of nonlocal problems, nonlocal Kirchhoff differ-ential equation arouse the attention of domestic and foreign scholars. In order to research equation's solution, we firstly think the equation of whether the equation should have solu-tion. In this paper, we mainly consider the influence of the parameters ? to the solution's existence of nonlocal Kirchhoff equations, while study the relationship between the size of the area, the first eigenvalue, and non-trivial solutions.According to the critical point theory of variational method, if we want to prove the question in this paper, firstly we can turn the equation's solution into the critical point of the energy functional I(u). When the parameter ? is positive, because the energy functional has a lower bound, taking advantage of the Sobolev space embedding theorem, norm weakly semi-continuous and Fatou Lemma, we will draw conclusions that energy functional exists critical points in space of H?/2; when the parameter ? is negative and p>3, we construct a new energy functional IT(u) through the nature of truncated function (?), meanwhile establishing the relationship between It(u) and I(u). According to the critical point theory, when the condition of (?)(?) D*u((?)D*u)dydx?1/? is satisfied, the energy functional have critical points. So a weak solution of the equation exists.