Multiple solutions for a class of non-local problems for semilinear elliptic equations (Q1318874)

We investigate the solvability of the following non-local problem for the semilinear elliptic equation \[ \begin{aligned} Lu=- \sum_{i,j=1}^ n a_{ij} (x) D_{ij} u+ \sum_{i=1}^ n b_ i(x) D_ i u+ c(x)u= g(x,u)+ t\Phi (x)+ f(x) &\quad \text{in } Q,\\ u(x)- \beta(x) u(\varphi (x)) =0 &\quad \text{on } \partial Q, \end{aligned} \tag \(N_ t\) \] in a bounded domain \(Q\) with a smooth boundary \(\partial Q\), where \(\varphi: \partial Q\to Q\) and \(\beta: \partial Q\to \mathbb{R}\) are given functions and \(t\) is real parameter. The problem of this type is often referred to as the boundary value problem with the Bitsadze-Samarskii condition. The most characteristic feature of a non-local problem is that the boundary condition relates values of a solution on the boundary to its values on some parts of the interior of the region. The main purpose of this article is to prove the existence result for the problem \((N_ t)\) under the assumption of the Ambrosetti-Prodi type.