A second order elliptic scalar equation with a nonlinear condition on the boundary of a three-dimensional body \(\Omega \) is considered. The nonlinearity is represented by \(bd(x,u)\), where \(b\) is a small positive constant, and the growth of \(d\) as well as of its partial derivatives and their differences is, roughly speaking, bounded by a polynomial in \(u\) characterized by an exponent \(\gamma \). As a special case, the boundary value problem (BVP) covers a time independent heat flow with a heat radiation boundary condition.
To prove the existence of a solution, the authors derive a fixed point problem from the BVP and show that the Schauder fixed point theorem is applicable if an auxiliary Sobolev space dependent on \(\gamma \) is introduced.
The existence theorem says that for any \(\gamma \in [1,\infty )\) a positive number \(b_0\) dependent on a linear part of the boundary condition exists such that the BVP has a solution \(u\in W^{2,q}(\Omega )\), \(q>3\gamma /(\gamma +1)\), for any \(b\) between 0 and \(b_0\).