The boundary value problem \[ A(x,D)u=f-g(x,u,...,D^{\beta}u)\text{ in }\Omega;\quad B_ j(x,D)u=\phi_ j\quad j=1,2,...,m\text{ in } \Gamma \] is considered where \(A(x,D)u\equiv P(D)u+Q(x,D)u\), with P an elliptic differential operator of order 2m with constant coefficients, Q(x,D) is a linear differential operator of order 2m at most with smooth compactly supported coefficients, \(D=(i\partial /\partial x_ 1,...,i\partial /\partial x_ n)\) and \(f\in L^ 2_ a({\mathbb{R}}^ n)=\{f\in L^ 2({\mathbb{R}}^ n):f(x)=0\) a.e. for \(| x| >a\}\). It is assumed that the dimension of the real zeros of the polynomial P is equal to (n-1) that grad\(P(\xi)\neq 0\) at real zeros of P and that the total curvature of the surface \(P(\xi)\) is nonzero at any point. The nonlinear function g satisfies the conditions; 1) g is continuously differentiable in \({\bar \Omega}\times {\mathbb{R}}^{n+N}\) (where N is the number of multi-indices \(\beta\) for which \(| \beta | \leq (2m-1)\); 2) \(g(x,u,...,D^{\beta}u...)=0\) if \(| x| >a\); 3) \((g(x,u,...,D^{\beta}u...) / (u,...,D^{\beta}u,...)\to 0\) uniformly with respect to x as \(| (u,...,D^{\beta}u,...)| \to \infty\); 4) the first derivatives of g are bounded. Finally \(\Omega \subset {\mathbb{R}}^ n\) is an unbounded domain with smooth boundary \(\Gamma \subset B_{\alpha}\) with \(B_{\alpha} = \{x\in {\mathbb{R}}^ n:| x| <\alpha \}.\)
Generalisations of the Sommerfeld radiation conditions are introduced for this class of problems. A theorem governing the solvability of this problem is then stated: solvability being settled on the assumption that the corresponding linear problem (i.e. \(g\equiv 0)\) has a solution. Theorems concerning the convergence of sequences of solutions of approximating problems are also stated.