Semilinear boundary value problems for unbounded domains
The Dirichlet problem for the equation (1) \(Au=f(x,u)\), \(x\in \Omega\), where A is a non-negative self adjoint elliptic operator of order m and \(\Omega\) is a domain in \(R^ n\), is considered. It is proved that (1) has a solution in \(\overset\circ W^ m_ 2\) provided the right-hand side of (1) satisfies certain conditions of growth which in particular include 1) \(f(x,u)=V(x)e^ u(W(x)\cos e^ u-1)\), \(V\in L_ 1\), \(W\in L_{\infty}\), \(V\geq 0;\) 2) \(f(x,u)=W(x)-V(x)ue^{u^ 2}\), \(W\in L_ t\), \(1/2\leq 1/t\leq 1/2+m/2;\) 3) \(f(x,u)=V(x)[W(x)e^ k \sin u^{k+1}-\sinh u+1]\) with V and W the same as in 1). Similar theorems are proved for the equation \(Au= \lambda f(x,u)\) for some \(\lambda \in (0,\lambda_ 0)\).
scientific article; zbMATH DE number 4021729