Positive solutions of semilinear elliptic problems in unbounded domains
In this paper boundary value problems of the type \(Lu=f(x,u)\) in D, \(u=0\) on \(\partial D\) are considered where L is a second order uniformly elliptic differential operator in divergence form, f(x,t) has polynomial growth in t, and D is unbounded. By means of variational methods a sequence of positive solutions \(u_ n\) is constructed in bounded subdomains \(D_ n\subset D\), vanishing on \(\partial D_ n\). By compactness arguments it is then shown that there exists a subsequence converging, locally uniformly in \(C^ 2(D)\), to a positive solution in the whole domain D. The behaviour of this solution as \(| x| \to \infty\) is also studied.
second order uniformly elliptic differential operator
scientific article; zbMATH DE number 3933606
scientific article