Nonnegative entire solutions of a class of degenerate semilinear elliptic equations (Q1813443)

This paper is concerned with the existence and qualitative behavior of nonnegative entire solutions of the degenerate elliptic equation \[ \Delta(u^ m)+u(1-u)(u-a)=0,\quad x\in R^ n,\quad, n\geq 2,\tag{*} \] where \(m\) and \(a\) are positive constants. By a radial entire solution of \((*)\) is meant a function \(u\in C(R^ n)\) depending only on \(| x|\) such that \(u^ m\in C^ 2(R^ n)\) and that \((*)\) is satisfied at every point of \(R^ n\). The one-dimensional case of \((*)\) has been studied by \textit{D. Aronson}, \textit{M. G. Crandall} and \textit{L. A. Peletier} [Nonlinear Anal., Theory Methods Appl. 6, 1001-1022 (1982; Zbl 0518.35050)], who have shown, among other things, that \((*)\) \((n=1)\) has nonnegative radial entire solutions \(u\) with compact support provided \(m>1\) and \(0<a<(m+1)/(m+3)\). Our purpose here is to extend some of the results of loc. cit. to the higher dimensional case \((n\geq 2)\) of \((*)\) by proving the Theorem: Let \(0<a<(m+1)/(m+3)\). Then, there exists a constant \(u_ *\in(0,1)\) such that \((*)\) has a nonnegative radial entire solution \(u(x)\) satisfying \(u(0)=u_ 0\) if \(0<u_ 0\leq u_ *\), and \((*)\) has no nonnegative entire solution \(u(x)\) satisfying \(u(0)=u_ 0\) if \(u_ *<u_ 0<1\). Furthermore, the following statements hold. (i) If \(0<u_ 0<u_ *\), the radial entire solution \(u(x)\) satisfying \(u(0)=u_ 0\) oscillates around \(a\) and converges to \(a\) as \(| x|\to\infty\). (ii) The radial entire solution \(u(x)\) satisfying \(u(0)=u_ *\) decreases monotonically to zero as \(| x|\to\infty\). This solution has compact support if \(m>1\).