Radial solutions of semilinear elliptic equations with prescribed asymptotic behavior

摘要

We consider semilinear elliptic equations of the form Δu + g(u) = 0 on R~N. Given a suitable positive root z of g, we discuss the existence of positive radial solutions u with u(∞) = z. Let N ≥ 2. We discuss the existence of radial solutions u of the problem Δu + g(u) = 0 in R~N, (1.1) with a prescribed limit z = u(∞) > 0 satisfying g(z) = 0. Radial solutions of (1.1) have been studied in connection with many questions of mathematical physics like the existence of solitary waves and the nonlinear field equations. The idea of having symmetric solutions for semilinear elliptic equations on general symmetric domains is now well understood by the moving plane method (see [3, 6]). If, for example, we consider a ball B = B_R(0), a suitable nonlinearity f and a sufficiently regular positive solution u of {Δu + f(u) = 0 in B, /u = 0 on {partial}B, (1.2) then u is radially symmetric u = u(r = |x|) and u'(r) ≤ 0. The proof of this result has the maximum principle for elliptic equations as its principal ingredient. When the radius R → ∞, it then becomes natural to ask about the existence of radially symmetric solutions of (1.1) that vanish at infinity. This question attracted a lot of attention (see [1, 2, 4, 5, 8, 10] and the references therein) where variational and topological methods have been used in order to establish the existence of positive radial solutions of (1.1) with u(∞) = 0. However, and up to our knowledge, none of the aforementioned works investigates the case with a prescribed limit different from 0.