Decaying solutions of semilinear elliptic equations in \(R^ N\). (English) Zbl 0696.35051

Noussair, Ezzat S.; Swanson, Charles A.

SIAM J. Math. Anal. 20, No. 6, 1336-1343 (1989).

The authors study existence and asymptotic behavior of positive classical solutions of the semilinear elliptic equation \(-\Delta u+bu=\lambda pf(u)\) in \({\mathbb{R}}^ N\), \(N\geq 2\), where b and p are radially symmetric, bounded and locally Hölder continuous functions in \({\mathbb{R}}^ N\), \(b\geq 0\), \(p>0\), f: \({\mathbb{R}}^+\to {\mathbb{R}}\) is locally Lipschitz continuous, \(f(t)>0\) iff \(0<t<T\), \(f(t)=O(t^{\gamma})\) as \(t\downarrow 0\), \(\gamma >1\), and the solutions are sought in the class of functions which decay uniformly to zero at \(\infty\). The existence of a solution pair \((\lambda,u_{\lambda})\) and an asymptotic estimate for \(u_{\lambda}\) is proved by a direct variational approach if \(b(r)\geq b_ 0>0\), and by an approximation procedure (in which the function b(r) is approximated by \(b(r)+1/k)\) if b(r)\(\geq 0\). The more important latter case is proved under the additional hypothesis \(N\geq 3\) and \(p(r)=O(r^{-a})\) as \(r\to \infty\), where \(\gamma >(N-2a+2)/(N-2)\).

Reviewer: P.Quittner

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35P30	Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
35B40	Asymptotic behavior of solutions to PDEs
35J20	Variational methods for second-order elliptic equations

Keywords:

positive classical solutions; semilinear; asymptotic estimate