A resonance phenomenon for ground states of an elliptic equation of Emden--Fowler type.
The author studies the existence of radial solutions of the elliptic equation \[ \Delta u+u^p+u^q=0\tag{1} \] in \(\mathbb R^N\), \(N\geq 3\), with \(0<u(x)\to+\infty\) as \(| x| \to+\infty\). By \textit{C. S. Lin} and \textit{W. M. Ni} [Proc. Am. Math. Soc. 102, 271--277 (1988; Zbl 0652.35085)] it was proved that if \(1<p<\frac{N+2}{N-2}<q\) and \(q=2p-1\), then there exists an explicit radial solution of the form \(u(r)=(\frac{A}{B+r^2})^{\frac{1}{p-1}}\). In the present paper, this result is generalized. It is proved that if the range of \(p\) is further restricted to \(p>\frac{N+2\sqrt{N-1}}{N+2\sqrt{N-1}-4}\), then for \(q=2p-1\), equation (1) has infinitely many radial solutions with fast decay \(O(r^{2-N})\) besides the explicit solution. If \(q\) is close to \(2p-1\), then equation (1) also has a large number of such solutions. It is known that the existence of positive radial solutions of (1) is equivalent to the existence of solutions of the ordinary differential equation \[ u''+\frac{N-1}{r}u'+u_+^p+u_+^q=0 \] with \(u'(0)=0\), \(0<u(r)\to+\infty\) where \(u_+=\max\{u, 0\}\). This equation can be further reduced to the second-order differential equation \[ x''-\alpha x'+x_+^p+e^{-\gamma t}x_+^q-\beta x=0 \] by using the classical transformation \(x(t)=r^{\frac{2}{p-1}}u(r)| _{r=e^t}\). On the basis of this preparation, the invariant manifold theory is used to prove the existence of solutions \(x(t)\) satisfying \(x(t)\to 0\) as \(t\to\pm\infty\).
scientific article; zbMATH DE number 2062712
scientific article