A resonance phenomenon for ground states of an elliptic equation of Emden--Fowler type.
type: Item
instance of: scholarly article
title: A resonance phenomenon for ground states of an elliptic equation of Emden--Fowler type.
zbMATH Open document ID: 1055.34031
DOI: 10.1016/S0022-0396(02)00015-3
author: Isabel Flores
published in: Journal of Differential Equations
publication date: 2004-03-29T00:00:00Z
review text: 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 \(t\to\pm\infty\).
reviewed by: Zaihong Wang
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B34
Mathematics Subject Classification ID: 34D05
Mathematics Subject Classification ID: 34C45
zbMATH DE Number: 2062712
MaRDI profile type: Publication
cites work: Positive solutions of elliptic equations in with a super-subcritical nonlinearity
cites work: Ground states of semilinear elliptic equations: A geometric approach
cites work: EQUIVALENCE AND NORMAL FORMS OF GERMS OF SMOOTH MAPPINGS
cites work: Q3938238
cites work: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth
cites work: Global and local behavior of positive solutions of nonlinear elliptic equations
cites work: Nonlinear oscillations, dynamical systems, and bifurcations of vector fields
cites work: Invariant manifolds
cites work: Positive solutions of super-critical elliptic equations and asymptotics
cites work: A Counterexample to the Nodal Domain Conjecture and a Related Semilinear Equation
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cites work: Symmetry of ground states of semilinear eliptic equations with mixed Sobolev growth
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description: scientific article; zbMATH DE number 2062712