Differential and Integral Equations

Precise exponential decay for solutions of semilinear elliptic equations and its effect on the structure of the solution set for a real analytic nonlinearity

Nils Ackermann and Norman Dancer

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Abstract

We are concerned with the properties of weak solutions of the stationary Schrödinger equation $-\Delta u + Vu = f(u)$, $u\in H^1(\mathbb{R}^N)\cap L^\infty(\mathbb{R}^N)$, where $V$ is Hölder continuous and $\inf V>0$. Assuming $f$ to be continuous and bounded near $0$ by a power function with exponent larger than $1,$ we provide precise decay estimates at infinity for solutions in terms of Green's function of the Schrödinger operator. In some cases this improves known theorems on the decay of solutions. If $f$ is also real analytic on $(0,\infty)$, we obtain that the set of positive solutions is locally path connected. For a periodic potential $V$ this implies that the standard variational functional has discrete critical values in the low energy range and that a compact isolated set of positive solutions exists, under additional assumptions.

Article information

Source
Differential Integral Equations Volume 29, Number 7/8 (2016), 757-774.

Dates
First available in Project Euclid: 3 May 2016

Permanent link to this document
https://projecteuclid.org/euclid.die/1462298684

Mathematical Reviews number (MathSciNet)
MR3498876

Zentralblatt MATH identifier
06604494

Subjects
Primary: 35B40: Asymptotic behavior of solutions 35J91: Semilinear elliptic equations with Laplacian, bi-Laplacian or poly- Laplacian 5J20

Citation

Ackermann, Nils; Dancer, Norman. Precise exponential decay for solutions of semilinear elliptic equations and its effect on the structure of the solution set for a real analytic nonlinearity. Differential Integral Equations 29 (2016), no. 7/8, 757--774.https://projecteuclid.org/euclid.die/1462298684


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