## Differential and Integral Equations

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

### 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

#### 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