Differential and Integral Equations

Schrödinger type equations with asymptotically linear nonlinearities

Abstract

We study the nonlinear Schrödinger type equation $$-\Delta u~+~(\lambda g(x)~+~1)u~=~f(u)$$ on the whole space ${{\mathbf R}^N}$. The nonlinearity $f$ is assumed to be asymptotically linear and $g(x)\geq 0$ has a potential well. We do not assume a limit for $g(x)$ as $|x|\to\infty$. Using variational techniques, we prove the existence of a positive solution for $\lambda$ large. In the case where $f$ is odd we obtain multiple pairs of solutions. The limiting behavior of solutions as $\lambda\to\infty$ is also considered.

Article information

Source
Differential Integral Equations, Volume 16, Number 3 (2003), 257-280.

Dates
First available in Project Euclid: 21 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.die/1356060671

Mathematical Reviews number (MathSciNet)
MR1947953

Zentralblatt MATH identifier
0476.03047

Citation

van Heerden, Francois A.; Wang, Zhi-Qiang. Schrödinger type equations with asymptotically linear nonlinearities. Differential Integral Equations 16 (2003), no. 3, 257--280. https://projecteuclid.org/euclid.die/1356060671