## Differential and Integral Equations

### A value function and applications to translation-invariant semilinear elliptic equations on unbounded domains

Ian Schindler

#### Abstract

We combine results from nonlinear functional analysis relating nonlinear eigenvalues of the type $g'(u) = \rho u$ to the derivatives of the critical value function $\gamma (t) := \sup_{\|u\|^2=t} g(u)$ with concentration compactness techniques to study the Dirichlet boundary value problem on $\Omega$, $$-\Delta u + u = \lambda f(x,u), \tag 0.1$$ where $\Omega$ is an unbounded cylindrical domain and the dependence on $x$ in the unbounded direction is periodic. We give sufficient conditions on $f$ to obtain an interval in which the $\lambda$'s for which (0.1) has a weak solution are dense.

#### Article information

Source
Differential Integral Equations, Volume 8, Number 4 (1995), 813-828.

Dates
First available in Project Euclid: 20 May 2013

https://projecteuclid.org/euclid.die/1369055612

Mathematical Reviews number (MathSciNet)
MR1306593

Zentralblatt MATH identifier
0820.35106

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35P05: General topics in linear spectral theory

#### Citation

Schindler, Ian. A value function and applications to translation-invariant semilinear elliptic equations on unbounded domains. Differential Integral Equations 8 (1995), no. 4, 813--828. https://projecteuclid.org/euclid.die/1369055612