### One-dimensional elliptic equation with concave and convex nonlinearities.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

Back to Simple Search
# Advanced Search

We study the existence, nonexistence and multiplicity of positive solutions for the family of problems $-\Delta u={f}_{\lambda}(x,u)$, $u\in {H}_{0}^{1}\left(\Omega \right)$, where $\Omega $ is a bounded domain in ${\mathbb{R}}^{N}$, $N\ge 3$ and $\lambda >0$ is a parameter. The results include the well-known nonlinearities of the Ambrosetti–Brezis–Cerami type in a more general form, namely $\lambda a\left(x\right){u}^{q}+b\left(x\right){u}^{p}$, where $0\le q<1<p\le {2}^{*}-1$. The coefficient $a\left(x\right)$ is assumed to be nonnegative but $b\left(x\right)$ is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential in this...

**Page 1**