On a class of sublinear quasilinear elliptic problems

Abstract:

We establish existence and multiplicity of positive solutions to the quasilinear boundary value problem \begin{align*} \operatorname {div}(|\nabla u|^{p-2}\nabla u) &= -\lambda f(u)\quad \text {in $\Omega $},\\ u &= 0\quad \text {on$\partial \Omega $}, \end{align*} where $\Omega$ is a bounded domain in $R^{n}$ with smooth boundary $\partial \Omega$, $f:[0,\infty )\rightarrow R$ is continuous and p-sublinear at $\infty ,$ and $\lambda$ is a large parameter.