Local ‘Superlinearity’ and ‘Sublinearity’ for the p-Laplacian

Abstract

We study the existence, nonexistence and multiplicity of positive solutions for a family of problems $- \Updelta_{p} u = f_{\lambda } \,(x,\,u),\,u \in \,W_{0}^{1,p} (\Upomega )$, where Ω is a bounded domain in ${\mathbb{R}}^{N} ,\,N > p$, and λ > 0 is a parameter. The family we consider includes the well-known nonlinearities of Ambrosetti–Brezis–Cerami type in a more general form, namely $\lambda a(x)u^{q} + b(x)u^{r}$, where $0 \leqslant q < p - 1 < r \leqslant p* - 1$. Here the coefficient a(x) is assumed to be nonnegative but b(x) is allowed to change sign, even in the critical case. Preliminary results of independent interest include the extension to the p-Laplacian context of the Brezis–Nirenberg result on local minimization in $W_{0}^{1,p} \,{\text{and}}\,C_{0}^{1} ,\,a\,C^{1,\alpha }$ estimate for equations of the form $- \Updelta_{p} u = h(x,u)$ with h of critical growth, a strong comparison result for the p-Laplacian, and a variational approach to the method of upper–lower solutions for the p-Laplacian.