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

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.