  The purpose of this paper is to prove some existence and non-existence
theorems for the nonlinear elliptic problems of the form
-{\Delta}_{p}u={\lambda}k(x)u^{q}\pmh(x)u^{{\sigma}} if x\in{\Omega}, subject
to the Dirichlet conditions u_{1}=u_{2}=0 on \partial{\Omega}. In the proofs of
our results we use the sub-super solutions method and variational arguments.
Related results as obtained here have been established in [Z. Guo and Z. Zhang,
W^{1,p} versus C^{1} local minimizers and multiplicity results for quasilinear
elliptic equations, Journal of Mathematical Analysis and Applications, Volume
286, Issue 1, Pages 32-50, 1 October 2003.] for the case k(x)=h(x)=1. Our
results reveal some interesting behavior of the solutions due to the
interaction between convex-concave nonlinearities and variable potentials.
