Quasilinear Problems with the Competition Between Convex and Concave\n Nonlinearities and Variable Potentials

The purpose of this paper is to prove some existence and non-existence\ntheorems for the nonlinear elliptic problems of the form\n-{\\Delta}_{p}u={\\lambda}k(x)u^{q}\\pmh(x)u^{{\\sigma}} if x\\in{\\Omega}, subject\nto the Dirichlet conditions u_{1}=u_{2}=0 on \\partial{\\Omega}. In the proofs of\nour results we use the sub-super solutions method and variational arguments.\nRelated results as obtained here have been established in [Z. Guo and Z. Zhang,\nW^{1,p} versus C^{1} local minimizers and multiplicity results for quasilinear\nelliptic equations, Journal of Mathematical Analysis and Applications, Volume\n286, Issue 1, Pages 32-50, 1 October 2003.] for the case k(x)=h(x)=1. Our\nresults reveal some interesting behavior of the solutions due to the\ninteraction between convex-concave nonlinearities and variable potentials.\n

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