| Abstract: |
| In this paper we consider a nonlinear parametric Dirichlet problem
driven by a nonhomogeneous differential operator (special cases
are the $p$-Laplacian and the $(p,q)$-differential operator) and
with a reaction which has the combined effects of concave
(($p-1)$-sublinear) and convex ($(p-1)$-superlinear) terms. We do
not employ the usual in such cases AR-condition. Using variational
methods based on critical point theory, together with truncation
and comparison techniques and Morse theory (critical groups), we
show that for all small $\lambda>0$ ($\lambda$ is a parameter),
the problem has at least five nontrivial smooth solutions (two
positive, two negative and the fifth nodal). We also prove two
auxiliary results of independent interest. The first is a strong
comparison principle and the second relates Sobolev and Holder
local minimizers for $C^1$ functionals.
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$\footnote{The publication of this paper has been partly supported by the University of Piraeus Research Center}$ |
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