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| We study the existence and multiplicity of positive solutions for the inhomogeneous Neumann boundary value problems involving the $p(x)$-Laplacian of the form
begin{equation*}
left{
begin{array}{c}
-divleft( leftvert nabla urightvert ^{p(x)-2}nabla
uright) +lambda leftvert urightvert ^{p(x)-2}u=f(x,u)quad text{ in }Omega
leftvert nabla urightvert ^{p(x)-2}frac{partial u}{partial eta }
=varphi text{ on }partial Omega ,
end{array}
right.
end{equation*}
where $Omega $ is a bounded smooth domain in $mathbf{R}^{N}$， $pin C^{1}(overline{Omega })$ and $p(x)>1$ for $xin overline{Omega }$, $varphi in C^{0,gamma }(partial Omega )$ with $gamma in (0,1)$, $varphi geq
0 $ and $varphi notequiv 0$ on $partial Omega$. Using the sub-supersolution method and the variational method, under appropriate assumptions on $f$, we prove that, there exists $lambda _{ast }>0$ such that the problem has at least two positive solutions if $lambda >lambda_{ast }$, has at least one positive solution if $lambda =lambda_{ast }$, and has no positive solution if $lambda <lambda _{ast }$. To prove the result we establish a special strong comparison principle for the Neumann problems. |
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| Keywords:$p(x)$-Laplacian equation; Neumann problem; positive solution; sub-supersolution method; variational method |
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