一类含p(x)-Laplacian的非齐次Neumann问题正解的多重性

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.