论文标题
略微超线性和sublinear椭圆形p-laplacian方程的先验界限和多样性结果
A priori bounds and multiplicity results for slightly superlinear and sublinear elliptic p-Laplacian equations
论文作者
论文摘要
我们考虑以下问题$-Δ_{p} u = h（x，x，u）\ mbox {in}ω$，$ u \ in w^{1，p} _ {0}（ω）$，其中$ω$是$ \ \ \ m varybb {r}^r}^{n} $ 1 <p Stample a $ \ mathbb {r}在本文中，我们假设$ h（x，u）= a（x）f（u）+b（x）g（u）$，以便$ f $定期改变索引$ p-1 $和无限的超级线。功能$ g $是零的$ p $ sublinear函数。对于某些$ k> \ frac {n} {p} $，系数$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a和b $属于$ l^{ω）$，它们没有符号条件。首先，我们显示了对解决方案的先验绑定，然后通过使用变分参数，证明至少存在两种非负解决方案。主要困难之一是非线性项$ h（x，u）$不满足标准的Ambrosetti和Rabinowitz条件。
We consider the following problem $ -Δ_{p}u= h(x,u) \mbox{ in }Ω$, $u\in W^{1,p}_{0}(Ω)$, where $Ω$ is a bounded domain in $\mathbb{R}^{N}$, $1<p<N$, with a smooth boundary. In this paper we assume that $h(x,u)=a(x)f(u)+b(x)g(u)$ such that $f$ is regularly varying of index $p-1$ and superlinear at infinity. The function $g$ is a $p$-sublinear function at zero. The coefficients $a$ and $b$ belong to $L^{k}(Ω)$ for some $k>\frac{N}{p}$ and they are without sign condition. Firstly, we show a priori bound on solutions, then by using variational arguments, we prove the existence of at least two nonnegative solutions. One of the main difficulties is that the nonlinearity term $h(x,u)$ does not satisfy the standard Ambrosetti and Rabinowitz condition.
