论文标题
$ p $ -laplace方程的存在结果，$ \ mathbb {r}^n $中的梯度非线性
An existence result for $p$-Laplace equation with gradient nonlinearity in $\mathbb{R}^N$
论文作者
论文摘要
我们证明了对问题的薄弱解决方案\ begin {equient*} \ begin {split}-Δ_{p} u+v（x）| u | u |^{p-2} u＆= f（u，| \ nabla u |^u |^{p-2} {p-2} {p-2} \ nabla u） x \ in \ Mathbb {r}^{n}，\ end {split} \ end {equation*}其中$δ_{p} u = \ hbox {div}（| \ nabla u |^u |^{p-2} {p-2} \ nabla u | $ f：\ mathbb {r} \ times \ mathbb {r}^{n} \ rightarrow \ rightarrow \ mathbb {r} $是连续的，这取决于解决方案的梯度。我们使用基于Mountain Pass定理的迭代技术来证明我们的存在结果。
We prove the existence of a weak solution to the problem \begin{equation*} \begin{split} -Δ_{p}u+V(x)|u|^{p-2}u & =f(u,|\nabla u|^{p-2}\nabla u), \ \ \ \\ u(x) & >0\ \ \forall x\in\mathbb{R}^{N}, \end{split} \end{equation*} where $Δ_{p}u=\hbox{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplace operator, $1<p<N$ and the nonlinearity $f:\mathbb{R}\times\mathbb{R}^{N}\rightarrow\mathbb{R}$ is continuous and it depends on gradient of the solution. We use an iterative technique based on the Mountain pass theorem to prove our existence result.
