论文标题
存在不连续非线性和渐近线性的一类椭圆方程的解决方案
Existence of solution for a class of elliptic equation with discontinuous nonlinearity and asymptotically linear
论文作者
论文摘要
本文涉及存在以下问题的非平凡解决方案 \ begin {equation} \ left \ {\ begin {aligned} -ΔU + v（x）u＆\ in \ partial_u f（x，u）\; \; \ mbox {a.e。 in} \; \; \ mathbb {r}^{n}，\ nonumber u \ in H^{1}（\ Mathbb {r}^{n}）， \ end {Aligned} \ right。\ leqno {（p）} 其中$ f（x，x，t）= \ int_ {0}^{t} f（x，x，s）\，ds $，ds $，$ f $是一个不连续的功能，在无限范围内是线性的，$λ= 0 $在$-Δ+$ v $ po $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ f $ n demient contient cons中。 $ t $。在这里，通过为局部Lipschitz功能采用各种方法，我们在$ f $是周期性且非周期性的情况下建立解决方案的存在
This paper concerns the existence of a nontrivial solution for the following problem \begin{equation} \left\{\begin{aligned} -Δu + V(x)u & \in \partial_u F(x,u)\;\;\mbox{a.e. in}\;\;\mathbb{R}^{N},\nonumber u \in H^{1}(\mathbb{R}^{N}), \end{aligned} \right.\leqno{(P)} \end{equation} where $F(x,t)=\int_{0}^{t}f(x,s)\,ds$, $f$ is a discontinuous function and asymptotically linear at infinity, $λ=0$ is in a spectral gap of $-Δ+V$, and $\partial_t F$ denotes the generalized gradient of $F$ with respect to variable $t$. Here, by employing Variational Methods for Locally Lipschitz Functionals, we establish the existence of solution when $f$ is periodic and non periodic
