Nonlinear scalar field equations with Berestycki?Lions’ nonlinearity on large domains
Nonlinear scalar field equations with Berestycki?Lions’ nonlinearity on large domains
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非线性标量场方程与 Berestycki？Lions 大域上的非线性
DOI：
10.1007/s41808-020-00079-5
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发表时间：
2020
影响因子：
0.8
通讯作者：
Shibata Masataka
中科院分区：
文献类型：
--
作者：
Sato Yohei;Shibata Masataka
我们证明了以下半线性椭圆方程解的存在性：$$\begin{aligned} - \Delta u = g(u) \quad \text {in } \Omega , \quad u \in H^1_0(\Omega ). \end{aligned}$$这是一个合适的大区域，满足与Berestycki-Lions条件完全相同的条件。中所定义的方程的非平凡解存在的“几乎充分必要条件”。证明该方程解的存在性的主要困难是不能得到有界的palais - small序列。为了克服这个困难，我们修改了相应的函数，这是我们在之前的文章中引入的一个新思路。
We prove the existence of solutions for the following semilinear elliptic equation: $$\begin{aligned} - \Delta u = g(u) \quad \text {in } \Omega , \quad u \in H^1_0(\Omega ). \end{aligned}$$Hereis a suitable large domain andgsatisfies the completely same conditions as Berestycki–Lions’ conditions. Those conditions ofgare known as “almost sufficient and necessary conditions” to the existence of nontrivial solutions of the equations defined in. The main difficulty to prove the existence of solutions of the equation is that we can not obtain bounded Palais–Smale sequences. To overcome this difficulty, we modify the corresponding functional, which is a new idea introduced in our previous paper.