On the existence of positive solutions to a certain class of semilinear elliptic equations
On the existence of positive solutions to a certain class of semilinear elliptic equations
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DOI：
10.1007/s42985-021-00079-7
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发表时间：
2021-03
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影响因子：
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通讯作者：
N. Ikoma
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作者：
N. Ikoma
本文研究了一类半线性椭圆方程$$\begin{aligned} \Delta u = \varphi \left( V(x) u - f(x,u(x)) \right) \quad \text {in} \,\, \mathbf {R}^N, \quad u \in H^1( \mathbf {R}^N ) \end{aligned}$$，其中V(x),f（x,s）为给定函数。在一定条件下，我们证明了正解的存在性。特别地，我们推广了Felmer和Ikoma （J Funct Anal 275(8):2162 - 2196,2018）的结果。在Felmer和Ikoma（数理学报，27(8):2162-2196,2018）中，用拓扑度理论证明了正解的存在性。在本文中，我们采用变分方法。
In this paper, we study the following semilinear elliptic equation $$\begin{aligned} \Delta u = \varphi \left( V(x) u - f(x,u(x)) \right) \quad \text {in} \,\, \mathbf {R}^N, \quad u \in H^1( \mathbf {R}^N ) \end{aligned}$$whereand,V(x),f(x,s) are given functions. Under some conditions on, we show the existence of positive solution. In particular, we extend the result of Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018). In Felmer and Ikoma (J Funct Anal 275(8):2162–2196, 2018), the existence of positive solution was proved by topological degree theoretic argument. In this paper, we employ the variational method.