On the existence of positive solutions to a certain class of semilinear elliptic equations

Abstract

In this paper, we study the following semilinear elliptic equation $$\begin{aligned} \Delta u = \varphi \left( V(x) u - f(x,u(x)) \right) \quad \text {in} \,\, \mathbf {R}^N, \quad u \in H^1( \mathbf {R}^N ) \end{aligned}$$ where $$N \ge 1$$ and $$\varphi (s)$$ , V(x), f(x, s) are given functions. Under some conditions on $$\varphi (s), V(x), f(x,s)$$ , 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.