Multiple ordered solutions for a class of elliptic problems involving fast increasing weights and nonlinearity with zeros

Abstract

In this paper, we study the existence of multiple “ordered solutions” to the following class of elliptic problems with fast increasing weights and oscillating nonlinearity

$$\begin{aligned} -\Delta u - \frac{1}{2}(x \cdot \nabla u)+\mu |u|^{q-2}u = \lambda h(x)f(u) \ \text{ in } \ {\mathbb {R}}^{N}, \end{aligned}$$

where $\mu \ge 0,$ $\lambda >0, q\in [2,\infty )$ are parameters, $h: {\mathbb {R}}^N\rightarrow [0,\infty )$ is a continuous function and $f:{\mathbb {R}}\rightarrow {\mathbb {R}}$ is of class $C^1$ that can change sign and satisfies an area condition. Our research combined truncation arguments with minimization technique and degree theory. The solutions are also ordered according to their $L^\infty$-norms.