Multiplicity of positive solutions for semilinear elliptic equations in $\mathbb{R}^{N}$

Abstract

In this paper, we study the multiplicity of positive solutions for the following semilinear elliptic equation:

\begin{alignat*}{2} -\Delta u+\lambda u&=f(x)u^{p-1}+h(x)u^{q-1} &\quad&\text{in }\mathbb{R}^{N}, \\ u&>0 &&\text{in }\mathbb{R}^{N}, \\ u&\in H^{1}(\mathbb{R}^{N}), \end{alignat*}

where $1\leq q<2<p<2^{\ast}$ ($2^{\ast}=2N/(N-2)$ if $N\geq3$ and $2^{\ast}=\infty$ if $N=1,2$), $\lambda>0,h\in L^{2/(2-q)}(\mathbb{R}^{N})\setminus\{0\}$ is non-negative and $f\in C(\mathbb{R}^{N})$. We will show how the shape of the graph of $f(x)$ affects the number of positive solutions.