Global behavior of the ground state energy of the nonlinear scalar field equation

Abstract

We study the global behavior of the ground state energy for the nonlinear scalar field equation (including the $p$-Laplacian version), and give a complete description of this energy in terms of nonlinearity power. More precisely,let$\mathcal{E}_{\lambda,q}(u)=\frac{1}{p}(\|\nabla~u\|_p^p+\lambda\|u\|_p^p)-\frac{1}{q}\|u\|_q^q$ bethe energy functional in $~W^{1,p}(\mathbb{R}^N)$,where $\lambda>0$, $1<p<N$, $p<q<p^*:=\frac{pN}{N-p}$, and $p^*$ is the critical Sobolev exponent. It is known that $\mathcal{E}_{\lambda,q}(u)$ has a unique radially symmetric mountain-pass critical point $U_{\lambda,q}$, which is called the ground state 解 to the corresponding nonlinear scalar field equation and whose energy $\mathcal{E}_{\lambda,q}(U_{\lambda,q})$ is called the ground state energy $\mathfrak{m}(\lambda,q)$.We show that there is a decreasing function $\lambda_0(q)$ defined in $q\in~[p,p^*]$ with $\lambda_0(p)=1~$ and $\lambda_0(p^*)=0$such that $\mathfrak{m}(\lambda,r)$ is strictly increasing for $r\in(p,~q)$ with $\lambda\in(0,~\lambda_0(q))$ being fixed, $\mathfrak{m}(\lambda,r)$ is strictly decreasing for $r\in(q,~p^*)$ with $\lambda\in(\lambda_0(q),~1)$ being fixed, and $\mathfrak{m}(\lambda,r)$ is strictly decreasing for all $r\in(p,~p^*)$ with $\lambda\in[1,~+\infty)$ being fixed.We also deduce the precise asymptotic behavior of $\lambda_0(q)$ as $q\to~p$ and $q\to~p^*$. This is done by establishing a relation between power-law scalar fieldequations and logarithmic-law scalar field equations,which is of the independent interest.