A nonexistence result for a class of quasilinear Schr\"{o}dinger equations with Berestycki-Lions conditions

Abstract

In this paper, we study the following quasilinear Schr\"{o}dinger equation
\begin{equation*}
-\Delta u+V(x)u-[\Delta(1+u^2)^{1/2}]\frac{u}{2(1+u^2)^{1/2}}=h(u),\,\, x\in\R^N,
\end{equation*}
where $N\geq3$, $2^*=\frac{2N}{N-2}$, $V(x)$ is a potential function. Unlike $V\in\mathcal{C}^2(\R^N)$, we only need to assume that $V\in\mathcal{C}^1(\R^N)$.
By using a change
of variable, we prove the non-existence of ground state solutions with Berestycki-Lions conditions, which contain the superliner case:
\begin{equation*}
\lim\limits_{s\rightarrow+\infty}\frac{h(s)}{s}=+\infty
\end{equation*}
and asymptotically linear case:
\begin{equation*}
\lim\limits_{s\rightarrow+\infty}\frac{h(s)}{s}=\eta.
\end{equation*}
Our results extend and complement the results in related literature.