Nonexistence and existence of positive radial solutions to a class of quasilinear Schrödinger equations in $\mathbb{R}^{N}$

Abstract

This paper aims to investigate the class of quasilinear Schrödinger equations $$\begin{aligned} \begin{aligned}[b] &-\Delta u-\bigl[\Delta \bigl(1+u^{2}\bigr)^{\frac{\gamma }{2}}\bigr] \frac{\gamma u}{2(1+u^{2})^{\frac{2-\gamma }{2}}}\\ &\quad =\alpha h\bigl( \vert x \vert \bigr) \vert u \vert ^{p-1}u+ \beta H\bigl( \vert x \vert \bigr) \vert u \vert ^{q-1}u, \quad x\in \mathbb{R}^{N}, \end{aligned} \end{aligned}$$ −Δu−[Δ(1+u2)γ2]γu2(1+u2)2−γ2=αh(|x|)|u|p−1u+βH(|x|)|u|q−1u,x∈RN, where $N >2$N>2, $1 \le \gamma \le 2$1≤γ≤2, $\alpha ,\beta \in \mathbb{R}$α,β∈R and either $0< p<1<q$0<p<1<q or $1< p< q$1<p<q. Functions $h(|x|)$h(|x|), $H(|x|)$H(|x|) are continuous and positive in $\mathbb{R}^{N} $RN. Relying on some special arguments and the Schauder–Tychonoff fixed point theorem, nonexistence criteria, existence of positive ground state solutions and blow-up solutions to Eq. (0.1) with $0< p<1<q$0<p<1<q or $1< p< q$1<p<q will be obtained.