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Infinite many Blow-up solutions for a Schrödinger quasilinear elliptic problem with a non-square diffusion term

In this paper, we consider existence of positive solutions for the Schrödinger quasilinear elliptic problem $$ \left\{ \begin{array}{l} Δ_pu+Δ_p(|u|^{2γ})|u|^{2γ-2}u = a(x)g(u)~ \mbox{on}~ \mathbb{R}^N,\\ u>0\ \mbox{in}~\mathbb{R}^N,\ u(x)\stackrel{\left|x\right|\rightarrow \infty}{\longrightarrow} \infty, \end{array} \right. $$ where $a(x), ~x\in \mathbb{R}^N$ and $g(s)~s>0$ are a nonnegative and continuous functions with $g$ being nonincreasing as well, $γ>{1}/{2}$, and $N \geq 1$. By a dual approach we establish sufficient conditions for existence and multiplicity of solutions for this problem.