Existence of weak solutions for quasilinear elliptic equations involving the p-Laplacian

This paper shows the existence of nontrivial weak solutions for the quasilinear elliptic equation $$ -ig(Delta_p u +Delta_p (u^2)ig) +V(x)|u|^{p-2}u= h(u) $$ in $mathbb{R}^N$. Here $V$ is a positive continuous potential bounded away from zero and $h(u)$ is a nonlinear term of subcritical type. Using minimax methods, we show the existence of a nontrivial solution in $C^{1,alpha}_{ m loc}(mathbb{R}^N)$ and then show that it decays to zero at infinity when $1 Keywords Quasilinear Schrodinger equation --- solitary waves --- p-Laplacian --- variational method --- mountain-pass theorem