Abstract
In this paper, we show that if $Q(x)$ satisfies some suitable conditions, then the quasilinear elliptic Dirichlet problem $-\Delta_p u+|u|^{p-2} u=Q(x)|u|^{q-2}u$ in an unbounded cylinder domain $\Omega$ has at least two solutions in which one is a positive ground state solution and the other is a nodal solution.
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