This paper shows the existence of infinitely many solutions for the quasilinear equations of the form
where
△Nu is the
N -Laplacian operator,
N≥3,
λ≥0,
1<q<N,
K∈Lθ(RN),
θ=N/(N−q) and
h is an odd continuous function having critical exponential growth. The potential function
V(x)∈C(RN) and
0<infx∈RNV(x)≤supx∈RNV(x)<∞. Using mountain-pass theorem and some special techniques, we demonstrate that there exists
λ0>0 such that problem
(0.1) admits infinitely many high-energy solutions provided that
λ∈[0,λ0].