For the quasilinear Schrödinger equation − Δ u + V ( x ) u + κ 2 Δ ( u 2 ) u = h ( u ) , u ∈ H 1 ( R N ) , where N ⩾ 3, κ is a real parameter, V(x) = V(|x|) is a potential allowed to be singular at the origin and h : R → R is a nonlinearity satisfying conditions similar to those in the paper (1983 Arch. Ration. Mech. Anal. 82 347–375) by Berestycki and Lions, we establish the existence of infinitely many radial solutions for κ < 0 and the existence of more and more radial solutions as κ ↓ 0. In the case κ < 0, we allow h(u) = |u| p−2 u for p in the whole range (2, 4N/(N − 2)) and this is in sharp contrast to most of the existing results which are only for p ∈ [4, 4N/(N − 2)). Moreover, our result in this case extends the result of Berestycki and Lions in the paper mentioned above to quasilinear equations with singular potentials. In the case κ ⩾ 0, our result extends and covers several related results in the literature, including the result of Berestycki and Lions.
Existence of sign-changing solutions to quasilinear elliptic equations of the form [Formula: see text] under the Dirichlet boundary condition, where [Formula: see text] ([Formula: see text]) is a bounded domain with smooth boundary and [Formula: see text] is a parameter, is studied. In particular, we examine how the number of sign-changing solutions depends on the parameter [Formula: see text]. In the case considered here, there exists no nontrivial solution for [Formula: see text] sufficiently small. We prove that, as [Formula: see text] becomes large, there exist both arbitrarily many sign-changing solutions with negative energy and arbitrarily many sign-changing solutions with positive energy. The results are proved via a variational perturbation method. We construct new invariant sets of descending flow so that sign-changing solutions to the perturbed equations outside of these sets are obtained, and then we take limits to obtain sign-changing solutions to the original equation.
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