LAUSR

We consider a class of generalized quasilinear Schrödinger equations − div ( l 2 ( u ) ∇ u ) + l ( u ) l ′ ( u ) | ∇ u | 2 + V ( x ) u = f ( u ) , x ∈ R N , $$ -\operatorname{div}\bigl(l^{2}(u)\nabla u\bigr)+l(u)l'(u) \vert \nabla u \vert ^{2}+V(x)u= f(u),\quad x\in \mathbb{R}^{N}, $$ where l ( t ) : R → R + $l(t): \mathbb{R}\to\mathbb{R}^{+}$ is a nondecreasing function with respect to | t | $|t|$ , the potential function V is allowed to be sign-changing so that the Schrödinger operator − Δ + V $-\Delta+V$ possesses a finite-dimensional negative space. We obtain existence and multiplicity results for the problem via the Symmetric Mountain Pass Theorem and Morse theory.