Quasilinear Schrödinger-Poisson equations involving a nonlocal term and an integral constraint

Abstract

In this paper, we consider a class of quasilinear Schrödinger-Poisson problems of the form

$$\left\{ {\matrix{{ - \left( {a + b\int_{{^N}} {{{\left| {\nabla u} \right|}^2}dx} } \right)\Delta u + V(x)u + \phi u - {1 \over 2}u\Delta ({u^2}) - \lambda {{\left| u \right|}^{p - 2}}u = 0} \hfill & {{\rm{in}}\,\,{\mathbb{R}^N},} \hfill \cr { - \Delta \phi = {u^2},\,\,\,\,\,u(x) \to 0,\,\,\,\,\,\left| x \right| \to \infty } \hfill & {{\rm{in}}\,\,{\mathbb{R}^N},} \hfill \cr {\int_{{\mathbb{R}^N}} {{{\left| u \right|}^p}dx = 1,} } \hfill & {} \hfill \cr } } \right.$$

where a > 0, b ⩾ 0, N ⩾ 3, λ appears as a Lagrangian multiplier, and $4 < p < 2 \cdot {2^\ast} = {{4N} \over {N - 2}}$. We deal with two different cases simultaneously, namely lim_{∣x∣→∞}V(x) = ∞ and lim_{∣x∣→∞}V(x) = V_∞. By using the method of invariant sets of the descending flow combined with the genus theory, we prove the existence of infinitely many sign-changing solutions. Our results extend and improve some recent work.