Quasilinear Schrödinger-Poisson equations involving a nonlocal term and an integral constraint. (English) Zbl 1501.35137
Summary: In this paper, we consider a class of quasilinear Schrödinger-Poisson problems of the form
\[
\begin{cases}
- \left({a + b\displaystyle\int_{{\mathbb{R}^N}} {{{\left| {\nabla u} \right|}^2}dx}} \right)\Delta u + V(x)u + \phi u - \displaystyle\frac{1}{2}u\Delta ({u^2}) - \lambda{{\left| u \right|}^{p - 2}}u = 0 & {{\text{in }}{\mathbb{R}^N},} \\
{ - \Delta \phi = {u^2},\,\,\,\,\,u(x) \to 0,\,\,\,\,\,\left| x \right| \to \infty }& {{\text{in }}{\mathbb{R}^N},}\\
\displaystyle{\int_{{\mathbb{R}^N}} {{\left| u \right|}^p}dx = 1,}
\end{cases}
\] where \(a > 0\), \(b \geq 0\), \(N \geq 3\), \(\lambda\) appears as a Lagrangian multiplier, and \(4 < p < 2 \cdot{2^\ast} = \frac{4N}{N - 2}\). We deal with two different cases simultaneously, namely \(\lim_{ \mid x \mid \rightarrow \infty }V(x) = \infty\) and \(\lim_{\mid x \mid \rightarrow \infty }V(x) = V_\infty \). 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.
MSC:
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 35J10 | Schrödinger operator, Schrödinger equation |
| 35J62 | Quasilinear elliptic equations |
| 35A01 | Existence problems for PDEs: global existence, local existence, non-existence |
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