Normalized solutions of quasilinear Schrödinger-Poisson system with critical nonlinear term in bounded domain

Abstract

This work examines a quasilinear Schrödinger-Poisson system involving a critical nonlinearity, expressed as \[ -Δu + φu + λu = |u|^{q-2} u + |u|^4 u, \quad x \in Ω_r, \] \[ -Δφ- \varepsilon^4 Δ_4 φ= u^2, \qquad\qquad\qquad\quad\ x \in Ω_r, \] \[ \enspace u = φ= 0, \qquad\qquad\qquad\qquad\qquad\enspace\ \ \,x \in \partial Ω_r \] subject to the normalized condition \[ \int_{Ω_r} |u|^2\, \mathrm d x = b^2. \] Here $\varepsilon > 0$, $q \in (2, 8/3)$, $Ω_r \subset \mathbb R^3$ is a bounded domain. By means of a truncation method combined with genus theory, we establish the existence of multiple families of normalized solutions. Due to the presence of a critical exponent in the nonlinear term, the associated energy functional fails to satisfy the usual compactness properties. To address this issue, we invoke the concentration-compactness principle. Furthermore, we derive the asymptotic result that the aforementioned system reduces to the classical Schrödinger-Poisson system (with $\varepsilon = 0$). Our findings extend several recent results concerning problems of this type.

Theorems & Definitions (14)

• Theorem 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3: MR4812883 Gagliardo-Nirenberg inequality
• Lemma 2.4
• Lemma 2.5: Chabrowski1995
• Proposition 2.1: Jeanjean2018
• Lemma 3.1
• proof