Normalized solutions of quasilinear Schrödinger equations in the general $L^2$-supercritical case

Abstract: This paper is devoted to studying the existence of normalized solutions for the following quasilinear Schrödinger equation \begin{equation*} \begin{aligned} -Δu-uΔu^2 +λu=h(u) \quad\mathrm{in}\ \mathbb{R}^{3}, \end{aligned} \end{equation*} where $λ$ appears as a Lagrange multiplier, $h$ is a $L<sup>2$-supercritical and Sobolev subcritical nonlinearity. The solutions correspond to critical points of the energy functional subject to the $L<sup>2$-norm constraint $\int_{\mathbb{R}<sup>3}|u|<sup>2dx=a<sup>2&gt;0$. Taking into account the Pohozaev manifold and perturbation method, we obtain the existence of ground state normalized solutions and infinitely many normalized solutions. Moreover, our results cover several relevant existing results in \cite{LZ2023}. And in the end, we get the asymptotic properties of energy as $a$ tends to $+\infty$ and $a$ tends to $0^+$.

• The paper demonstrates the existence of positive, radially symmetric normalized solutions for any prescribed mass using a perturbation method on the Pohozaev manifold.
• It establishes an infinite sequence of normalized solutions with strictly increasing energies, detailing the asymptotic behavior of the minimal constrained energy as the mass varies.
• The work introduces innovative variational techniques to overcome analytical challenges posed by non-smooth quasilinear terms in the L2-supercritical regime.

Normalized Solutions of Quasilinear Schrödinger Equations in the $L^2$-Supercritical Regime

Problem Setting and Motivation

This paper addresses the existence, multiplicity, and asymptotic energy properties of normalized solutions to the quasilinear stationary Schrödinger equation

$$-\Delta u - u \Delta u^2 + \lambda u = h(u) \quad \text{in} \; \mathbb{R}^3,$$

subject to the mass constraint

$\int_{\mathbb{R}^3} |u|^2 \, dx = a^2 > 0,$

where $\lambda \in \mathbb{R}$ is a Lagrange multiplier and $h$ exhibits $L^2$-supercritical but Sobolev subcritical growth. Such models arise in the analysis of standing waves of time-dependent quasilinear Schrödinger equations, relevant for superfluidity, plasma physics, and nonlinear optics.

Compared to the classical semilinear Schrödinger problem, the inclusion of the quasilinear term $u \Delta u^2$ introduces non-smoothness and severe analytical obstacles. In $L^2$-supercritical regimes—i.e., when the nonlinearity exponent $p > 4 + \frac{4}{N}$ for $h(t) = |t|^{p-2} t$—existing methods (Nehari manifold, change of variables) are generally inapplicable due to lack of differentiability and unboundedness from below on constraint manifolds. This work extends recent results to general nonlinearities $$-\Delta u - u \Delta u^2 + \lambda u = h(u) \quad \text{in} \; \mathbb{R}^3,$$0, provides new characterizations of ground states, multiplicity, and energy asymptotics, and overcomes issues left open in prior literature.

Main Analytical Contributions

Existence and Multiplicity of Normalized Solutions

The principal technical achievement of the paper is to prove the existence of positive, radially symmetric, ground state normalized solutions in the general $$-\Delta u - u \Delta u^2 + \lambda u = h(u) \quad \text{in} \; \mathbb{R}^3,$$1-supercritical regime with minimal structure assumed on $$-\Delta u - u \Delta u^2 + \lambda u = h(u) \quad \text{in} \; \mathbb{R}^3,$$2. Under regularity, symmetry, and growth assumptions ($$-\Delta u - u \Delta u^2 + \lambda u = h(u) \quad \text{in} \; \mathbb{R}^3,$$3–$$-\Delta u - u \Delta u^2 + \lambda u = h(u) \quad \text{in} \; \mathbb{R}^3,$$4), two main results are established:

• For each mass $$-\Delta u - u \Delta u^2 + \lambda u = h(u) \quad \text{in} \; \mathbb{R}^3,$$5, there exists a ground state solution $$-\Delta u - u \Delta u^2 + \lambda u = h(u) \quad \text{in} \; \mathbb{R}^3,$$6 which is a maximizer of the constrained energy functional on the Pohozaev manifold and, in particular, solves the constraint problem with minimal energy.
• There exists an infinite sequence $$-\Delta u - u \Delta u^2 + \lambda u = h(u) \quad \text{in} \; \mathbb{R}^3,$$7 of normalized solutions with strictly increasing energies, $$-\Delta u - u \Delta u^2 + \lambda u = h(u) \quad \text{in} \; \mathbb{R}^3,$$8.

The construction and compactness of critical points hinge upon a perturbation/regularization technique originally due to Jeanjean et al. The authors introduce a parameterized, differentiable energy functional $$-\Delta u - u \Delta u^2 + \lambda u = h(u) \quad \text{in} \; \mathbb{R}^3,$$9 for $\int_{\mathbb{R}^3} |u|^2 \, dx = a^2 > 0,$0, analyze the geometry and Palais–Smale behavior on radial mass balls, and then pass to the limit as $\int_{\mathbb{R}^3} |u|^2 \, dx = a^2 > 0,$1. This approach is glued to a precise variational analysis on the associated Pohozaev manifold, exploiting monotonicity and symmetrization arguments to produce the desired solutions.

Pohozaev Manifold and Variational Structure

The analysis is formulated on the natural constrained set—where the mass is fixed—intersected with the Pohozaev manifold, defined by a generalized Pohozaev identity valid for constrained critical points. The paper provides a rigorous treatment of the properties of this manifold, establishing existence and uniqueness of energy-maximizing scaling for each $\int_{\mathbb{R}^3} |u|^2 \, dx = a^2 > 0,$2, continuity of this scaling map, and coercivity of the energy functional restricted to the manifold. For general nonlinearities, technical conditions on $\int_{\mathbb{R}^3} |u|^2 \, dx = a^2 > 0,$3 guarantee abstract monotonicity and lower semicontinuity properties necessary for minimax arguments.

Asymptotic Behavior of Energy Levels

A further notable result is the detailed analysis of the asymptotic behavior of the minimal constrained energy $\int_{\mathbb{R}^3} |u|^2 \, dx = a^2 > 0,$4 as the prescribed mass $\int_{\mathbb{R}^3} |u|^2 \, dx = a^2 > 0,$5 varies:

• $\int_{\mathbb{R}^3} |u|^2 \, dx = a^2 > 0,$6 as $\int_{\mathbb{R}^3} |u|^2 \, dx = a^2 > 0,$7,
• $\int_{\mathbb{R}^3} |u|^2 \, dx = a^2 > 0,$8 as $\int_{\mathbb{R}^3} |u|^2 \, dx = a^2 > 0,$9,
• $\lambda \in \mathbb{R}$0 is lower semicontinuous in $\lambda \in \mathbb{R}$1.

This implies, in particular, that the ground state energy can be made arbitrarily small or large by tuning the mass constraint, a phenomenon with implications for stability and blow-up thresholds in the underlying dynamical equations.

Strong Numerical and Structural Claims

• Existence of a positive, radially symmetric ground state for any mass $\lambda \in \mathbb{R}$2, under broad conditions on the $\lambda \in \mathbb{R}$3-supercritical nonlinearity.
• Existence of infinitely many distinct normalized solutions with diverging energies.
• Novel asymptotics for the minimal constrained energy, rigorously characterized in the supercritical regime for the first time.
• For general nonlinearities—not only power-law cases—full coercivity and compactness properties of the energy on the Pohozaev manifold are established.

The paper's claims enlarge the class of admissible nonlinearities beyond those previously analyzed (notably, Li and Zou [LZ2023]), closing technical gaps regarding the geometry of the constraint and the limiting behavior of solutions.

Implications and Directions for Further Research

Practically, these results have ramifications for the analysis of quasilinear matter waves in plasma physics, nonlinear optics, and Bose–Einstein condensates at high densities (when $\lambda \in \mathbb{R}$4-supercritical effects become relevant). The existence and explicit energy scaling of normalized states inform the phase portrait and stability analysis of quasilinear systems modeling supercritical interactions.

Theoretically, the paper strengthens the variational approach to strongly non-smooth, constraint problems in PDE, particularly in the context of mass-conserving dynamics where standard methods often break down. The combination of perturbation methods, Pohozaev manifold analysis, and homotopy/minimax theory here may be adapted to other quasilinear or degenerate equations in mathematical physics.

Future developments could target the dynamical (in)stability of the constructed normalized states, finer qualitative properties (e.g., uniqueness modulo symmetries, concentration phenomena), and extensions to more general physical settings (e.g., inclusion of external potentials, fractional dispersion, or systems of equations). There is also scope for exploring non-radial ground states or sign-changing solutions using variants of the methods introduced.

Conclusion

This work provides a rigorous, comprehensive existence and multiplicity theory for normalized solutions to $\lambda \in \mathbb{R}$5-supercritical quasilinear Schrödinger equations with broad nonlinearities. The deployment of perturbation techniques and Pohozaev manifold analysis overcomes longstanding obstacles in the supercritical regime, yielding sharp structural and asymptotic energy results with both analytic and application-oriented significance (2606.28806).