Normalized Semiclassical Solutions to Magnetic Schrödinger-Poisson Systems with Critical Local and Nonlocal Interactions

Abstract: We study the existence, multiplicity, and concentration of normalized semiclassical states for a magnetic Schrödinger--Poisson system in $\mathbb{R}<sup>3$ featuring both the Sobolev-critical local nonlinearity $|u|<sup>4u$ and a critical nonlocal Poisson interaction. The problem is considered under the prescribed mass constraint $$\int_{\mathbb{R}<sup>3}|u|<sup>2\,dx=a<sup>2\varepsilon<sup>3,$$ where $a&gt;0$ denotes the prescribed mass and $\varepsilon&gt;0$ is the semiclassical parameter. By combining constrained variational methods, a suitable penalization scheme, concentration--compactness arguments, and Ljusternik--Schnirelmann theory, we first prove the existence of a normalized semiclassical solution for sufficiently small $a$ and $\varepsilon$. We then establish a multiplicity result showing that, for every sufficiently small $\varepsilon&gt;0$, the number of distinct normalized solutions is bounded from below by the Ljusternik--Schnirelmann category of the minimum set [ \mathcal M = {x\in\mathbb{R}^3:V(x)=\min_{\mathbb{R}^3}V}. ] Finally, we describe the semiclassical concentration phenomenon by showing that the maximum points of the resulting solutions approach $\mathcal M$ as $\varepsilon\to0$.

• The paper establishes normalized solutions for sufficiently small mass and semiclassical parameter by combining constrained variational methods, double penalization, and critical compactness estimates for magnetic and Poisson terms.
• It proves at least \(\operatorname{cat}_{\mathcal{M}_\delta}(\mathcal{M})\) distinct solutions, linking multiplicity to the Ljusternik–Schnirelmann category of the electric potential’s minimum set.
• The resulting states concentrate near the minimum set \(\mathcal{M}\) as \(\varepsilon\to0\), while the analysis depends on assumed strict subadditivity and an energy gap below the critical bubbling threshold.

This paper studies the existence, multiplicity, and semiclassical concentration of normalized solutions to a magnetic Schrödinger–Poisson system in $\mathbb{R}^3$ that simultaneously contains a Sobolev-critical local nonlinearity $|u|^4u$ and a critical nonlocal Poisson interaction. The system is posed under the prescribed mass constraint $\int_{\mathbb{R}^3}|u|^2\,dx = a^2\varepsilon^3$, where $a>0$ is the mass and $\varepsilon>0$ is the semiclassical parameter. The frequency $\lambda$ is not fixed but emerges as a Lagrange multiplier. The author establishes two main theorems: an existence result for sufficiently small $a$ and $\varepsilon$, and a multiplicity result whose lower bound is given by the Ljusternik–Schnirelmann category of the minimum set of the electric potential, together with concentration of the solutions near that set as $\varepsilon \to 0$.

Problem setting and novelty

The system under consideration is

$$(-i\varepsilon\nabla-A(x))^2u+V(x)u-\phi|u|^3u = \lambda u+\mu |u|^{q-2}u+|u|^4u, \qquad -\varepsilon^2\Delta\phi=|u|^5,$$

with $|u|^4u$0. The assumptions on the potentials are mild: $|u|^4u$1–$|u|^4u$2 require $|u|^4u$3 with positive infimum $|u|^4u$4, a bounded open set $|u|^4u$5 on which $|u|^4u$6, and a nonempty minimum set $|u|^4u$7; assumption $|u|^4u$8 requires only linear growth of the magnetic potential $|u|^4u$9.

The paper's stated contribution is the simultaneous treatment of five features—magnetic structure, prescribed mass, Sobolev-critical local term, critical nonlocal Poisson coupling, and semiclassical concentration—that have previously been handled only separately or in non-magnetic settings. The author asserts that normalized semiclassical solutions for such a combination have not been studied before; this claim appears well supported by the literature review, which covers Feng's subcritical semiclassical problem, Gao–He–Rădulescu's critical constrained problem concentrating at a potential well, Meng–He's doubly critical normalized system without magnetic field or potential, and He–Liu–Meng's potential case.

Variational framework

The analysis takes place in the semiclassical magnetic Sobolev space $\int_{\mathbb{R}^3}|u|^2\,dx = a^2\varepsilon^3$0 equipped with the norm induced by $\int_{\mathbb{R}^3}|u|^2\,dx = a^2\varepsilon^3$1. The diamagnetic inequality, $\int_{\mathbb{R}^3}|u|^2\,dx = a^2\varepsilon^3$2 a.e., is the central device for transferring real-valued estimates to the complex setting; it yields the embedding into $\int_{\mathbb{R}^3}|u|^2\,dx = a^2\varepsilon^3$3 for $\int_{\mathbb{R}^3}|u|^2\,dx = a^2\varepsilon^3$4 with constant scaling like $\int_{\mathbb{R}^3}|u|^2\,dx = a^2\varepsilon^3$5 at the critical exponent.

The electrostatic potential is eliminated via the Lax–Milgram theorem: for each $\int_{\mathbb{R}^3}|u|^2\,dx = a^2\varepsilon^3$6, there is a unique $\int_{\mathbb{R}^3}|u|^2\,dx = a^2\varepsilon^3$7 solving $\int_{\mathbb{R}^3}|u|^2\,dx = a^2\varepsilon^3$8, represented as a Newton potential scaled by $\int_{\mathbb{R}^3}|u|^2\,dx = a^2\varepsilon^3$9. The resulting nonlocal energy satisfies $a>0$0 and obeys the estimate $a>0$1, obtained from Hardy–Littlewood–Sobolev combined with the diamagnetic inequality. The reduced functional $a>0$2 is shown to be $a>0$3, and constrained critical points on the mass sphere $a>0$4 are characterized precisely as weak solutions of the original system via the Lagrange multiplier rule.

A notable structural point is the mass-preserving dilation $a>0$5, under which the Poisson interaction scales as $a>0$6 while the kinetic part scales as $a>0$7. Since $a>0$8 gives $a>0$9, the negative $\varepsilon>0$0 perturbation dominates near $\varepsilon>0$1, which drives both the unboundedness below of the energy on the mass sphere and the construction of paths crossing any energy barrier.

The autonomous limit problem

The reference profile is built from the autonomous problem at level $\varepsilon>0$2. Because both critical terms make $\varepsilon>0$3 unbounded below on $\varepsilon>0$4, minimization is performed locally over $\varepsilon>0$5. A careful choice of $\varepsilon>0$6 and $\varepsilon>0$7 produces a strict mountain-pass geometry: the boundary level exceeds $\varepsilon>0$8 while the interior infimum $\varepsilon>0$9 lies strictly below it. Compactness of minimizing sequences is obtained through the concentration–compactness principle, excluding vanishing (which would force the energy to at least $\lambda$0, contradicting $\lambda$1) and dichotomy (excluded by assuming strict subadditivity $\lambda$2). The limit $\lambda$3 is then a nonnegative constrained ground state satisfying a Pohozaev identity derived by differentiating along the dilation curve:

$\lambda$4

It should be noted that the strict subadditivity condition is assumed rather than proved within the paper; the compactness lemma is conditional on it, and no independent verification (for instance via a Pohozaev-based argument) is provided.

Penalization and mountain-pass geometry

Following del Pino–Felmer, the nonlinearities are truncated outside $\lambda$5: the sources become linear ($\lambda$6 and $\lambda$7) for large amplitudes, so that exterior quadratic contributions can be absorbed into the coercive potential term for small truncation threshold $\lambda$8. Crucially, the Poisson source itself is also penalized, defining $\lambda$9 via the truncated source $a$0; the factor $a$1 in the derivative of the nonlocal term ensures exact recovery of $a$2 where the penalization is inactive.

On $a$3, the penalized functional exhibits mountain-pass geometry at scale $a$4: an energy barrier of height $a$5 on the sphere $a$6, an interior point below the barrier, and an exterior point with negative energy reached by dilating a compactly supported test function until the $a$7 nonlocal term dominates. Localized magnetic test functions

$a$8

carry the correct magnetic phase and satisfy, uniformly for $a$9,

$\varepsilon$0

which yields the sharp upper bound $\varepsilon$1 on the minimax level.

Compactness and concentration

Boundedness of low-energy constrained Palais–Smale sequences, together with boundedness of the Lagrange multipliers, is established by combining the energy relation with growth estimates for the truncated nonlinearities. Nonvanishing follows from Lions' vanishing lemma applied to rescaled functions: if mass spread out uniformly, all nonlinear contributions would vanish and the rescaled energy would tend to zero, contradicting $\varepsilon$2.

Compactness below the first critical threshold rests on two hypotheses made explicit in the key proposition: boundedness of Palais–Smale sequences at $\varepsilon$3 levels, and a uniform energy-defect lower bound $\varepsilon$4 carried by any nontrivial critical bubble of the limiting local–nonlocal critical problem. Under the gap condition $\varepsilon$5, the Palais–Smale condition holds at every level $\varepsilon$6. This is the analytic heart of the paper: since the mountain-pass level satisfies $\varepsilon$7, the critical point produced by the constrained mountain-pass theorem lies in the compactness range.

Concentration is then established in three steps. First, concentration points $\varepsilon$8 exist with $\varepsilon$9. Second, after gauge correction $\varepsilon \to 0$0, the sequence converges strongly in $\varepsilon \to 0$1 to an autonomous ground state; the points $\varepsilon \to 0$2 remain bounded (escape to infinity would leave only quadratic exterior nonlinearities, admitting no nontrivial limit), and $\varepsilon \to 0$3 follows from monotonicity of the autonomous level in the constant potential together with the energy bound. Third, uniform decay away from the concentration region is obtained via Moser iteration plus a second-bubble exclusion argument based on the full energy being carried by the first profile. As a consequence, the penalization becomes inactive: $\varepsilon \to 0$4 outside $\varepsilon \to 0$5, so every low-energy critical point of the penalized functional solves the original problem. This de-penalization step is what converts the auxiliary variational problem into Theorem 1.

Multiplicity and localization at the potential well

Multiplicity uses the standard barycenter scheme adapted to the mass constraint. The barycenter map $\varepsilon \to 0$6 is continuous, maps the low-energy sublevel $\varepsilon \to 0$7 into $\varepsilon \to 0$8 (by the concentration analysis above), and the localized test map $\varepsilon \to 0$9 satisfies $$(-i\varepsilon\nabla-A(x))^2u+V(x)u-\phi|u|^3u = \lambda u+\mu |u|^{q-2}u+|u|^4u, \qquad -\varepsilon^2\Delta\phi=|u|^5,$$0 in $$(-i\varepsilon\nabla-A(x))^2u+V(x)u-\phi|u|^3u = \lambda u+\mu |u|^{q-2}u+|u|^4u, \qquad -\varepsilon^2\Delta\phi=|u|^5,$$1. The category inequality then gives

$$(-i\varepsilon\nabla-A(x))^2u+V(x)u-\phi|u|^3u = \lambda u+\mu |u|^{q-2}u+|u|^4u, \qquad -\varepsilon^2\Delta\phi=|u|^5,$$2

and the Ljusternik–Schnirelmann theorem, combined with the Palais–Smale condition on the relevant range, produces at least $$(-i\varepsilon\nabla-A(x))^2u+V(x)u-\phi|u|^3u = \lambda u+\mu |u|^{q-2}u+|u|^4u, \qquad -\varepsilon^2\Delta\phi=|u|^5,$$3 distinct normalized solutions. Finally, global maximum points $$(-i\varepsilon\nabla-A(x))^2u+V(x)u-\phi|u|^3u = \lambda u+\mu |u|^{q-2}u+|u|^4u, \qquad -\varepsilon^2\Delta\phi=|u|^5,$$4 of $$(-i\varepsilon\nabla-A(x))^2u+V(x)u-\phi|u|^3u = \lambda u+\mu |u|^{q-2}u+|u|^4u, \qquad -\varepsilon^2\Delta\phi=|u|^5,$$5 satisfy $$(-i\varepsilon\nabla-A(x))^2u+V(x)u-\phi|u|^3u = \lambda u+\mu |u|^{q-2}u+|u|^4u, \qquad -\varepsilon^2\Delta\phi=|u|^5,$$6 relative to concentration points, hence $$(-i\varepsilon\nabla-A(x))^2u+V(x)u-\phi|u|^3u = \lambda u+\mu |u|^{q-2}u+|u|^4u, \qquad -\varepsilon^2\Delta\phi=|u|^5,$$7 and $$(-i\varepsilon\nabla-A(x))^2u+V(x)u-\phi|u|^3u = \lambda u+\mu |u|^{q-2}u+|u|^4u, \qquad -\varepsilon^2\Delta\phi=|u|^5,$$8.

An implication worth emphasizing is that the multiplicity bound depends only on the topology of $$(-i\varepsilon\nabla-A(x))^2u+V(x)u-\phi|u|^3u = \lambda u+\mu |u|^{q-2}u+|u|^4u, \qquad -\varepsilon^2\Delta\phi=|u|^5,$$9: for instance, if $|u|^4u$00 consists of $|u|^4u$01 connected components, the category lower bound is $|u|^4u$02, yielding at least $|u|^4u$03 distinct concentrating states for all sufficiently small $|u|^4u$04.

Limitations and open questions

Several restrictions qualify the results. The existence theorem requires both $|u|^4u$05 and $|u|^4u$06; nothing is claimed about large masses, and the smallness of $|u|^4u$07 enters through multiple simultaneous conditions (the barrier estimate, the subadditivity regime, and the gap $|u|^4u$08). The strict subadditivity of $|u|^4u$09 and the energy-defect bound involving $|u|^4u$10 are assumed as hypotheses in the compactness argument rather than derived, so the chain of implications is complete only modulo these facts. The restriction $|u|^4u$11 excludes the $|u|^4u$12-critical and supercritical regimes treated elsewhere in the non-magnetic literature. The concentration result identifies limits of maximum points but does not address uniqueness, orbital stability, or the sign and asymptotics of the Lagrange multipliers $|u|^4u$13. Whether the category lower bound is sharp, and whether solutions exist for potentials $|u|^4u$14 not bounded away from zero, remain open.

Conclusion

The paper extends the theory of normalized semiclassical states to a magnetic Schrödinger–Poisson system carrying both a Sobolev-critical local nonlinearity and a critical nonlocal Poisson term. Its technical contribution lies in combining a double penalization (of the local sources and of the Poisson source itself), diamagnetic-inequality-based estimates, a low-energy compactness threshold calibrated against the autonomous ground-state level $|u|^4u$15, and a barycenter/category argument—all carried out in the complex magnetic framework. The outcome is a complete qualitative picture for small mass and small $|u|^4u$16: existence, multiplicity governed by $|u|^4u$17, and concentration of maxima on the minimum set of the electric potential.