Sihua Liang

In this paper, we study a class of normalized solutions for the following doubly critical p -Laplacian Schrödinger–Bopp–Podolsky system in R 3 : { − Δ p u − ϕ | u | p ♯ − 2 u = λ | u | p − 2 u + μ | u | q − 2 u + | u | p ∗ − 2 u in  ℝ 3 , [ 4 p t ] − Δ ϕ + a 2 Δ 2 ϕ = 4 π | u | p ♯ in  ℝ 3 , subject to the mass constraint ∫ R 3 | u | p d x = ϱ p > 0 , the main purpose of the paper is to establish existence and multiplicity results for normalized solutions under both the L p -subcritical and L p -supercritical regimes. In the subcritical case, multiplicity is obtained by combining a truncation procedure with genus arguments. In the supercritical setting, we prove the existence of two distinct normalized solutions through the construction of a suitable auxiliary functional. In both situations, a concentration-compactness analysis plays a crucial role in recovering compactness in the presence of the doubly critical nonlinear structure. We also examine two singular limit processes. More precisely, we describe the asymptotic behaviour of solutions as the Lagrange multiplier λ → 0 and as the Bopp–Podolsky parameter a → 0 , and show that the system converges, in the corresponding limit, to the classical Schrödinger-Poisson model. The numerical part of the paper presents an approximation scheme that preserves the normalization constraint. The resulting computations, reported in tabular form, are consistent with the theoretical analysis and support the predicted existence of normalized solutions. In this way, the paper provides a unified variational and numerical treatment of the problem. This work seems to be the first attempt to prove multiplicity of normalized solutions for a doubly critical p -Laplacian Schrödinger–Bopp–Podolsky system in R 3 . The novelty of the paper mainly lies in the introduction of new analytical techniques adapted to this nonlocal quasilinear framework.

In this paper, we study the following fractional ‐Kirchhoff equation with critical growth: where is a small parameter, denotes the fractional ‐Laplacian operator with , and is the fractional critical Sobolev exponent. is the Kirchhoff function, is a continuous function with subcritical growth and is a positive continuous potential. Using the penalization method and Ljusternik‐Schnirelmann theory, we show that the existence, multiplicity, and concentration of nontrivial solutions as sufficiently small.

The paper provides some qualitative properties of normalized solutions of the following critical double phase problems with competing potentials: [Formula: see text] where [Formula: see text], with [Formula: see text], is the [Formula: see text]-Laplace operator, [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] is the Sobolev critical exponent associated to [Formula: see text], [Formula: see text] is a Lagrange multiplier, [Formula: see text] is the absorption potential, [Formula: see text] is the reaction potential and [Formula: see text] is a continuous function with subcritical growth. To address the challenges stemming from the presence of a critical growth and the competing potentials of the equation, we conduct meticulous analyses. More specifically, on the one hand, because of the appearance of the Sobolev critical nonlinearity the energy functional corresponding ( [Formula: see text] ) is unbounded from below on the given sphere, so that we have to use the trick of truncation and to prove some of its nice properties. Moreover, in order to show the crucial compactness condition, we make a clever use of the concentration compactness principle. On the other hand, since the [Formula: see text]-Laplacian and the competing potentials appear simultaneously in ( [Formula: see text] ), we need to make a detailed estimation of these competing terms. With the aid of minimization techniques, variational methods and the Lusternik–Schnirelmann category theory, we show existence of normalized solutions and the relation between the number of normalized solutions and the topology of the set where [Formula: see text] attains its global minimum and [Formula: see text] attains its global maximum. Next, we determine two concrete sets related to the potentials [Formula: see text] and [Formula: see text] as the concentration positions and describe the concentration of normalized solutions as [Formula: see text]. Finally, we also give a sufficient condition for non-existence of normalized solutions of ( [Formula: see text] ). The note complements and extends the papers, e.g., [C. O. Alves and N. V. Thin, On existence of multiple normalized solution to a class of elliptic problems in whole [Formula: see text] via Lusternik–Schnirelmann category, SIAM J. Math. Anal. 55 (2023) 1264–1283] due to Alves and Thin, [Double phase problems with competing potentials: Concentration and multiplication of ground states, Math. Z. 301 (2022) 4037–4078] by Zhang et al. and [Existence and concentration of normalized solutions for [Formula: see text]-Laplacian equations with logarithmic nonlinearity, J. Differ. Equ. 421 (2025) 1–49] by Shen and Squassina. Indeed, we deal with the effect of competing potentials and the critical nonlinearity on some qualitative properties of normalized solutions.

In this paper, we study the following critical Schrödinger–Bopp–Podolsky system with competing nonlinearities: [ ] and prescribed mass [ ] where [ ] [ ] [ ], and [ ]. The potential [ ] is a bounded and continuous nonnegative function, satisfying some suitable global conditions. The main feature of this paper consists in the competing effects of critical terms, nonlocal terms, and potential functions. To address this issue, the concentration-compactness principle is needed to overcome the loss of compactness of the energy functional due to critical growth. Meanwhile, by using the minimization techniques and the truncated argument, we show that the number of normalized solutions is not less than the number of global minimum points of [ ] when the parameter [ ] is sufficiently small. Furthermore, we study the asymptotic behaviors of normalized solutions as [ ] and as [ ], respectively. To the best of our knowledge, this study seems to be the first contribution regarding the concentration and asymptotic behavior for Schrödinger–Bopp–Podolsky systems.

The paper deals with the following critical Schrödinger–Bopp–Podolsky system [ ] where [ ] is a small parameter, [ ] is the Bopp–Podolsky constant, [ ] appears as a Lagrange multiplier, [ ], [ ] is the Sobolev critical exponent, [ ] is the [ ]-Laplace operator, the absorption potential [ ] and the reaction potential [ ] are continuous functions, and [ ] is a continuous function with subcritical growth at infinity. To our best knowledge, this paper is the first study on normalized solutions for a Schrödinger–Bopp–Podolsky system involving the [ ]-Laplacian, the Sobolev critical exponent and competing potentials. With the aid of minimization techniques and the Lusternik–Schnirelmann category theory, the existence, concentration, multiplicity and nonexistence of normalized solutions for system [ ] are obtained. To some extent, our main theorems complement and extend the results of Alves and Thin [On existence of multiple normalized solution to a class of elliptic problems in whole [ ] via Lusternik–Schnirelmann category, SIAM J. Math. Anal. 55 (2023) 1264–1283], D’Avenia and Siciliano [Nonlinear Schrödinger equation in the Bopp-Podolsky electrodynamics: Solutions in the electrostatic case, J. Differential Equations 267 (2019) 1025–1065], Santos et al. [Critical Schrödinger–Bopp–Podolsky systems: solutions in the semiclassical limit, Calc. Var. Partial Differential Equations 63 (2024) 155] and Zhang et al. [Double phase problems with competing potentials: concentration and multiplication of ground states, Math. Z. 301 (2022) 4037–4078].