Some qualitative properties of normalized solutions for critical double phase problems with competing potentials

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.