Normalized solutions beyond the Hilbert setting
Normalized solutions have become an important topic in the study of nonlinear Schrödinger equations. Instead of prescribing the frequency parameter in advance, one looks for solutions satisfying a prescribed norm, while the corresponding Lagrange multiplier becomes part of the solution.
In our paper, “Normalized solution for quasilinear Schrödinger operators with general potentials: the Lᵖ-supercritical case”, we investigate this question for a quasilinear elliptic equation involving the p-Laplacian in ℝᴺ.
The prescribed constraint is
∫ℝᴺ |u|ᵖ dx = mᵖ,
so the problem is studied on the constraint
S(m) = {u ∈ W¹,ᵖ(ℝᴺ) : ‖u‖ₚ = m}.
Our main interest is the Lᵖ-supercritical regime
p + p²/N < q < p*.
The problem also includes a general nonnegative potential V.
Why is the quasilinear case more delicate?
A substantial part of the existing theory of normalized Schrödinger equations is developed in the Hilbert space H¹(ℝᴺ). In the quasilinear problem considered here, however, the natural framework is the Banach space W¹,ᵖ(ℝᴺ).
This distinction is important.
In a Hilbert space, closed subspaces admit orthogonal complements, a property that is frequently used in constrained variational arguments. This structure is no longer available in W¹,ᵖ(ℝᴺ) when p is different from 2.
As a consequence, constructing suitable Palais–Smale sequences for the constrained functional becomes more involved.
A Palais–Smale–Pohozaev approach
To overcome this difficulty, we use a variational framework based on the energy functional restricted to S(m), together with the associated Pohozaev structure.
An important ingredient is the construction of a locally Lipschitz pseudo-gradient vector field. Through the corresponding deformation argument, we obtain a Palais–Smale–Pohozaev sequence at the relevant variational level.
We then prove, under appropriate assumptions on the potential, that this sequence converges strongly to a nontrivial function. The Lagrange Multiplier Theorem finally provides a normalized weak solution of the original equation.
The role of the prescribed mass
The variational problem has mountain-pass geometry on the prescribed-mass constraint S(m). Another important part of the analysis is therefore to understand how the corresponding variational level depends on the mass m.
This dependence plays an important role in recovering compactness and obtaining a nontrivial critical point.
For the zero-potential problem, we prove the existence of a normalized solution for every sufficiently small prescribed mass m > 0 in the Lᵖ-supercritical range.
General nonnegative potentials
The analysis is then extended to a class of nonnegative potentials.
Under the assumptions introduced in the paper, we prove that the problem admits a normalized solution for sufficiently small mass also in the presence of a general potential V.
This makes the compactness analysis and the variational construction particularly important, since the presence of the potential introduces additional difficulties compared with the autonomous problem.
A variational framework for quasilinear normalized problems
One of the main features of the work is that the method provides a variational framework for normalized quasilinear Schrödinger problems in a genuine Banach-space setting.
The combination of the prescribed Lᵖ constraint, mountain-pass geometry, Pohozaev identities, pseudo-gradient deformation and strong convergence of Palais–Smale–Pohozaev sequences makes it possible to treat the Lᵖ-supercritical regime with general nonnegative potentials.
Our paper, “Normalized solution for quasilinear Schrödinger operators with general potentials: the Lᵖ-supercritical case”, was published in the Journal of Fixed Point Theory and Applications.
Read the paper:
https://doi.org/10.1007/s11784-026-01292-w