Prof. Existence of normalized solutions for elliptic equation with Hardy potential and nonautonomous term

Abstract

The present work is devoted to the existence of normalized solutions for elliptic equations with a Hardy potential and a nonautonomous term. Such equations are closely connected with standing wave solutions of nonlinear Schrödinger equations with singular potentials, which arise widely in mathematical physics. Here, normalized solutions are understood as solutions satisfying a prescribed L^2 norm constraint, this constraint fundamentally distinguishes the problem from classical unconstrained variational problems. The L^2 supercritical case is considered, for which the energy functional is unbounded from below on ℵ(κ). To overcome this difficulty, a suitable manifold is constructed via an L^2-norm-preserving scaling transformation, and the analysis is carried out on this natural constraint. Several analytical tools, including the Hardy inequality, the Gagliardo–Nirenberg inequality, and variational methods, are combined to systematically examine the variational structure of the constrained minimization problem. Since the problem is set on R^N, a primary difficulty arises from the lack of compactness. To address this issue, the Schwarz symmetric decreasing rearrangement technique is employed, and the problem is reduced to the radial symmetric function space. This space possesses a favorable compact embedding property in the subcritical range 2 < q < 2N/(N+2) which compensates for the loss of compactness in the whole space and allows us to extract a strongly convergent subsequence from the constructed radially rearranged minimizing sequence.