Existence of Normalized Solutions for Mass Super-Critical Quasilinear Schrödinger Equation with Potentials.

Gao, Fengshuang; Guo, Yuxia

This paper is concerned with the existence of normalized solutions to a mass-supercritical quasilinear Schrödinger equation: 0.1 - Δ u - u Δ u 2 V (x) u λ u = g (u) , in R N , u ≥ 0 , satisfying the constraint ∫ R N u 2 = a . We will investigate how the potential and the nonlinearity effect the existence of the normalized solution. As a consequence, under a smallness assumption on V(x) and a relatively strict growth condition on g, we obtain a normalized solution for N = 2 , 3. Moreover, when V(x) is not too small in some sense, we show the existence of a normalized solution for N ≥ 2 and g (u) = u q - 2 u with 4 4 N < q < 2 · 2 ∗ .

Journal of Geometric Analysis, 2024, Vol 34, Issue 11, p1

1050-6926

Academic Journal

10.1007/s12220-024-01779-3