Elliptic Schrödinger Equations with Gradient-Dependent Nonlinearity and Hardy Potential Singular on Manifolds

• Published in: The Journal of geometric analysis Vol. 35; no. 7
• Main Authors: Gkikas, Konstantinos T., Nguyen, Phuoc-Tai
• Format: Journal Article
• Language: English
• Published: New York Springer Nature B.V 01.07.2025
• Subjects: Boundary conditions; Boundary value problems; Elliptic functions; Manifolds (mathematics); Schrodinger equation
• ISSN: 1050-6926; 1559-002X
• DOI: 10.1007/s12220-025-02046-9

Let Ω⊂RN (N≥3) be a C2 bounded domain and Σ⊂Ω is a C2 compact boundaryless submanifold in RN of dimension k, 0≤k<N-2. For μ≤(N-k-22)2, put Lμ:=Δ+μdΣ-2 where dΣ(x)=dist(x,Σ). We study boundary value problems for equation -Lμu=g(u,|∇u|) in Ω\Σ, subject to the boundary condition u=ν on ∂Ω∪Σ, where g:R×R+→R+ is a continuous and nondecreasing function with g(0,0)=0, ν is a given nonnegative measure on ∂Ω∪Σ. When g satisfies a so-called subcritical integral condition, we establish an existence result for the problem under a smallness assumption on ν. If g(u,|∇u|)=|u|p|∇u|q, there are ranges of p, q, called subcritical ranges, for which the subcritical integral condition is satisfied, hence the problem admits a solution. Beyond these ranges, where the subcritical integral condition may be violated, we establish various criteria on ν for the existence of a solution to the problem expressed in terms of appropriate Bessel capacities.