Elliptic Schrödinger Equations with Gradient-Dependent Nonlinearity and Hardy Potential Singular on Manifolds
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...
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Published in | The Journal of geometric analysis Vol. 35; no. 7 |
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Main Authors | , |
Format | Journal Article |
Language | English |
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New York
Springer Nature B.V
01.07.2025
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ISSN | 1050-6926 1559-002X |
DOI | 10.1007/s12220-025-02046-9 |
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Abstract | 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. |
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AbstractList | 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. |
ArticleNumber | 212 |
Author | Nguyen, Phuoc-Tai Gkikas, Konstantinos T. |
Author_xml | – sequence: 1 givenname: Konstantinos T. surname: Gkikas fullname: Gkikas, Konstantinos T. – sequence: 2 givenname: Phuoc-Tai surname: Nguyen fullname: Nguyen, Phuoc-Tai |
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Snippet | 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... |
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SubjectTerms | Boundary conditions Boundary value problems Elliptic functions Manifolds (mathematics) Schrodinger equation |
Title | Elliptic Schrödinger Equations with Gradient-Dependent Nonlinearity and Hardy Potential Singular on Manifolds |
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