Multiple solutions for some Neumann problems in exterior domains

In this paper, we show that if q(x) satisfies suitable conditions, then the Neumann problem -Δu+u = q(x)ⅠuⅠp−2u in Ω has at least two solutions of which one is positive and the other changes sign.

Download Full-text

Multiple Solutions for Nonlinear Neumann Problems with Asymmetric Reaction, via Morse Theory

Advanced Nonlinear Studies ◽

10.1515/ans-2011-0402 ◽

2011 ◽

Vol 11 (4) ◽

Cited By ~ 4

Author(s):

Leszek Gasiński ◽

Nikolaos S. Papageorgiou

Keyword(s):

Variational Methods ◽

Neumann Problem ◽

Morse Theory ◽

Multiple Solutions ◽

Smooth Solutions ◽

Asymmetric Reaction ◽

Neumann Problems ◽

Nonlinear Neumann Problem ◽

Asymmetric Behaviour ◽

Nonlinear Neumann Problems

AbstractWe consider a nonlinear Neumann problem driven by the p-Laplacian and with a reaction which exhibits an asymmetric behaviour near +∞ and near −∞. Namely, it is (p − 1)- superlinear near +∞ (but need not satisfy the Ambrosetti-Rabinowitz condition) and it is (p − 1)-linear near −∞. Combining variational methods with Morse theory, we show that the problem has at least three nontrivial smooth solutions.

Download Full-text

Duality by reproducing kernels

International Journal of Mathematics and Mathematical Sciences ◽

10.1155/s0161171203206037 ◽

2003 ◽

Vol 2003 (6) ◽

pp. 327-395 ◽

Cited By ~ 6

Author(s):

A. Shlapunov ◽

N. Tarkhanov

Keyword(s):

Differential Operator ◽

Neumann Problem ◽

Compact Manifold ◽

Reproducing Kernels ◽

Elliptic Differential Operator ◽

Regularity Theorem ◽

Neumann Problems ◽

Smooth Compact Manifold ◽

Convexity Properties ◽

The Neumann Problem

LetAbe a determined or overdetermined elliptic differential operator on a smooth compact manifoldX. Write𝒮A(𝒟)for the space of solutions of the systemAu=0in a domain𝒟⋐X. Using reproducing kernels related to various Hilbert structures on subspaces of𝒮A(𝒟), we show explicit identifications of the dual spaces. To prove the regularity of reproducing kernels up to the boundary of𝒟, we specify them as resolution operators of abstract Neumann problems. The matter thus reduces to a regularity theorem for the Neumann problem, a well-known example being the∂¯-Neumann problem. The duality itself takes place only for those domains𝒟which possess certain convexity properties with respect toA.

Download Full-text

Multiple solutions for superlinear double phase Neumann problems

Revista de la Real Academia de Ciencias Exactas Físicas y Naturales Serie A Matemáticas ◽

10.1007/s13398-021-01164-7 ◽

2021 ◽

Vol 116 (1) ◽

Author(s):

Nikolaos S. Papageorgiou ◽

Vicenţiu D. Rădulescu ◽

Youpei Zhang

Keyword(s):

Neumann Problem ◽

Ground State ◽

Multiple Solutions ◽

Nehari Manifold ◽

Ground State Solutions ◽

Neumann Problems ◽

Double Phase ◽

The Nehari Manifold ◽

Superlinear Reaction

AbstractWe study a double phase Neumann problem with a superlinear reaction which need not satisfy the Ambrosetti-Rabinowitz condition. Using the Nehari manifold method, we show that the problem has at least three nontrivial bounded ground state solutions, all with sign information (positive, negative and nodal).

Download Full-text