Very weak solutions to elliptic equations with nonlinear Neumann boundary conditions

2008 ◽  
Vol 32 (4) ◽  
pp. 429-452 ◽  
Author(s):  
P. Quittner ◽  
W. Reichel
Keyword(s):  
Boundary Conditions ◽  
Weak Solutions ◽  
Elliptic Equations ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Very Weak Solutions ◽  
Nonlinear Neumann Boundary Conditions

scholarly journals Multiple solutions for a class of bi-nonlocal problems with nonlinear Neumann boundary conditions

2022 ◽  
Vol 40 ◽  
pp. 1-11
Author(s):  
Ghasem A. Afrouzi ◽  
Z. Naghizadeh ◽  
Nguyen Thanh Chung
Keyword(s):  
Boundary Conditions ◽  
Variational Methods ◽  
Weak Solutions ◽  
Multiple Solutions ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Nonlocal Problems ◽  
Mapping Theory ◽  
Nonlinear Neumann Boundary Conditions

In this paper, we are interested in a class of bi-nonlocal problems with nonlinear Neumann boundary conditions and sublinear terms at infinity. Using $(S_+)$ mapping theory and variational methods, we establish the existence of at least two non-trivial weak solutions for the problem provied that the parameters are large enough. Our result complements and improves some previous ones for the superlinear case when the Ambrosetti-Rabinowitz type conditions are imposed on the nonlinearities.

scholarly journals Extremal solutions to Liouville–Gelfand type elliptic problems with nonlinear Neumann boundary conditions

2015 ◽  
Vol 17 (03) ◽  
pp. 1450016
Author(s):  
Futoshi Takahashi
Keyword(s):  
Boundary Conditions ◽  
Weak Solutions ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Extremal Solutions ◽  
Smooth Bounded Domain ◽  
Suitable Notion ◽  
Extremal Parameter ◽  
Nonlinear Neumann Boundary Conditions ◽  
Increasing Function

Consider the Liouville–Gelfand type problems with nonlinear Neumann boundary conditions [Formula: see text] where Ω ⊂ ℝN, N ≥ 2, is a smooth bounded domain, f : [0, +∞) → (0, +∞) is a smooth, strictly positive, convex, increasing function with superlinear at +∞, and λ > 0 is a parameter. In this paper, after introducing a suitable notion of weak solutions, we prove several properties of extremal solutions u* corresponding to λ = λ*, called an extremal parameter, such as regularity, uniqueness, and the existence of weak eigenfunctions associated to the linearized extremal problem.

Symmetry properties of positive solutions of elliptic equations in an infinite sectorial cone

1992 ◽  
Vol 122 (1-2) ◽  
pp. 137-160
Author(s):  
Chie-Ping Chu ◽  
Hwai-Chiuan Wang
Keyword(s):  
Boundary Conditions ◽  
Positive Solution ◽  
Positive Solutions ◽  
Elliptic Equations ◽  
Neumann Boundary ◽  
Neumann Boundary Conditions ◽  
Semilinear Elliptic Equations ◽  
Spherically Symmetric ◽  
Semilinear Elliptic ◽  
Symmetry Properties

SynopsisWe prove symmetry properties of positive solutions of semilinear elliptic equations Δu + f(u) = 0 with Neumann boundary conditions in an infinite sectorial cone. We establish that any positive solution u of such equations in an infinite sectorial cone ∑α in ℝ3 is spherically symmetric if the amplitude α of ∑α is not greater than π.

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