Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions

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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On the Number of Solutions to Semilinear Boundary Value Problems

Advanced Nonlinear Studies ◽

10.1515/ans-2004-0301 ◽

2004 ◽

Vol 4 (3) ◽

Cited By ~ 4

Author(s):

Markus Kunze ◽

Rafael Ortega

Keyword(s):

Boundary Conditions ◽

Boundary Value Problems ◽

Upper Bound ◽

Dirichlet Boundary ◽

Neumann Boundary ◽

Neumann Boundary Conditions ◽

Dirichlet Boundary Conditions ◽

Number Of Solutions ◽

Semilinear Elliptic ◽

Semilinear Elliptic Problems

AbstractWe consider semilinear elliptic problems of the form Δu + g(u) = f(x) with Neumann boundary conditions or Δu+λ1u+g(u) = f(x) with Dirichlet boundary conditions, and we derive conditions on g and f under which an upper bound on the number of solutions can be obtained.

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On the existence of positive solutions for semilinear elliptic equations with Neumann boundary conditions

Probability Theory and Related Fields ◽

10.1007/bf01375828 ◽

1995 ◽

Vol 101 (2) ◽

pp. 251-276

Author(s):

Z. Q. Chen ◽

R. J. Williams ◽

Z. Zhao

Keyword(s):

Boundary Conditions ◽

Positive Solutions ◽

Elliptic Equations ◽

Neumann Boundary ◽

Neumann Boundary Conditions ◽

Semilinear Elliptic Equations ◽

Semilinear Elliptic ◽

Existence Of Positive Solutions

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EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS TO ELLIPTIC PROBLEMS WITH SUBLINEAR MIXED BOUNDARY CONDITIONS

Communications in Contemporary Mathematics ◽

10.1142/s0219199709003508 ◽

2009 ◽

Vol 11 (04) ◽

pp. 585-613 ◽

Cited By ~ 11

Author(s):

JORGE GARCÍA-MELIÁN ◽

JULIO D. ROSSI ◽

JOSÉ C. SABINA DE LIS

Keyword(s):

Boundary Conditions ◽

Positive Solutions ◽

Nonlinear Boundary ◽

Mixed Boundary ◽

Elliptic Problems ◽

Mixed Boundary Conditions ◽

Nonlinear Boundary Conditions ◽

Semilinear Elliptic ◽

Semilinear Elliptic Problems ◽

Outward Unit

In this work, we consider a class of semilinear elliptic problems with nonlinear boundary conditions of mixed type. Under some monotonicity properties of the nonlinearities involved, we show that positive solutions are unique, and that their existence is characterized by the sign of some associated eigenvalues. One of the most important contributions of this work relies on the fact that we deal with boundary conditions of the form ∂u/∂ν = g(x,u) on Γ and u = 0 on Γ', where ν is the outward unit normal to Γ while Γ,Γ' are open, Γ ∩ Γ' = ∅, [Formula: see text], but [Formula: see text] need not be disjoint.

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