Optimal bounds in semilinear elliptic problems with nonlinear boundary conditions

1993 ◽  
Vol 44 (4) ◽  
pp. 639-653 ◽  
Author(s):  
Ren� P. Sperb
Keyword(s):  
Boundary Conditions ◽  
Nonlinear Boundary ◽  
Elliptic Problems ◽  
Nonlinear Boundary Conditions ◽  
Semilinear Elliptic ◽  
Optimal Bounds ◽  
Semilinear Elliptic Problems

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

2009 ◽  
Vol 11 (04) ◽  
pp. 585-613 ◽  
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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