scholarly journals On bounded solutions of nonlinear first- and second-order equations with a Carathéodory function

2007 ◽  
Vol 334 (2) ◽  
pp. 1462-1480
Author(s):  
Wioletta Karpińska
Keyword(s):  
Second Order ◽  
Bounded Solutions ◽  
Carathéodory Function ◽  
Second Order Equations

On a structure of the set of positive solutions to second-order equations with a super-linear non-linearity

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Jiří Šremr
Keyword(s):  
Positive Solution ◽  
Positive Solutions ◽  
Periodic Problem ◽  
Second Order ◽  
Solution Set ◽  
Existence And Multiplicity ◽  
Carathéodory Function ◽  
Multiplicity Of Positive Solutions ◽  
Lower And Upper Functions ◽  
Second Order Equations

Abstract We study the existence and multiplicity of positive solutions to the periodic problem u ′′ = p ⁢ ( t ) ⁢ u - q ⁢ ( t , u ) ⁢ u + f ⁢ ( t ) ; u ⁢ ( 0 ) = u ⁢ ( ω ) , u ′ ⁢ ( 0 ) = u ′ ⁢ ( ω ) , u^{\prime\prime}=p(t)u-q(t,u)u+f(t);\quad u(0)=u(\omega),\quad u^{\prime}(0)=u^{\prime}(\omega), where p , f ∈ L ⁢ ( [ 0 , ω ] ) p,f\in L([0,\omega]) and q : [ 0 , ω ] × R → R q\colon[0,\omega]\times\mathbb{R}\to\mathbb{R} is a Carathéodory function. By using the method of lower and upper functions, we show some properties of the solution set of the considered problem and, in particular, the existence of a minimal positive solution.

scholarly journals A Note on the Strong Maximum Principle for Fully Nonlinear Equations on Riemannian Manifolds

2021 ◽  
Author(s):  
Alessandro Goffi ◽  
Francesco Pediconi
Keyword(s):  
Maximum Principle ◽  
Riemannian Manifolds ◽  
Mean Curvature ◽  
Strong Maximum Principle ◽  
Second Order ◽  
Fully Nonlinear Equations ◽  
Nonlinear Operators ◽  
Fully Nonlinear ◽  
Second Order Equations ◽  
Curvature Type

AbstractWe investigate strong maximum (and minimum) principles for fully nonlinear second-order equations on Riemannian manifolds that are non-totally degenerate and satisfy appropriate scaling conditions. Our results apply to a large class of nonlinear operators, among which Pucci’s extremal operators, some singular operators such as those modeled on the p- and $$\infty $$ ∞ -Laplacian, and mean curvature-type problems. As a byproduct, we establish new strong comparison principles for some second-order uniformly elliptic problems when the manifold has nonnegative sectional curvature.

scholarly journals Boundary value problems for singular second order equations

2018 ◽  
Vol 2018 (1) ◽  
Author(s):  
Alessandro Calamai ◽  
Cristina Marcelli ◽  
Francesca Papalini
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Second Order ◽  
Second Order Equations
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