scholarly journals Positive solutions for (p, q)-equations with convection and a sign-changing reaction

2021 ◽  
Vol 11 (1) ◽  
pp. 40-57
Author(s):  
Shengda Zeng ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Smooth Solution ◽  
Variable Technique ◽  
Topological Methods ◽  
Nonlinear Dirichlet Problem

Abstract We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction which is dependent on the gradient. We look for positive solutions and we do not assume that the reaction is nonnegative. Using a mixture of variational and topological methods (the "frozen variable" technique), we prove the existence of a positive smooth solution.

Multiple positive solutions for a nonlinear Dirichlet problem with non-convex vector-valued response

2005 ◽  
Vol 135 (1) ◽  
pp. 105-117 ◽  
Author(s):  
Andrzej Nowakowski ◽  
Andrzej Rogowski
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Multiple Positive Solutions ◽  
Nonlinear Dirichlet Problem ◽  
Image Position ◽  
Vector Valued

In this paper we establish the existence of many positive solutions to with Vx a vector-valued nonlinearity.

scholarly journals Nonlinear nonhomogeneous Dirichlet problems with singular and convection terms

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Youpei Zhang
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Smooth Solution ◽  
Singular Term ◽  
Variable Method ◽  
Combined Effects ◽  
Dirichlet Problems ◽  
Nonlinear Dirichlet Problem ◽  
Nonhomogeneous Differential Operator

Abstract We consider a nonlinear Dirichlet problem driven by a general nonhomogeneous differential operator and with a reaction exhibiting the combined effects of a parametric singular term plus a Carathéodory perturbation $f(z,x,y)$ f ( z , x , y ) which is only locally defined in $x \in {\mathbb {R}} $ x ∈ R . Using the frozen variable method, we prove the existence of a positive smooth solution, when the parameter is small.

Dirichlet problems with singular and superlinear terms

2015 ◽  
Vol 17 (06) ◽  
pp. 1550056
Author(s):  
Sergiu Aizicovici ◽  
Nikolaos S. Papageorgiou ◽  
Vasile Staicu
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Dirichlet Problems ◽  
Nonlinear Dirichlet Problem ◽  
Truncation Techniques ◽  
Superlinear Perturbation

We consider a parametric nonlinear Dirichlet problem driven by the p-Laplacian, with a singular term and a p-superlinear perturbation, which need not satisfy the usual Ambrosetti–Rabinowitz condition. Using variational methods together with truncation techniques, we prove a bifurcation-type theorem describing the behavior of the set of positive solutions as the parameter varies.

scholarly journals Positive Solutions for Resonant (p, q)-equations with convection

2020 ◽  
Vol 10 (1) ◽  
pp. 217-232 ◽  
Author(s):  
Zhenhai Liu ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Fixed Point ◽  
Dirichlet Problem ◽  
Positive Solutions ◽  
Smooth Solution ◽  
Singular Term ◽  
Phase Problem ◽  
Drift Term ◽  
Double Phase ◽  
Double Phase Problem ◽  
Point Argument

Abstract We consider a nonlinear parametric Dirichlet problem driven by the (p, q)-Laplacian (double phase problem) with a reaction exhibiting the competing effects of three different terms. A parametric one consisting of the sum of a singular term and of a drift term (convection) and of a nonparametric perturbation which is resonant. Using the frozen variable method and eventually a fixed point argument based on an iterative asymptotic process, we show that the problem has a positive smooth solution.

scholarly journals Existence and Nonexistence of Positive Solutions for Singular (p, q)-Equations with Superdiffusive Perturbation

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Patrick Winkert
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Singular Term ◽  
Combined Effects ◽  
Nonlinear Dirichlet Problem ◽  
Nonexistence Result ◽  
Existence And Nonexistence

AbstractWe consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction which is parametric and exhibits the combined effects of a singular term and of a superdiffusive one. We prove an existence and nonexistence result for positive solutions depending on the value of the parameter $$\lambda \in \overset{\circ }{{\mathbb {R}}}_+=(0,+\infty )$$ λ ∈ R ∘ + = ( 0 , + ∞ ) .

scholarly journals Positive Solutions of the Dirichlet Problem for the One-dimensional Minkowski-Curvature Equation

2012 ◽  
Vol 12 (3) ◽  
Author(s):  
Isabel Coelho ◽  
Chiara Corsato ◽  
Franco Obersnel ◽  
Pierpaolo Omari
Keyword(s):  
Differential Equation ◽  
Dirichlet Problem ◽  
Positive Solutions ◽  
Singular Problem ◽  
Topological Methods ◽  
One Dimensional ◽  
Curvature Equation ◽  
Existence And Multiplicity ◽  
The One ◽  
Multiplicity Of Positive Solutions

AbstractWe discuss existence and multiplicity of positive solutions of the Dirichlet problem for the quasilinear ordinary differential equation.Depending on the behaviour of f = f (t, s) near s = 0, we prove the existence of either one, or two, or three, or infinitely many positive solutions. In general, the positivity of f is not required. All results are obtained by reduction to an equivalent non-singular problem to which variational or topological methods apply in a classical fashion.

scholarly journals (p, q)-Equations with Singular and Concave Convex Nonlinearities

2020 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Patrick Winkert
Keyword(s):  
Dirichlet Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Nonlinear Dirichlet Problem ◽  
Concave And Convex Nonlinearities

Abstract We consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian with $$1<q<p$$ 1 < q < p . The reaction is parametric and exhibits the competing effects of a singular term and of concave and convex nonlinearities. We are looking for positive solutions and prove a bifurcation-type theorem describing in a precise way the set of positive solutions as the parameter varies. Moreover, we show the existence of a minimal positive solution and we study it as a function of the parameter.

Multiplicity of Positive Solutions for Second Order Nonlinear Dirichlet Problem with One-dimension Minkowski-Curvature Operator

2015 ◽  
Vol 15 (4) ◽  
Author(s):  
Ruyun Ma ◽  
Yanqiong Lu
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Second Order ◽  
One Dimension ◽  
Curvature Operator ◽  
Multiple Positive Solutions ◽  
Nonlinear Dirichlet Problem ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

AbstractIn this work, we study the existence and multiplicity of positive solutions for nonlinear Dirichlet problem with one-dimension Minkowski-curvature operator,where k > 0 is a constant, λ > 0 is a parameter and f : [0,∞) → ℝ is continuous. We apply the quadrature arguments to prove how changes in the sign of f (u) lead to multiple positive solutions of the above problem for sufficiently large λ.

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