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 Singular Dirichlet (p, q)-Equations

2021 ◽  
Vol 18 (4) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Patrick Winkert
Keyword(s):  
Dirichlet Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Combined Effects ◽  
Type Result ◽  
Nonlinear Dirichlet Problem ◽  
Continuity Properties ◽  
Superlinear Perturbation

AbstractWe consider a nonlinear Dirichlet problem driven by the (p, q)-Laplacian and with a reaction having the combined effects of a singular term and of a parametric $$(p-1)$$ ( p - 1 ) -superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $$\lambda >0$$ λ > 0 varies. Moreover, we prove the existence of a minimal positive solution $$u^*_\lambda $$ u λ ∗ and study the monotonicity and continuity properties of the map $$\lambda \rightarrow u^*_\lambda $$ λ → u λ ∗ .

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 (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.

scholarly journals A singular eigenvalue problem for the Dirichlet (p, q)-Laplacian

2021 ◽  
Author(s):  
Yunru Bai ◽  
Nikolaos S. Papageorgiou ◽  
Shengda Zeng
Keyword(s):  
Dirichlet Problem ◽  
Eigenvalue Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Singular Term ◽  
Type Theorem ◽  
Solution Map ◽  
Continuity Properties ◽  
Variational Tools ◽  
Superlinear Reaction

AbstractWe consider a parametric nonlinear, nonhomogeneous Dirichlet problem driven by the (p, q)-Laplacian with a reaction involving a singular term plus a superlinear reaction which does not satisfy the Ambrosetti–Rabinowitz condition. The main goal of the paper is to look for positive solutions and our approach is based on the use of variational tools combined with suitable truncations and comparison techniques. We prove a bifurcation-type theorem describing in a precise way the dependence of the set of positive solutions on the parameter $$\lambda $$ λ . Moreover, we produce minimal positive solutions and determine the monotonicity and continuity properties of the minimal positive solution map.

scholarly journals Positive solutions for anisotropic singular $$\varvec{(p,q)}$$-equations

2020 ◽  
Vol 71 (5) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Andrea Scapellato
Keyword(s):  
Dirichlet Problem ◽  
Positive Solutions ◽  
Combined Effects ◽  
Smooth Solutions ◽  
Nonlinear Elliptic ◽  
Variational Tools

Abstract We consider a nonlinear elliptic Dirichlet problem driven by the anisotropic (p, q)-Laplacian and with a reaction which is nonparametric and has the combined effects of a singular and of a superlinear terms. Using variational tools together with truncation and comparison techniques, we show that the problem has at least two positive smooth solutions.

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 MULTIPLE POSITIVE SOLUTIONS FOR A CRITICAL GROWTH PROBLEM WITH HARDY POTENTIAL

2006 ◽  
Vol 49 (1) ◽  
pp. 53-69 ◽  
Author(s):  
Pigong Han
Keyword(s):  
Dirichlet Problem ◽  
Positive Solution ◽  
Positive Solutions ◽  
Variational Approach ◽  
Critical Growth ◽  
Multiple Positive Solutions ◽  
Hardy Potential ◽  
Unique Positive Solution ◽  
Existence And Nonexistence ◽  
Growth Problem

AbstractIn this paper we study the existence and nonexistence of multiple positive solutions for the Dirichlet problem:$$ -\Delta{u}-\mu\frac{u}{|x|^2}=\lambda(1+u)^p,\quad u\gt0,\quad u\in H^1_0(\varOmega), \tag{*} $$where $0\leq\mu\lt(\frac{1}{2}(N-2))^2$, $\lambda\gt0$, $1\ltp\leq(N+2)/(N-2)$, $N\geq3$. Using the sub–supersolution method and the variational approach, we prove that there exists a positive number $\lambda^*$ such that problem (*) possesses at least two positive solutions if $\lambda\in(0,\lambda^*)$, a unique positive solution if $\lambda=\lambda^*$, and no positive solution if $\lambda\in(\lambda^*,\infty)$.

scholarly journals Asymptotic behavior of positive solutions of a singular nonlinear Dirichlet problem

2010 ◽  
Vol 369 (2) ◽  
pp. 719-729 ◽  
Author(s):  
Sabrine Gontara ◽  
Habib Mâagli ◽  
Syrine Masmoudi ◽  
Sameh Turki
Keyword(s):  
Asymptotic Behavior ◽  
Dirichlet Problem ◽  
Positive Solutions ◽  
Nonlinear Dirichlet Problem

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.

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