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 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 On a Parametric Nonlinear Dirichlet Problem with Subdiffusive and Equidiffusive Reaction

2014 ◽  
Vol 14 (3) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Patrick Winkert
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Elliptic Equation ◽  
Eigenvalue Problem ◽  
Mean Curvature ◽  
Nonlinear Eigenvalue Problem ◽  
Nodal Solution ◽  
Smooth Solutions ◽  
Nonlinear Dirichlet Problem ◽  
Nonhomogeneous Differential Operator

AbstractWe study a nonlinear parametric elliptic equation (nonlinear eigenvalue problem) driven by a nonhomogeneous differential operator. Our setting incorporates equations driven by the p-Laplacian, the (p, q)-Laplacian, and the generalized p-mean curvature differential operator. Applying variational methods we show that for λ > 0 (the parameter) sufficiently large the problem has at least three nontrivial smooth solutions whereby one is positive, one is negative and the last one has changing sign (nodal). In the particular case of (p, 2)-equations, using Morse theory, we produce another nodal solution for a total of four nontrivial smooth solutions.

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 Positive solutions for nonlinear parametric singular Dirichlet problems

2019 ◽  
Vol 09 (03) ◽  
pp. 1950011 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Positive Solutions ◽  
Singular Term ◽  
Principal Eigenvalue ◽  
Type Theorem ◽  
Dirichlet Problems

We consider a nonlinear parametric Dirichlet problem driven by the [Formula: see text]-Laplace differential operator and a reaction which has the competing effects of a parametric singular term and of a Carathéodory perturbation which is ([Formula: see text])-linear near [Formula: see text]. The problem is uniformly nonresonant with respect to the principal eigenvalue of [Formula: see text]. We look for positive solutions and prove a bifurcation-type theorem describing in an exact way the dependence of the set of positive solutions on the parameter [Formula: see text].

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 On the Dirichlet problem of elliptic type

2007 ◽  
Vol 75 (2) ◽  
pp. 169-177
Author(s):  
Marek Galewski
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Existence Of Solutions ◽  
Elliptic Type ◽  
Numerical Parameter ◽  
Dirichlet Problems ◽  
Dual Action

We investigate the existence of solutions and their stability for elliptic Dirichlet problems with nonlinearity of a convex-concave type. By relating the primal action and the dual action functionals on certain subsets of their domains we get the existence of solutions which are further stable with respect to a numerical parameter. We allow also for the differential operator to depend on a numerical parameter.

scholarly journals Nonlinear Dirichlet problems with unilateral growth on the reaction

2019 ◽  
Vol 31 (2) ◽  
pp. 319-340
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Dirichlet Problem ◽  
Critical Groups ◽  
Constant Sign ◽  
Nodal Solution ◽  
Nontrivial Solutions ◽  
Dirichlet Problems ◽  
Multiplicity Results ◽  
Subcritical Growth ◽  
Nonlinear Dirichlet Problem ◽  
The Third

AbstractWe consider a nonlinear Dirichlet problem driven by the p-Laplace differential operator with a reaction which has a subcritical growth restriction only from above. We prove two multiplicity theorems producing three nontrivial solutions, two of constant sign and the third nodal. The two multiplicity theorems differ on the geometry near the origin. In the semilinear case (that is, {p=2}), using Morse theory (critical groups), we produce a second nodal solution for a total of four nontrivial solutions. As an illustration, we show that our results incorporate and significantly extend the multiplicity results existing for a class of parametric, coercive Dirichlet problems.

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 Solutions For p-Laplacian Equations Which Need Not be Coercive

2008 ◽  
Vol 8 (2) ◽  
Author(s):  
Michael E. Filippakis ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Critical Point Theory ◽  
Point Theory ◽  
Smooth Solutions ◽  
Nonlinear Dirichlet Problem ◽  
Noncoercive Problems ◽  
Truncation Techniques ◽  
Laplacian Equations ◽  
Semilinear Problem

AbstractIn this paper, we consider nonlinear Dirichlet problem driven by the p-Laplacian differential operator. Using variational methods based on the critical point theory and truncation techniques, we prove the existence of at least three nontrivial smooth solutions. The hypotheses on the nonlinearity incorporate in our framework of analysis both coercive and noncoercive problems. For the semilinear problem (p = 2), using Morse theory, we show the existence of four nontrivial smooth solutions.

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