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 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.

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.

Positive Solutions for the Generalized Nonlinear Logistic Equations

2016 ◽  
Vol 59 (01) ◽  
pp. 73-86 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Elliptic Equation ◽  
Positive Solutions ◽  
Type Result ◽  
Logistic Equations ◽  
Nonhomogeneous Differential Operator

AbstractWe consider a nonlinear parametric elliptic equation driven by a nonhomogeneous differential operator with a logistic reaction of the superdiòusive type. Using variationalmethods coupled with suitable truncation and comparison techniques, we prove a bifurcation type result describing the set of positive solutions as the parameter varies.

scholarly journals The Dirichlet problem for gradient dependent prescribed mean curvature equations in the Lorentz–Minkowski space

2017 ◽  
Vol 24 (1) ◽  
pp. 113-134 ◽  
Author(s):  
Chiara Corsato ◽  
Franco Obersnel ◽  
Pierpaolo Omari
Keyword(s):  
Dirichlet Problem ◽  
Differential Operator ◽  
Minkowski Space ◽  
Mean Curvature ◽  
Differential Operators ◽  
Curvature Equation ◽  
Prescribed Mean Curvature ◽  
Mean Curvature Equation ◽  
Prescribed Mean Curvature Equation ◽  
Mean Curvature Equations

AbstractWe discuss existence, multiplicity, localisation and stability properties of solutions of the Dirichlet problem associated with the gradient dependent prescribed mean curvature equation in the Lorentz–Minkowski space$\left\{\begin{aligned} \displaystyle{-}\operatorname{div}\biggl{(}\frac{\nabla u% }{\sqrt{1-|\nabla u|^{2}}}\biggr{)}&\displaystyle=f(x,u,\nabla u)&&% \displaystyle\phantom{}\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega.\end{aligned}\right.$The obtained results display various peculiarities, which are due to the special features of the involved differential operator and have no counterpart for elliptic problems driven by other quasilinear differential operators. This research is also motivated by some recent achievements in the study of prescribed mean curvature graphs in certain Friedmann–Lemaître–Robertson–Walker, as well as Schwarzschild–Reissner–Nordström, spacetimes.

A Multiplicity Theorem for p-Superlinear Neumann Problems with a Nonhomogeneous Differential Operator

2014 ◽  
Vol 14 (4) ◽  
Author(s):  
Giuseppina Barletta ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Differential Operator ◽  
Critical Point Theory ◽  
Point Theory ◽  
Smooth Solutions ◽  
Neumann Problems ◽  
Nonhomogeneous Differential Operator ◽  
Nonlinear Neumann Problem ◽  
Multiplicity Theorem ◽  
Truncation Techniques ◽  
Special Case

AbstractWe consider a nonlinear Neumann problem driven by a nonhomogeneous differential operator (special case is the p-Laplacian) with a (p − 1)-superlinear Carathéodory reaction term, which need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition. Using variational methods based on the critical point theory coupled with suitable truncation techniques, we show that the problem has at least five nontrivial smooth solutions.

Positive solutions of Kirchhoff-type non-local elliptic equation: a bifurcation approach

2017 ◽  
Vol 147 (4) ◽  
pp. 875-894 ◽  
Author(s):  
Zhanping Liang ◽  
Fuyi Li ◽  
Junping Shi
Keyword(s):  
Elliptic Equation ◽  
Eigenvalue Problem ◽  
Positive Solutions ◽  
Nonlinear Eigenvalue Problem ◽  
Global Bifurcation ◽  
Integral Term ◽  
Kirchhoff Type ◽  
Nonlinear Elliptic ◽  
Linear Eigenvalue Problem ◽  
Non Local

Positive solutions of a Kirchhoff-type nonlinear elliptic equation with a non-local integral term on a bounded domain in ℝN, N ⩾ 1, are studied by using bifurcation theory. The parameter regions of existence, non-existence and uniqueness of positive solutions are characterized by the eigenvalues of a linear eigenvalue problem and a nonlinear eigenvalue problem. Local and global bifurcation diagrams of positive solutions for various parameter regions are obtained.

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 Eigenvalue problem for the weighted p-Laplacian

2015 ◽  
Vol 63 (1) ◽  
pp. 269-281
Author(s):  
Ewa Szlachtowska ◽  
Dominik Mielczarek
Keyword(s):  
Dirichlet Problem ◽  
Sobolev Space ◽  
Eigenvalue Problem ◽  
Nonlinear Eigenvalue Problem ◽  
Nonlinear Eigenvalue

Abstract In this paper we are concerned with the nonlinear eigenvalue problem for the weighted p-Laplacian. The main result of the work is the existence of the eigenpair for the Dirichlet problem provided that the weights are bounded. Furthermore, under this assumption the eigenvector belongs to the Sobolev space W1,p0 (Ω).

scholarly journals CONSTANT SIGN AND NODAL SOLUTIONS FOR NONLINEAR ROBIN EQUATIONS WITH LOCALLY DEFINED SOURCE TERM

2020 ◽  
Vol 25 (3) ◽  
pp. 374-390
Author(s):  
Nikolaos S. Papageorgiou ◽  
Calogero Vetro ◽  
Francesca Vetro
Keyword(s):  
Differential Operator ◽  
Source Term ◽  
Constant Sign ◽  
Robin Problem ◽  
Smooth Solutions ◽  
Nodal Solutions ◽  
Nonhomogeneous Differential Operator ◽  
Special Cases ◽  
Variational Tools ◽  
Nonlinear Nonhomogeneous Differential Operator

We consider a parametric Robin problem driven by a nonlinear, nonhomogeneous differential operator which includes as special cases the p-Laplacian and the (p,q)-Laplacian. The source term is parametric and only locally defined (that is, in a neighborhood of zero). Using suitable cut-off techniques together with variational tools and comparison principles, we show that for all big values of the parameter, the problem has at least three nontrivial smooth solutions, all with sign information (positive, negative and nodal).

Sign in / Sign up

Export Citation Format

Share Document