On p-Laplace equations with concave terms and asymmetric perturbations

2011 ◽  
Vol 141 (1) ◽  
pp. 171-192 ◽  
Author(s):  
D. Motreanu ◽  
V. V. Motreanu ◽  
N. S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Variational Principle ◽  
Point Theory ◽  
Smooth Solutions ◽  
Precise Information ◽  
Nonlinear Dirichlet Problem ◽  
Laplace Equations ◽  
Asymmetric Behaviour ◽  
Truncation Techniques ◽  
Concave Term

We consider a nonlinear Dirichlet problem driven by the p-Laplace differential operator with a concave term and a nonlinear perturbation, which exhibits an asymmetric behaviour near +∞ and near −∞. Namely, it is (p − 1)-superlinear on ℝ+ and (p − 1)-(sub)linear on ℝ−. Using variational methods based on the critical point theory together with truncation techniques, Ekeland's variational principle, Morse theory and the lower-and-upper-solutions approach, we show that the problem has at least four non-trivial smooth solutions. Also, we provide precise information about the sign of these solutions: two are positive, one is negative and one is nodal (sign changing).

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.

scholarly journals Dirichlet Problems with an Indefinite and Unbounded Potential and Concave-Convex Nonlinearities

2012 ◽  
Vol 2012 ◽  
pp. 1-36 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Smooth Solutions ◽  
Dirichlet Problems ◽  
Unbounded Potential ◽  
Indefinite Potential ◽  
Truncation Techniques ◽  
Concave Term

We consider a parametric semilinear Dirichlet problem with an unbounded and indefinite potential. In the reaction we have the competing effects of a sublinear (concave) term and of a superlinear (convex) term. Using variational methods coupled with suitable truncation techniques, we prove two multiplicity theorems for small values of the parameter. Both theorems produce five nontrivial smooth solutions, and in the second theorem we provide precise sign information for all the solutions.

scholarly journals Multiple Solutions for Nonlinear Dirichlet Problems with Concave Terms

2013 ◽  
Vol 113 (2) ◽  
pp. 206 ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Dirichlet Problem ◽  
Multiple Solutions ◽  
Critical Case ◽  
Critical Value ◽  
Constant Sign ◽  
Smooth Solutions ◽  
Dirichlet Problems ◽  
Truncation Techniques ◽  
Semilinear Problem ◽  
Concave Term

We consider a nonlinear parametric Dirichlet problem with parameter $\lambda>0$, driven by the $p$-Laplacian and with a concave term $\lambda|u|^{q-2}u$, $1<q<p$ and a Carathéodory perturbation $f(z,\zeta)$ which is asymptotically $(p-1)$-linear at infinity. Using variational methods combined with Morse theory and truncation techniques, we show that there is a critical value $\lambda^*>0$ of the parameter such that for $\lambda\in (0,\lambda^*)$ the problem has five nontrivial smooth solutions, four of constant sign (two positive and two negative) and the fifth nodal. In the semilinear case ($p=2$), we show that there is a sixth nontrivial smooth solution, but we cannot provide information about its sign. Finally for the critical case $\lambda=\lambda^*$, we show that the nonlinear problem ($p\ne 2$) still has two nontrivial constant sign smooth solutions and the semilinear problem ($p=2$) has three nontrivial smooth solutions, two of which have constant sign.

scholarly journals A MULTIPLICITY THEOREM FOR A VARIABLE EXPONENT DIRICHLET PROBLEM

2008 ◽  
Vol 50 (2) ◽  
pp. 335-349 ◽  
Author(s):  
NIKOLAOS S. PAPAGEORGIOU ◽  
EUGÉNIO M. ROCHA
Keyword(s):  
Dirichlet Problem ◽  
Variable Exponent ◽  
Point Theory ◽  
Critical Groups ◽  
Constant Sign ◽  
Nontrivial Solutions ◽  
Nonlinear Dirichlet Problem ◽  
Multiplicity Theorem ◽  
Noncoercive Problems ◽  
Truncation Techniques

AbstractWe consider a nonlinear Dirichlet problem driven by thep(ċ)-Laplacian. Using variational methods based on the critical point theory, together with suitable truncation techniques and the use of upper-lower solutions and of critical groups, we show that the problem has at least three nontrivial solutions, two of which have constant sign (one positive, the other negative). The hypotheses on the nonlinearity incorporates in our framework of analysis, both coercive and noncoercive problems.

scholarly journals Noncoercive resonant (p,2)-equations with concave terms

2018 ◽  
Vol 9 (1) ◽  
pp. 228-249 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Chao Zhang
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Morse Theory ◽  
Critical Groups ◽  
Smooth Solutions ◽  
Nonlinear Dirichlet Problem ◽  
Concave Term

Abstract We consider a nonlinear Dirichlet problem driven by the sum of a p-Laplace and a Laplacian (a {(p,2)} -equation). The reaction exhibits the competing effects of a parametric concave term plus a Caratheodory perturbation which is resonant with respect to the principle eigenvalue of the Dirichlet p-Laplacian. Using variational methods together with truncation and comparison techniques and Morse theory (critical groups), we show that for all small values of the parameter, the problem has as least six nontrivial smooth solutions all with sign information (two positive, two negative and two nodal (sign changing)).

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.

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.

Positive, Negative, and Nodal Solutions to Elliptic Differential Inclusions Depending on a Parameter

2013 ◽  
Vol 13 (2) ◽  
Author(s):  
Antonio Iannizzotto ◽  
Salvatore A. Marano ◽  
Dumitru Motreanu
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Differential Inclusion ◽  
Differential Inclusions ◽  
Smooth Solutions ◽  
Nodal Solutions ◽  
Homogeneous Dirichlet Problem ◽  
Partial Differential Inclusion ◽  
Truncation Techniques ◽  
Elliptic Differential Inclusions

AbstractThe homogeneous Dirichlet problem for a partial differential inclusion involving the p- Laplace operator and depending on a parameter λ > 0 is investigated. The existence of three smooth solutions, a smallest positive, a biggest negative, and a nodal one, is obtained for any λ sufficiently large by combining variational methods with truncation techniques.

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 TWO NONTRIVIAL WEAK SOLUTIONS FOR THE DIRICHLET PROBLEM ON THE SIERPIŃSKI GASKET

2011 ◽  
Vol 85 (3) ◽  
pp. 395-414 ◽  
Author(s):  
DENISA STANCU-DUMITRU
Keyword(s):  
Dirichlet Problem ◽  
Variational Principle ◽  
Critical Point ◽  
Weak Solutions ◽  
Critical Point Theory ◽  
Sierpinski Gasket ◽  
Point Theory ◽  
Sierpiński Gasket ◽  
Variational Techniques ◽  
Weak Laplacian

AbstractWe study a Dirichlet problem involving the weak Laplacian on the Sierpiński gasket, and we prove the existence of at least two distinct nontrivial weak solutions using Ekeland’s Variational Principle and standard tools in critical point theory combined with corresponding variational techniques.

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