A multiplicity theorem for p-superlinear p-Laplacian equations using critical groups and morse theory

2010 ◽  
Vol 163 (4) ◽  
pp. 471-491
Author(s):  
Sophia Th. Kyritsi ◽  
Donal O’Regan ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Morse Theory ◽  
Critical Groups ◽  
Multiplicity Theorem ◽  
Laplacian Equations

Nonlinear Parametric Robin Problems with Combined Nonlinearities

2015 ◽  
Vol 15 (3) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Growth Condition ◽  
Morse Theory ◽  
Constant Sign ◽  
Robin Problem ◽  
Three Solutions ◽  
Multiplicity Theorem ◽  
Solutions Of Constant Sign ◽  
Semilinear Problem ◽  
Concave Term

AbstractWe consider a nonlinear parametric Robin problem driven by the p-Laplacian. We assume that the reaction exhibits a concave term near the origin. First we prove a multiplicity theorem producing three solutions with sign information (positive, negative and nodal) without imposing any growth condition near ±∞ on the reaction. Then, for problems with subcritical reaction, we produce two more solutions of constant sign, for a total of five solutions. For the semilinear problem (that is, for p = 2), we generate a sixth solution but without any sign information. Our approach is variational, coupled with truncation, perturbation and comparison techniques and with Morse theory.

Morse Theory and Critical Groups

2019 ◽  
pp. 457-555
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rădulescu ◽  
Dušan D. Repovš
Keyword(s):  
Morse Theory ◽  
Critical Groups

NONTRIVIAL SOLUTIONS FOR N-LAPLACIAN EQUATIONS WITH SUB-EXPONENTIAL GROWTH

2013 ◽  
Vol 11 (03) ◽  
pp. 1350005 ◽  
Author(s):  
ZHONG TAN ◽  
FEI FANG
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Exponential Growth ◽  
Morse Theory ◽  
Smooth Boundary ◽  
Orlicz Spaces ◽  
Nontrivial Solutions ◽  
Laplacian Equations

Let Ω be a bounded domain in RNwith smooth boundary ∂Ω. In this paper, the following Dirichlet problem for N-Laplacian equations (N > 1) are considered: [Formula: see text] We assume that the nonlinearity f(x, t) is sub-exponential growth. In fact, we will prove the mapping f(x, ⋅): LA(Ω) ↦ LÃ(Ω) is continuous, where LA(Ω) and LÃ(Ω) are Orlicz spaces. Applying this result, the compactness conditions would be obtained. Hence, we may use Morse theory to obtain existence of nontrivial solutions for problem (N).

Critical groups and multiple solutions for Kirchhoff type equations with critical exponents

2020 ◽  
pp. 2050031
Author(s):  
Mingzheng Sun ◽  
Jiabao Su ◽  
Binlin Zhang
Keyword(s):  
Critical Exponents ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Nonlinear Term ◽  
Type Equation ◽  
Critical Groups ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
Kirchhoff Type ◽  
The First Eigenvalue

In this paper, by Morse theory we will study the Kirchhoff type equation with an additional critical nonlinear term, and the main results are to compute the critical groups including the cases where zero is a mountain pass solution and the nonlinearity is resonant at zero. As an application, the multiplicity of nontrivial solutions for this equation with the parameter across the first eigenvalue is investigated under appropriate assumptions. To our best knowledge, estimates of our critical groups are new even for the Kirchhoff type equations with subcritical nonlinearities.

Three non-trivial solutions for not necessarily coercive p-Laplacian equations

2009 ◽  
Vol 52 (3) ◽  
pp. 679-688
Author(s):  
Shouchuan Hu ◽  
Nikolas S. Papageorgiou
Keyword(s):  
Upper And Lower Solutions ◽  
Elliptic Problems ◽  
Critical Groups ◽  
Constant Sign ◽  
Smooth Solutions ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic ◽  
Truncation Techniques ◽  
Laplacian Equations ◽  
Coercive Problems

AbstractWe consider the existence of three non-trivial smooth solutions for nonlinear elliptic problems driven by the p-Laplacian. Using variational arguments, coupled with the method of upper and lower solutions, critical groups and suitable truncation techniques, we produce three non-trivial smooth solutions, two of which have constant sign. The hypotheses incorporate both coercive and non-coercive problems in our framework of analysis.

Extremal, Nodal and Stable Solutions for Nonlinear Elliptic Equations

2015 ◽  
Vol 15 (3) ◽  
Author(s):  
Leszek Gasiński ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Eigenvalue Problems ◽  
Principal Eigenvalue ◽  
Nonlinear Elliptic Equations ◽  
Nodal Solutions ◽  
The Past ◽  
Nonlinear Elliptic ◽  
Multiplicity Theorem ◽  
The Stability ◽  
Laplacian Equations ◽  
Special Case

AbstractWe consider Dirichlet p-Laplacian equations which may be resonant with respect to the principal eigenvalue at ±∞. We show the existence of extremal nontrivial constant sign solutions and of nodal solutions. In the semilinear case (p = 2) we produce additional nodal solutions. We show that certain parametric equations (eigenvalue problems) studied in the past, are a special case of our multiplicity theorem. Finally, we establish the stability of the extremal solutions.

scholarly journals Stationary Boltzmann equation: an approach via Morse theory

2021 ◽  
Vol 40 (6) ◽  
pp. 1473-1487
Author(s):  
Rafael Galeano Andrades ◽  
Joel Torres del Valle
Keyword(s):  
Boltzmann Equation ◽  
Critical Points ◽  
Morse Theory ◽  
Morse Index ◽  
Critical Groups

In this paper we study the unidimensional Stationary Boltzmann Equation by an approach via Morse theory. We define a functional J whose critical points coincide with the solutions of the Stationary Boltzmann Equation. By the calculation of Morse index of J’’0(0)h and the critical groups C2(J, 0) and C2(J, ∞) we prove that J has two different critical points u1 and u2 different from 0, that is, solutions of Boltzmann Equation.

scholarly journals Existence results for double-phase problems via Morse theory

2017 ◽  
Vol 20 (02) ◽  
pp. 1750023 ◽  
Author(s):  
Kanishka Perera ◽  
Marco Squassina
Keyword(s):  
Morse Theory ◽  
Existence Results ◽  
Critical Groups ◽  
Nontrivial Solutions ◽  
Direct Sum ◽  
Double Phase ◽  
Direct Sum Decomposition

We obtain nontrivial solutions for a class of double-phase problems using Morse theory. In the absence of a direct sum decomposition, we use a cohomological local splitting to get an estimate of the critical groups at zero.

scholarly journals Anisotropic double-phase problems with indefinite potential: multiplicity of solutions

2020 ◽  
Vol 10 (4) ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Dongdong Qin ◽  
Vicenţiu D. Rădulescu
Keyword(s):  
Maximum Principle ◽  
Critical Groups ◽  
Phase Problem ◽  
Smooth Solutions ◽  
Double Phase ◽  
Multiplicity Theorem ◽  
Variational Tools ◽  
Indefinite Potential ◽  
Double Phase Problem ◽  
Strong Comparison Principle

AbstractWe consider an anisotropic double-phase problem plus an indefinite potential. The reaction is superlinear. Using variational tools together with truncation, perturbation and comparison techniques and critical groups, we prove a multiplicity theorem producing five nontrivial smooth solutions, all with sign information and ordered. In this process we also prove two results of independent interest, namely a maximum principle for anisotropic double-phase problems and a strong comparison principle for such solutions.

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