scholarly journals Fractional Schrödinger-Poisson systems with indefinite potentials

2021 ◽  
Author(s):  
Jun Wang ◽  
li wang ◽  
Qiao Zhong
Keyword(s):  
Morse Theory ◽  
Nontrivial Solutions

This paper is devoted to the following fractional Schrödinger-Poisson systems: \begin{equation*} \left\{\aligned &(-\Delta)^{s} u+V(x)u+\phi(x)u= f(x,u) \,\,\,&\text{in } \mathbb{R}^3, \\ & (-\Delta)^{t} \phi(x)=u^2 \,\,\,&\text{in } \mathbb{R}^3, \endaligned \right. \end{equation*} where $(-\Delta)^{s}$ is the fractional Lapalcian, $s, t \in (0, 1),$ $V : \R^3 \to \R$ is continuous. In contrast to most studies, we consider that the potentials $V$ is indefinite. With the help of Morse theory, the existence of nontrivial solutions for the above problem is obtained.

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.

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 NONTRIVIAL SOLUTIONS OF p-SUPERLINEAR p-LAPLACIAN PROBLEMS VIA A COHOMOLOGICAL LOCAL SPLITTING

2010 ◽  
Vol 12 (03) ◽  
pp. 475-486 ◽  
Author(s):  
MARCO DEGIOVANNI ◽  
SERGIO LANCELOTTI ◽  
KANISHKA PERERA
Keyword(s):  
Laplace Operator ◽  
Nontrivial Solution ◽  
Morse Theory ◽  
Quasilinear Equation ◽  
Mountain Pass ◽  
Nontrivial Solutions ◽  
Superlinear Nonlinearity ◽  
Sign Condition ◽  
Mountain Pass Geometry

We consider a quasilinear equation, involving the p-Laplace operator, with a p-superlinear nonlinearity. We prove the existence of a nontrivial solution, also when there is no mountain pass geometry, without imposing a global sign condition. Techniques of Morse theory are employed.

scholarly journals Three nontrivial solutions of boundary value problems for semilinear $\Delta_{\gamma}-$Laplace equation

2022 ◽  
Vol 40 ◽  
pp. 1-10
Author(s):  
Duong Trong Luyen ◽  
Le Thi Hong Hanh
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Bounded Domain ◽  
Morse Theory ◽  
Multiple Solutions ◽  
Smooth Boundary ◽  
Boundary Value ◽  
Nontrivial Solutions ◽  
Subcritical Growth ◽  
Subelliptic Operator

In this paper, we study the existence of multiple solutions for the boundary value problem\begin{equation}\Delta_{\gamma} u+f(x,u)=0 \quad \mbox{ in } \Omega, \quad \quad u=0 \quad \mbox{ on } \partial \Omega, \notag\end{equation}where $\Omega$ is a bounded domain with smooth boundary in $\mathbb{R}^N \ (N \ge 2)$ and $\Delta_{\gamma}$ is the subelliptic operator of the type $$\Delta_\gamma: =\sum\limits_{j=1}^{N}\partial_{x_j} \left(\gamma_j^2 \partial_{x_j} \right), \ \partial_{x_j}=\frac{\partial }{\partial x_{j}}, \gamma = (\gamma_1, \gamma_2, ..., \gamma_N), $$the nonlinearity $f(x , \xi)$ is subcritical growth and may be not satisfy the Ambrosetti-Rabinowitz (AR) condition. We establish the existence of three nontrivial solutions by using Morse theory.

Multiple Nontrivial Solutions for Neumann Problems Involving the p-Laplacian: a Morse Theoretical Approach

2010 ◽  
Vol 10 (1) ◽  
Author(s):  
Alexandru Kristály ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Theoretical Approach ◽  
Morse Theory ◽  
Nonlinear Term ◽  
Critical Groups ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Neumann Problems ◽  
Variational Techniques ◽  
Nonlinear Elliptic ◽  
Hopf Formula

AbstractWe consider nonlinear elliptic Neumann problems driven by the p-Laplacian. Using variational techniques together with Morse theory (in particular, critical groups and the Poincaré-Hopf formula), we prove some multiplicity results: either three or four distinct nontrivial solutions are guaranteed, depending on the geometry and smoothness of the nonlinear term.

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