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).

N-Laplacian Equations in ℝN with Subcritical and Critical GrowthWithout the Ambrosetti-Rabinowitz Condition

2013 ◽  
Vol 13 (2) ◽  
Author(s):  
Nguyen Lam ◽  
Guozhen Lu
Keyword(s):  
Bounded Domain ◽  
Exponential Growth ◽  
Nonnegative Solution ◽  
Nontrivial Solutions ◽  
Critical Exponential Growth ◽  
Trudinger Inequality ◽  
Laplacian Equations

AbstractLet Ω be a bounded domain in ℝwhen f is of subcritical or critical exponential growth. This nonlinearity is motivated by the Moser-Trudinger inequality. In fact, we will prove the existence of a nontrivial nonnegative solution to (0.1) without the Ambrosetti-Rabinowitz (AR) condition. Earlier works in the literature on the existence of nontrivial solutions to N−Laplacian in ℝ

On a nonlocal asymmetric Kirchhoff problem

2019 ◽  
Vol 13 (05) ◽  
pp. 2030001 ◽  
Author(s):  
Mohamed Karim Hamdani
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Exponential Growth ◽  
Smooth Boundary ◽  
Nontrivial Solutions ◽  
Subcritical Exponential Growth ◽  
Trudinger Inequality ◽  
Nonlocal Asymmetric ◽  
Kirchhoff Problem ◽  
Text Condition

This work is devoted to study the existence of nontrivial solutions to nonlocal asymmetric problems involving the [Formula: see text]-Laplacian. [Formula: see text] where [Formula: see text] is a bounded domain with smooth boundary, [Formula: see text] is a Kirchhoff function, [Formula: see text] and [Formula: see text] is of subcritical polynomial or subcritical exponential growth. Moreover, the existence of nontrivial solutions for the above problem is obtained by using variational methods combined with the Moser–Trudinger inequality. Our interest then is to study [Formula: see text] without the analogue of Ambrosetti–Rabinowitz superquadratic condition ([Formula: see text] condition for short) in the positive semi-axis. To the best of our best knowledge, our results are new even in the asymmetric Kirchhoff Laplacian and [Formula: see text]-Laplacian cases.

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.

scholarly journals On fractional p-Laplacian problems with local conditions

2018 ◽  
Vol 7 (4) ◽  
pp. 485-496 ◽  
Author(s):  
Anran Li ◽  
Chongqing Wei
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Nontrivial Solutions ◽  
Local Conditions ◽  
Existence And Multiplicity ◽  
Laplacian Equations

AbstractIn this paper, we deal with fractional p-Laplacian equations of the form\left\{\begin{aligned} \displaystyle(-\Delta)_{p}^{s}u&\displaystyle=\lambda f% (x,u),&&\displaystyle x\in\Omega,\\ \displaystyle u(x)&\displaystyle=0,&&\displaystyle x\in\mathbb{R}^{N}\setminus% \Omega,\end{aligned}\right.where {\lambda\in(0,+\infty)}, {0<s<1<p<+\infty} and {\Omega\subset\mathbb{R}^{N}}, {N\geqslant 2}, is a bounded domain with smooth boundary. With assumptions on {f(x,t)} just in {\Omega\times(-\delta,\delta)}, where {\delta>0} is small, existence and multiplicity of nontrivial solutions are obtained via variational methods.

scholarly journals On Dirichlet problem for fractional p-Laplacian with singular non-linearity

2016 ◽  
Vol 8 (1) ◽  
pp. 52-72 ◽  
Author(s):  
Tuhina Mukherjee ◽  
Konijeti Sreenadh
Keyword(s):  
Dirichlet Problem ◽  
Variational Methods ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Smooth Boundary ◽  
Critical Growth ◽  
Laplacian Equation ◽  
Existence And Multiplicity ◽  
Multiplicity Of Positive Solutions

Abstract In this article, we study the following fractional p-Laplacian equation with critical growth and singular non-linearity: (-\Delta_{p})^{s}u=\lambda u^{-q}+u^{\alpha},\quad u>0\quad\text{in }\Omega,% \qquad u=0\quad\text{in }\mathbb{R}^{n}\setminus\Omega, where Ω is a bounded domain in {\mathbb{R}^{n}} with smooth boundary {\partial\Omega} , {n>sp} , {s\in(0,1)} , {\lambda>0} , {0<q\leq 1} and {1<p<\alpha+1\leq p^{*}_{s}} . We use variational methods to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter λ.

scholarly journals Three nontrivial solutions for nonlinear fractional Laplacian equations

2018 ◽  
Vol 7 (2) ◽  
pp. 211-226 ◽  
Author(s):  
Fatma Gamze Düzgün ◽  
Antonio Iannizzotto
Keyword(s):  
Boundary Value Problem ◽  
Morse Theory ◽  
Fractional Laplacian ◽  
Mountain Pass Theorem ◽  
Pseudodifferential Equation ◽  
Nontrivial Solutions ◽  
Reaction Term ◽  
Second Deformation Theorem ◽  
Type Boundary ◽  
Laplacian Equations

AbstractWe study a Dirichlet-type boundary value problem for a pseudodifferential equation driven by the fractional Laplacian, proving the existence of three non-zero solutions. When the reaction term is sublinear at infinity, we apply the second deformation theorem and spectral theory. When the reaction term is superlinear at infinity, we apply the mountain pass theorem and Morse theory.

scholarly journals Multiple positive solutions for a p-Laplace Benci–Cerami type problem (1 < p < 2), via Morse theory

2021 ◽  
pp. 2150065
Author(s):  
Giuseppina Vannella
Keyword(s):  
Continuous Function ◽  
Bounded Domain ◽  
Morse Theory ◽  
Smooth Boundary ◽  
Quasilinear Equation ◽  
Multiple Positive Solutions ◽  
Type Problem ◽  
Poincaré Polynomial ◽  
Subcritical Growth ◽  
Quasilinear Problem

Let us consider the quasilinear problem [Formula: see text] where [Formula: see text] is a bounded domain in [Formula: see text] with smooth boundary, [Formula: see text], [Formula: see text], [Formula: see text] is a parameter and [Formula: see text] is a continuous function with [Formula: see text], having a subcritical growth. We prove that there exists [Formula: see text] such that, for every [Formula: see text], [Formula: see text] has at least [Formula: see text] solutions, possibly counted with their multiplicities, where [Formula: see text] is the Poincaré polynomial of [Formula: see text]. Using Morse techniques, we furnish an interpretation of the multiplicity of a solution, in terms of positive distinct solutions of a quasilinear equation on [Formula: see text], approximating [Formula: see text].

scholarly journals Four Solutions for Fractional p-Laplacian Equations with Asymmetric Reactions

2021 ◽  
Vol 18 (5) ◽  
Author(s):  
Antonio Iannizzotto ◽  
Roberto Livrea
Keyword(s):  
Morse Theory ◽  
Critical Point Theory ◽  
Point Theory ◽  
Positive Parameter ◽  
Type Problem ◽  
Nontrivial Solutions ◽  
Comparison Result ◽  
Asymmetric Perturbation ◽  
Laplacian Equations ◽  
Dirichlet Type Problem

AbstractWe consider a Dirichlet type problem for a nonlinear, nonlocal equation driven by the degenerate fractional p-Laplacian, whose reaction combines a sublinear term depending on a positive parameter and an asymmetric perturbation (superlinear at positive infinity, at most linear at negative infinity). By means of critical point theory and Morse theory, we prove that, for small enough values of the parameter, such problem admits at least four nontrivial solutions: two positive, one negative, and one nodal. As a tool, we prove a Brezis-Oswald type comparison result.

scholarly journals Existence theorems on the Dirichlet problem for the equation Δu + f(x, u)=0

1996 ◽  
Vol 39 (1) ◽  
pp. 31-36
Author(s):  
Gabriele Bonanno
Keyword(s):  
Dirichlet Problem ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Lower Order ◽  
Smooth Boundary ◽  
Existence Theorems ◽  
Suitable Condition ◽  
Strong Solutions ◽  
Order Perturbation

In this note we consider the Dirichlet problem Δu + f(x, u)=0 in Ω, u = 0 on ∂Ω here Ω is a bounded domain in ℝn(n≧3), with smooth boundary ∂Ω. We prove the existence of strong solutions to the previous problem, which are positive if f satisfies a suitable condition. As a consequence we find that the problem with , may have positive solutions even if g is not a lower-order perturbation of Next We examine the case .

The existence and asymptotic behaviour of the unique solution near the boundary to a singular Dirichlet problem with a convection term

2006 ◽  
Vol 136 (1) ◽  
pp. 209-222 ◽  
Author(s):  
Zhijun Zhang
Keyword(s):  
Dirichlet Problem ◽  
Asymptotic Behaviour ◽  
Unique Solution ◽  
Bounded Domain ◽  
Smooth Boundary ◽  
Convection Term ◽  
Nonlinear Dirichlet Problem ◽  
Singular Dirichlet Problem

We show the existence and exact asymptotic behaviour of the unique solution u ∈ C2(Ω)∩C(Ω̄) near the boundary to the singular nonlinear Dirichlet problem −Δu = k(x)g(u) + λ|∇u|q, u > 0, x ∈ Ω, u|∂Ω = 0, where Ω is a bounded domain with smooth boundary in RN, λ ∈ R, q ∈ [0, 2], g(s) is non-increasing and positive in (0, ∞), lims→0+g(s) = +∞, k ∈ Cα(Ω) is non-negative non-trivial on Ω, which may be singular on the boundary.

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