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articleNo Access
A singular elliptic system involving the $p (x)$ $p (x)$ -Laplacian and generalized Lebesgue–Sobolev spaces
- Kamel Saoudi
International Journal of Mathematics01 Nov 2019
Preview Abstract
The purpose of this work is to study a class of singular elliptic system involving the $p (x)$ $p (x)$ -Laplace operator of the form
$\{\begin{matrix} - Δ_{p (x)} u = λ a (x) | u |^{q (x) - 2} u (x) + \frac{1 - α (x)}{2 - α (x) - β (x)} c (x) | u |^{- α (x)} | v |^{1 - β (x)} & in Ω, \\ - Δ_{p (x)} v = μ b (x) | v |^{q (x) - 2} v (x) + \frac{1 - β (x)}{2 - α (x) - β (x)} c (x) | u |^{1 - α (x)} | v |^{- β (x)} & in Ω, \\ u = v = 0, & on \partial Ω, \end{matrix}$
where $Ω \subset ℝ^{N},$ $(N \geq 2)$ is a bounded domain with $C^{2}$ boundary, $λ, μ$ are two parameters, $a, b, c \in C (\bar{Ω})$ are non-negative weight functions with compact support in $Ω$ and $p, q, α, β \in C (\bar{Ω})$ are assumed to satisfy the assumptions (A0)–(A2) in Sec. 1. We employ the Nehari manifold approach combined with some variational techniques in order to show the existence and the multiplicity of positive solutions.
articleNo Access
A multiplicity results for a singular quasilinear elliptic equation
- Mounir Hsini,
- Kamel Saoudi, and
- Mouldi Seddik
International Journal of Mathematics10 Jul 2021
Preview Abstract
The purpose of this work is to study the following singular problem:
$(P_{λ}) \{\begin{matrix} - div (a (x, \nabla u)) = u^{- α} + λ u^{q} & in Ω; \\ u > 0, & in Ω; \\ u = 0, & in ℝ^{n} ∖ Ω; \end{matrix}$
where $Ω \subset ℝ^{N}$ , $N \geq 2$ be a bounded smooth domain, $λ$ is a positive parameter, $p \geq 2$ such that $N \geq p$ and $0 < α \leq 1$ , $p - 1 < q \leq p^{*} - 1,$ where $p^{*} = \frac{N p}{N - p} .$ We employ the Nehari manifold approach and the fibering maps in order to show the existence of $T_{q, α}$ such that for all $λ \in (0, T_{q, α})$ , problem $(P_{λ})$ has at least two solutions.
articleNo Access
Multiple solutions of a nonlocal system with singular nonlinearities
- Mouna Kratou,
- Kamel Saoudi, and
- Aisha AlShehri
International Journal of Mathematics05 Jul 2021
Preview Abstract
In this work, we study the fractional Laplacian equation with singular nonlinearity:
$\{\begin{matrix} {(- Δ)}^{s} u = λ a (x) | u |^{q - 2} u + \frac{1 - α}{2 - α - β} c (x) | u |^{- α} | v |^{1 - β} & in Ω, \\ {(- Δ)}^{s} v = μ b (x) | v |^{q - 2} v + \frac{1 - β}{2 - α - β} c (x) | u |^{1 - α} | v |^{- β} & in Ω, \\ u = v = 0 & in ℝ^{N} ∖ Ω, \end{matrix}$
where $Ω$ is a bounded domain in $ℝ^{n}$ with smooth boundary $\partial Ω$ , $N > 2 s$ , $s \in (0, 1)$ , $0 < α < 1, 0 < β < 1,$ $1 < q < 2 < 2_{s}^{*},$ $2_{s}^{*} = \frac{2 N}{N - 2 s}$ is the fractional Sobolev exponent, $λ, μ$ are two parameters, $a, b, c \in C (\bar{Ω})$ are nonnegative weight functions, and ${(- Δ)}^{s}$ is the fractional Laplace operator. We use the Nehari manifold approach and some variational techniques in order to show the existence and multiplicity of positive solutions of the above problem with respect to the parameter $λ$ and $μ$ .
articleNo Access
ASYMPTOTICS AND SYMMETRIES OF LEAST ENERGY NODAL SOLUTIONS OF LANE–EMDEN PROBLEMS WITH SLOW GROWTH
Communications in Contemporary Mathematics01 Aug 2008
Preview Abstract
In this paper, we consider the Lane–Emden problem
where Ω is a bounded domain in ℝ^N and p > 2. First, we prove that, for p close to 2, the solution is unique once we fix the projection on the second eigenspace. From this uniqueness property, we deduce partial symmetries of least energy nodal solutions. We also analyze the asymptotic behavior of least energy nodal solutions as p goes to 2. Namely, any accumulation point of sequences of (renormalized) least energy nodal solutions is a second eigenfunction that minimizes a reduced functional on a reduced Nehari manifold. From this asymptotic behavior, we also deduce an example of symmetry breaking. We use numerics to illustrate our results.
articleNo Access
LEAST ENERGY NODAL SOLUTIONS OF THE BREZIS–NIRENBERG PROBLEM IN DIMENSION N = 5
- PAOLO ROSELLI and
- MICHEL WILLEM
Communications in Contemporary Mathematics01 Feb 2009
Preview Abstract
We prove the existence of (a pair of) least energy sign changing solutions of
when Ω is a bounded domain in ℝ^N, N = 5 and λ is slightly smaller than λ₁, the first eigenvalue of -Δ with homogeneous Dirichlet boundary conditions on Ω.
articleNo Access
Least energy radial sign-changing solutions for a singular elliptic equation in lower dimensions
- Shuangjie Peng and
- Yanfang Peng
Communications in Contemporary Mathematics14 Jul 2014
Preview Abstract
We study the following singular elliptic equation
with Dirichlet boundary condition, which is related to the well-known Caffarelli–Kohn–Nirenberg inequalities. By virtue of variational method and Nehari manifold, we obtain least energy sign-changing solutions in some ranges of the parameters μ and λ. In particular, our result generalizes the existence results of sign-changing solutions to lower dimensions 5 and 6.
articleNo Access
Standing waves for weakly coupled nonlinear Schrödinger systems
- Claudiney Goulart and
- Elves A. B. Silva
Communications in Contemporary Mathematics19 Feb 2019
Preview Abstract
This paper is concerned with the application of variational methods in the study of positive solutions for a system of weakly coupled nonlinear Schrödinger equations in the Euclidian space. The results on multiplicity of positive solutions are established under the hypothesis that the coupling is either sublinear or superlinear with respect to one of the variables. Conditions for the existence or nonexistence of a positive least energy solution are also considered.
articleNo Access
On a coupled system of a Ginzburg–Landau equation with a quasilinear conservation law
- João-Paulo Dias,
- Filipe Oliveira, and
- Hugo Tavares
Communications in Contemporary Mathematics13 Jun 2019
Preview Abstract
We study a coupled system of a complex Ginzburg–Landau equation with a quasilinear conservation law
$\{\begin{matrix} e^{- i 𝜃} u_{t} = u_{x x} - | u |^{2} u - α g (v) u, \\ v_{t} + {(f (v))}_{x} = α {(g^{'} (v) | u |^{2})}_{x}, \end{matrix} x \in ℝ, t \geq 0$
which can describe the interaction between a laser beam and a fluid flow (see [I.-S. Aranson and L. Kramer, The world of the complex Ginzburg–Landau equation, Rev. Mod. Phys.74 (2002) 99–143]). We prove the existence of a local in time strong solution for the associated Cauchy problem and, for a certain class of flux functions, the existence of global weak solutions. Furthermore, we prove the existence of standing wave solutions of the form $(u (t, x), v (t, x)) = (U (x), V (x))$ in several cases.
articleNo Access
Regularity and multiplicity results for fractional $(p, q)$ -Laplacian equations
- Divya Goel,
- Deepak Kumar, and
- K. Sreenadh
Communications in Contemporary Mathematics20 Aug 2019
Preview Abstract
This paper deals with the study of the following nonlinear doubly nonlocal equation:
${(- Δ)}_{p}^{s_{1}} u + β {(- Δ)}_{q}^{s_{2}} u = λ a (x) | u |^{δ - 2} u + b (x) | u |^{r - 2} u, in Ω, u = 0 in ℝ^{n} ∖ Ω,$
where $Ω$ is a bounded domain in $ℝ^{n}$ with smooth boundary, $1 < δ \leq q \leq p < r \leq p_{s_{1}}^{*}$ , with $p_{s_{1}}^{*} = \frac{n p}{n - p s_{1}}$ , $0 < s_{2} < s_{1} < 1$ , $n > p s_{1}$ and $λ, β > 0$ are parameters. Here $a \in L^{\frac{r}{r - δ}} (Ω)$ and $b \in L^{\infty} (Ω)$ are sign-changing functions. We prove $L^{\infty}$ estimates, weak Harnack inequality and Interior Hölder regularity of the weak solutions of the above problem in the subcritical case $(r < p_{s_{1}}^{*}) .$ Also, by analyzing the fibering maps and minimizing the energy functional over suitable subsets of the Nehari manifold, we prove existence and multiplicity of weak solutions to above convex–concave problem. In case of $δ = q$ , we show the existence of a solution.
articleNo Access
Multiplicity of negative-energy solutions for singular-superlinear Schrödinger equations with indefinite-sign potential
Communications in Contemporary Mathematics17 May 2021
Preview Abstract
We are concerned with the multiplicity of positive solutions for the singular superlinear and subcritical Schrödinger equation
$- Δ u + V (x) u = λ a (x) u^{- γ} + b (x) u^{p} in ℝ^{N},$
beyond the Nehari extremal value, as defined in [Y. Il’yasov, On extreme values of Nehari manifold via nonlinear Rayleigh’s quotient, Topol. Methods Nonlinear Anal. 49 (2017) 683–714], when the potential $b \in L^{\infty} (ℝ^{N})$ may change its sign, $0 < a \in L^{\frac{2}{1 + γ}} (ℝ^{N})$ , $V$ is a positive continuous function, $N \geq 3$ and $λ > 0$ is a real parameter. The main difficulties come from the non-differentiability of the energy functional and the fact that the intersection of the boundaries of the connected components of the Nehari set is non-empty. We overcome these difficulties by exploring topological structures of that boundary to build non-empty sets whose boundaries have empty intersection and minimizing over them by controlling the energy level.
articleNo Access
Positive solutions for parametric equations with unbalanced growth and indefinite perturbation
- Zhenhai Liu and
- Nikolaos S. Papageorgiou
Analysis and Applications01 Jun 2024
Preview Abstract
We consider a nonlinear Dirichlet problem driven by the double phase operator. The reaction has the combined effects of parametric concave term and of an indefinite convex one (“concave-convex” problem with indefinite weight). Using the Nehari method we prove the existence of two bounded, positive ground state solutions.
articleOpen Access
Existence and multiplicity of solutions to fractional $p$ -Laplacian systems with concave–convex nonlinearities
Bulletin of Mathematical Sciences30 Mar 2020
Preview Abstract
This paper is concerned with a fractional $p$ -Laplacian system with both concave–convex nonlinearities. The existence and multiplicity results of positive solutions are obtained by variational methods and the Nehari manifold.
articleNo Access
Multiplicity of solutions for $p$ -Laplacian Kirchhoff problems with singular nonlinearity
- Haikel Ouerghi and
- Khaled Benali
Asian-European Journal of Mathematics09 Nov 2024
Preview Abstract
In this paper, we prove the existence of solutions for $p$ -Laplacian Kirchhoff problems with singular nonlinearity. More precisely, we use the Nehari manifold method and the analysis of the fibering maps in order to prove the existence of nonnegative solutions for such problem.

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A singular elliptic system involving the $p (x)$ $p (x)$ -Laplacian and generalized Lebesgue–Sobolev spaces

A multiplicity results for a singular quasilinear elliptic equation

Multiple solutions of a nonlocal system with singular nonlinearities

ASYMPTOTICS AND SYMMETRIES OF LEAST ENERGY NODAL SOLUTIONS OF LANE–EMDEN PROBLEMS WITH SLOW GROWTH

LEAST ENERGY NODAL SOLUTIONS OF THE BREZIS–NIRENBERG PROBLEM IN DIMENSION N = 5

Least energy radial sign-changing solutions for a singular elliptic equation in lower dimensions

Standing waves for weakly coupled nonlinear Schrödinger systems

On a coupled system of a Ginzburg–Landau equation with a quasilinear conservation law

Regularity and multiplicity results for fractional $(p, q)$ -Laplacian equations

Multiplicity of negative-energy solutions for singular-superlinear Schrödinger equations with indefinite-sign potential

Positive solutions for parametric equations with unbalanced growth and indefinite perturbation

Existence and multiplicity of solutions to fractional $p$ -Laplacian systems with concave–convex nonlinearities

Multiplicity of solutions for $p$ -Laplacian Kirchhoff problems with singular nonlinearity