scholarly journals A weak solution for a $(p(x),q(x))$-Laplacian elliptic problem with a singular term

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
MirKeysaan Mahshid ◽  
Abdolrahman Razani
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Variational Method ◽  
Bounded Domain ◽  
Elliptic Problem ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Singular Term

AbstractHere, we consider the following elliptic problem with variable components: $$ -a(x)\Delta _{p(x)}u - b(x) \Delta _{q(x)}u+ \frac{u \vert u \vert ^{s-2}}{|x|^{s}}= \lambda f(x,u), $$ − a ( x ) Δ p ( x ) u − b ( x ) Δ q ( x ) u + u | u | s − 2 | x | s = λ f ( x , u ) , with Dirichlet boundary condition in a bounded domain in $\mathbb{R}^{N}$ R N with a smooth boundary. By applying the variational method, we prove the existence of at least one nontrivial weak solution to the problem.

On a family of torsional creep problems involving rapidly growing operators in divergence form

2018 ◽  
Vol 149 (2) ◽  
pp. 495-510 ◽  
Author(s):  
Maria Fărcăşeanu ◽  
Mihai Mihăilescu
Keyword(s):  
Boundary Condition ◽  
Uniform Convergence ◽  
Bounded Domain ◽  
Distance Function ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Smooth Boundary ◽  
Divergence Form ◽  
Homogeneous Dirichlet Boundary Condition ◽  
Real Numbers

Let Ω⊂ℝN (N≥2) be a bounded domain with smooth boundary and {pn} be a sequence of real numbers converging to+∞ as n→∞. For each integer n>1, we define the function $\varphi_{n}(t)=p_{n} \vert t \vert^{p_{n}-2}te^{ \vert t \vert^{p_{n}}}$, for all t∈ℝ, and we prove the existence of a unique nonnegative variational solution for the problem−div(((φn(|∇ u(x)|))/(|∇ u(x)|))∇ u(x))=φn(1), when x∈Ω, subject to the homogeneous Dirichlet boundary condition. Next, we establish the uniform convergence in Ω of the sequence of solutions for the above family of equations to the distance function to the boundary of Ω. Our result complements the earlier developments on the topic obtained by Payne and Philippin [26], Kawohl [21], Bhattacharya, DiBenedetto and Manfredi [2], Perez-Llanos and Rossi [27] and Bocea and Mihăilescu [4].

scholarly journals Infinitely many solutions for a fourth order singular elliptic problem

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 5003-5010 ◽  
Author(s):  
Chaharlang Makvand ◽  
A. Razani
Keyword(s):  
Boundary Condition ◽  
Fourth Order ◽  
Sobolev Spaces ◽  
Critical Points ◽  
Elliptic Problem ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Singular Term ◽  
Infinitely Many Solutions ◽  
Biharmonic Operator

Here, a fourth order singular elliptic problem involving p-biharmonic operator with Dirichlet boundary condition is established where the exponent in the singular term is different from that in the p-biharmonic operator. The existence of infinitely many solutions is proved by the variational methods in Sobolev spaces and the critical points principle of Ricceri. Finally, an example is presented.

scholarly journals The sub-supersolution method for a nonhomogeneous elliptic equation involving Lebesgue generalized spaces

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Giovany M. Figueiredo ◽  
A. Razani
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Elliptic Equation ◽  
Bounded Domain ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Logistic Equation ◽  
Positive Weak Solution

AbstractIn this paper, a nonhomogeneous elliptic equation of the form $$\begin{aligned}& - \mathcal{A}\bigl(x, \vert u \vert _{L^{r(x)}}\bigr) \operatorname{div}\bigl(a\bigl( \vert \nabla u \vert ^{p(x)}\bigr) \vert \nabla u \vert ^{p(x)-2} \nabla u\bigr) \\& \quad =f(x, u) \vert \nabla u \vert ^{\alpha (x)}_{L^{q(x)}}+g(x, u) \vert \nabla u \vert ^{ \gamma (x)}_{L^{s(x)}} \end{aligned}$$ − A ( x , | u | L r ( x ) ) div ( a ( | ∇ u | p ( x ) ) | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , u ) | ∇ u | L q ( x ) α ( x ) + g ( x , u ) | ∇ u | L s ( x ) γ ( x ) on a bounded domain Ω in ${\mathbb{R}}^{N}$ R N ($N >1$ N > 1 ) with $C^{2}$ C 2 boundary, with a Dirichlet boundary condition is considered. Using the sub-supersolution method, the existence of at least one positive weak solution is proved. As an application, the existence of at least one solution of a generalized version of the logistic equation and a sublinear equation are shown.

The Bahri–Coron Theorem for Fractional Yamabe-Type Problems

2018 ◽  
Vol 18 (2) ◽  
pp. 393-407 ◽  
Author(s):  
Wael Abdelhedi ◽  
Hichem Chtioui ◽  
Hichem Hajaiej
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Dirichlet Boundary Condition ◽  
Fractional Laplacian ◽  
Dirichlet Boundary ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Laplacian Operator ◽  
Fractional Laplacian Operator

AbstractWe study the following fractional Yamabe-type equation:\left\{\begin{aligned} \displaystyle A_{s}u&\displaystyle=u^{\frac{n+2s}{n-2s}% },\\ \displaystyle u&\displaystyle>0&&\displaystyle\text{in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\text{on }\partial\Omega,\end{% aligned}\right.Here Ω is a regular bounded domain of{\mathbb{R}^{n}},{n\geq 2}, and{A_{s}},{s\in(0,1)}, represents the fractional Laplacian operator{(-\Delta)^{s}}in Ω with zero Dirichlet boundary condition. We investigate the effect of the topology of Ω on the existence of solutions. Our result can be seen as the fractional counterpart of the Bahri–Coron theorem [3].

scholarly journals Multiplicity of solutions for singular semilinear elliptic equations with critical Hardy-Sobolev exponents

2008 ◽  
Vol 2 (2) ◽  
pp. 158-174 ◽  
Author(s):  
Qianqiao Guo ◽  
Pengcheng Niu ◽  
Jingbo Dou
Keyword(s):  
Boundary Condition ◽  
Variational Methods ◽  
Elliptic Problem ◽  
Dirichlet Boundary Condition ◽  
Elliptic Equations ◽  
Dirichlet Boundary ◽  
Nontrivial Solutions ◽  
Existence And Multiplicity ◽  
Semilinear Elliptic Problem ◽  
Semilinear Elliptic

We consider the semilinear elliptic problem with critical Hardy-Sobolev exponents and Dirichlet boundary condition. By using variational methods we obtain the existence and multiplicity of nontrivial solutions and improve the former results.

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

2014 ◽  
Vol 16 (04) ◽  
pp. 1350048
Author(s):  
Shuangjie Peng ◽  
Yanfang Peng
Keyword(s):  
Boundary Condition ◽  
Elliptic Equation ◽  
Variational Method ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Nehari Manifold ◽  
Existence Results ◽  
Singular Elliptic Equation ◽  
Sign Changing Solutions ◽  
Least Energy

We study the following singular elliptic equation [Formula: see text] 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.

Infinitely many arbitrarily small solutions for singular elliptic problems with critical Sobolev–Hardy exponents

2009 ◽  
Vol 52 (1) ◽  
pp. 97-108 ◽  
Author(s):  
Xiaoming He ◽  
Wenming Zou
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Elliptic Problems ◽  
Infinitely Many Solutions ◽  
Critical Problem

AbstractLet Ω ⊂ ℝN be a bounded domain such that 0 ∈ Ω, N ≥ 3, 2*(s) = 2(N − s)/(N − 2), 0 ≤ s < 2, $0\leq\mu\lt\bar{\mu}=\frac{14}(N-2)^{2}$. We obtain the existence of infinitely many solutions for the singular critical problem $\smash{-\Delta u-\mu(u/|x|^2)=(|u|^{2^*(s)-2/|x|^s)u+\lambda f(x,u)$ with Dirichlet boundary condition for suitable positive number λ.

scholarly journals Global weak solutions for a kinetic-fluid model with local alignment force in a bounded domain

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Fucai Li ◽  
Yue Li
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Weak Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Stokes Equations ◽  
Fluid Model ◽  
Navier Stokes ◽  
Local Alignment ◽  
Fokker Planck

<p style='text-indent:20px;'>We study a kinetic-fluid model in a <inline-formula><tex-math id="M1">\begin{document}$ 3D $\end{document}</tex-math></inline-formula> bounded domain. More precisely, this model is a coupling of the Vlasov-Fokker-Planck equation with the local alignment force and the compressible Navier-Stokes equations with nonhomogeneous Dirichlet boundary condition. We prove the global existence of weak solutions to it for the isentropic fluid (adiabatic coefficient <inline-formula><tex-math id="M2">\begin{document}$ \gamma&gt; 3/2 $\end{document}</tex-math></inline-formula>) and hence extend the existence result of Choi and Jung [Asymptotic analysis for a Vlasov-Fokker-Planck/Navier-Stokes system in a bounded domain, arXiv: 1912.13134v2], where the velocity of the fluid is supplemented with homogeneous Dirichlet boundary condition.</p>

scholarly journals The existence of nontrivial solutions for a critically coupled Schr\”{o}dinger system in a bounded domain of $\R^3$

2021 ◽  
Author(s):  
Hongyu Ye ◽  
Lina Zhang
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Nontrivial Solution ◽  
Critical Exponents ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
First Eigenvalue ◽  
Nontrivial Solutions ◽  
The First Eigenvalue ◽  
Nirenberg Problem

In this paper, we consider the following coupled Schr\”{o}dinger system with doubly critical exponents, which can be seen as a counterpart of the Brezis-Nirenberg problem $$\left\{% \begin{array}{ll} -\Delta u+\lambda_1 u=\mu_1 u^5+ \beta u^2v^3, & \hbox{$x\in \Omega$}, \\ -\Delta v+\lambda_2 v=\mu_2 v^5+ \beta v^2u^3, & \hbox{$x\in \Omega$}, \\ u=v=0,& \hbox{$x\in \partial\Omega$}, \\ \end{array}% \right.$$ where $\Omega$ is a ball in $\R^3,$ $-\lambda_1(\Omega)<\lambda_1,\lambda_2<-\frac14\lambda_1(\Omega)$, $\mu_1,\mu_2>0$ and $\beta>0$. Here $\lambda_1(\Omega)$ is the first eigenvalue of $-\Delta$ with Dirichlet boundary condition in $\Omega$. We show that the problem has at least one nontrivial solution for all $\beta>0$.

scholarly journals A New Proof of Existence of Positive Weak Solutions for Sublinear Kirchhoff Elliptic Systems with Multiple Parameters

Complexity ◽  
2020 ◽  
Vol 2020 ◽  
pp. 1-6 ◽  
Author(s):  
Salah Mahmoud Boulaaras ◽  
Rafik Guefaifia ◽  
Bahri Cherif ◽  
Sultan Alodhaibi
Keyword(s):  
Boundary Condition ◽  
Bounded Domain ◽  
Positive Solutions ◽  
Weak Solutions ◽  
Dirichlet Boundary Condition ◽  
Dirichlet Boundary ◽  
Elliptic Systems ◽  
Multiple Parameters

This paper deals with the study of the existence of weak positive solutions for sublinear Kirchhoff elliptic systems with zero Dirichlet boundary condition in bounded domain Ω⊂ℝN by using the subsuper solutions method.

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