scholarly journals Existence of Solutions for thep(x)-Laplacian Problem with the Critical Sobolev-Hardy Exponent

2012 ◽  
Vol 2012 ◽  
pp. 1-17
Author(s):  
Yu Mei ◽  
Fu Yongqiang ◽  
Li Wang
Keyword(s):  
Bounded Domain ◽  
Existence Of Solutions ◽  
Growth Conditions ◽  
Concentration Compactness ◽  
Laplacian Equation

This paper deals with thep(x)-Laplacian equation involving the critical Sobolev-Hardy exponent. Firstly, a principle of concentration compactness inW01,p(x)(Ω)space is established, then by applying it we obtain the existence of solutions for the followingp(x)-Laplacian problem:-div (|∇u|p(x)-2∇u)+|u|p(x)-2u=(h(x)|u|ps*(x)-2u/|x|s(x))+f(x,u),  x∈Ω,  u=0,  x∈∂Ω,whereΩ⊂ℝNis a bounded domain,0∈Ω,1<p-≤p(x)≤p+<N, andf(x,u)satisfiesp(x)-growth conditions.

scholarly journals EXISTENCE OF SOLUTIONS FOR CRITICAL SYSTEMS WITH VARIABLE EXPONENTS

2018 ◽  
Vol 23 (4) ◽  
pp. 596-610 ◽  
Author(s):  
Hadjira Lalilia ◽  
Saadia Tas ◽  
Ali Djellit
Keyword(s):  
Existence Result ◽  
Existence Of Solutions ◽  
Elliptic Systems ◽  
Growth Conditions ◽  
Critical Growth ◽  
Concentration Compactness ◽  
Variable Exponents ◽  
Critical Systems ◽  
Concentration Compactness Principle ◽  
Compactness Principle

In this work, we deal with elliptic systems under critical growth conditions on the nonlinearities. Using a variant of concentration-compactness principle, we prove an existence result.

scholarly journals Solvability of Some Boundary Value Problems for Fractional -Laplacian Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Taiyong Chen ◽  
Wenbin Liu
Keyword(s):  
Fixed Point ◽  
Fixed Point Theorem ◽  
Boundary Value Problems ◽  
Fractional Laplacian ◽  
Existence Of Solutions ◽  
Boundary Value ◽  
Growth Conditions ◽  
Existence Results ◽  
Laplacian Equation ◽  
Nonlinear Growth

This paper considers the existence of solutions for two boundary value problems for fractional -Laplacian equation. Under certain nonlinear growth conditions of the nonlinearity, two new existence results are obtained by using Schaefer's fixed point theorem. As an application, an example to illustrate our results is given.

scholarly journals Existence and multiplicity of solutions for a fractional p-Laplacian equation with perturbation

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Zhen Zhi ◽  
Lijun Yan ◽  
Zuodong Yang
Keyword(s):  
Variational Methods ◽  
Bounded Domain ◽  
Analytical Techniques ◽  
Nontrivial Solutions ◽  
Multiplicity Results ◽  
Multiplicity Of Solutions ◽  
Laplacian Equation ◽  
Existence And Multiplicity

AbstractIn this paper, we consider the existence of nontrivial solutions for a fractional p-Laplacian equation in a bounded domain. Under different assumptions of nonlinearities, we give existence and multiplicity results respectively. Our approach is based on variational methods and some analytical techniques.

scholarly journals Quasilinear Riccati-Type Equations with Oscillatory and Singular Data

2020 ◽  
Vol 20 (2) ◽  
pp. 373-384
Author(s):  
Quoc-Hung Nguyen ◽  
Nguyen Cong Phuc
Keyword(s):  
Elliptic Operator ◽  
Bounded Domain ◽  
Existence Of Solutions ◽  
Type Equation ◽  
Linear Range ◽  
Measure Data ◽  
Regularity Conditions ◽  
Compactly Supported ◽  
Singular Data ◽  
Quasilinear Elliptic

AbstractWe characterize the existence of solutions to the quasilinear Riccati-type equation\left\{\begin{aligned} \displaystyle-\operatorname{div}\mathcal{A}(x,\nabla u)% &\displaystyle=|\nabla u|^{q}+\sigma&&\displaystyle\phantom{}\text{in }\Omega,% \\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{on }\partial% \Omega,\end{aligned}\right.with a distributional or measure datum σ. Here {\operatorname{div}\mathcal{A}(x,\nabla u)} is a quasilinear elliptic operator modeled after the p-Laplacian ({p>1}), and Ω is a bounded domain whose boundary is sufficiently flat (in the sense of Reifenberg). For distributional data, we assume that {p>1} and {q>p}. For measure data, we assume that they are compactly supported in Ω, {p>\frac{3n-2}{2n-1}}, and q is in the sub-linear range {p-1<q<1}. We also assume more regularity conditions on {\mathcal{A}} and on {\partial\Omega\Omega} in this case.

Multiple solutions for a class of p ( x )-Laplacian equations in involving the critical exponent

2010 ◽  
Vol 466 (2118) ◽  
pp. 1667-1686 ◽  
Author(s):  
Yongqiang Fu ◽  
Xia Zhang
Keyword(s):  
Critical Exponent ◽  
Multiple Solutions ◽  
Existence Of Solutions ◽  
Concentration Compactness ◽  
Concentration Compactness Principle ◽  
Radially Symmetric Solutions ◽  
Compactness Principle ◽  
Laplacian Equations

In this paper, we first establish a principle of concentration compactness in . Then, based on this concentration compactness principle, we study the existence of solutions for a class of p ( x )-Laplacian equations in involving the critical exponent. Under suitable assumptions, we obtain a sequence of radially symmetric solutions associated with a sequence of positive energies going towards infinity.

scholarly journals Existence Results for Differential Inclusions with Nonlinear Growth Conditions in Banach Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-7
Author(s):  
Messaoud Bounkhel
Keyword(s):  
Banach Space ◽  
Hilbert Spaces ◽  
Differential Inclusions ◽  
Existence Of Solutions ◽  
Growth Conditions ◽  
Existence Results ◽  
Nonlinear Growth ◽  
Upper Semicontinuous ◽  
Set Valued Mapping ◽  
Sweeping Processes

In the Banach space setting, the existence of viable solutions for differential inclusions with nonlinear growth; that is,ẋ(t)∈F(t,x(t))a.e. onI,x(t)∈S,∀t∈I,x(0)=x0∈S, (*), whereSis a closed subset in a Banach space𝕏,I=[0,T],(T>0),F:I×S→𝕏, is an upper semicontinuous set-valued mapping with convex values satisfyingF(t,x)⊂c(t)x+xp𝒦,∀(t,x)∈I×S, wherep∈ℝ, withp≠1, andc∈C([0,T],ℝ+). The existence of solutions for nonconvex sweeping processes with perturbations with nonlinear growth is also proved in separable Hilbert spaces.

scholarly journals On the stability of solutions for thep(x)-Laplacian equation and some applications to optimisation problems with state constraints

2006 ◽  
Vol 48 (2) ◽  
pp. 245-257 ◽  
Author(s):  
Elżbieta Galewska ◽  
Marek Galewski
Keyword(s):  
Control Problem ◽  
Continuous Dependence ◽  
State Constraints ◽  
Optimal Solution ◽  
Growth Conditions ◽  
Functional Parameter ◽  
Stability Of Solutions ◽  
Dirichlet Problems ◽  
Laplacian Equation ◽  
The Stability

AbstractWe consider the stability of solutions for a family of Dirichlet problems with (p, q)-growth conditions. We apply the results obtained to show continuous dependence on a functional parameter and the existence of an optimal solution in a control problem with state constraints governed by thep(x)-Laplacian equation.

scholarly journals Multiple solutions for some strongly degenerate second order elliptic equations

2021 ◽  
pp. 1-12
Author(s):  
João R. Santos ◽  
Gaetano Siciliano
Keyword(s):  
Boundary Value Problem ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Elliptic Equations ◽  
Multiple Solutions ◽  
Existence Of Solutions ◽  
Weighted Sobolev Spaces ◽  
Degenerate Operator ◽  
Second Order Elliptic Equations ◽  
Strongly Degenerate

We consider a boundary value problem in a bounded domain involving a degenerate operator of the form L ( u ) = − div ( a ( x ) ∇ u ) and a suitable nonlinearity f. The function a vanishes on smooth 1-codimensional submanifolds of Ω where it is not allowed to be C 2 . By using weighted Sobolev spaces we are still able to find existence of solutions which vanish, in the trace sense, on the set where a vanishes.

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