scholarly journals Some Remarks on Biharmonic Elliptic Problems with a Singular Nonlinearity

2014 ◽  
Vol 2014 ◽  
pp. 1-11
Author(s):  
Baishun Lai
Keyword(s):  
Weak Solution ◽  
Unique Solution ◽  
Biharmonic Equation ◽  
Final Section ◽  
Elliptic Problems ◽  
Weak Sense ◽  
Extremal Case ◽  
Singular Nonlinearity

We study the following semilinear biharmonic equation  , which satisfies0<u̲λ<1. In the extremal caseλ=λ*, we prove the existence of a weak solution which is the unique solution even in a very weak sense. Besides, several new difficulties arise and many problems still remain to be solved. We list those of particular interest in the final section.

scholarly journals Existence of a Unique Solution to a Fractional Partial Differential Equation and Its Continuous Dependence on Parameters

Entropy ◽  
2021 ◽  
Vol 23 (7) ◽  
pp. 851
Author(s):  
Robert Stegliński
Keyword(s):  
Differential Equation ◽  
Partial Differential Equation ◽  
Weak Solution ◽  
Variational Methods ◽  
Unique Solution ◽  
Continuous Dependence ◽  
Fractional Laplacian ◽  
Unique Weak Solution ◽  
Fractional Partial Differential Equation ◽  
Integrodifferential Operator

In the present paper we give conditions under which there exists a unique weak solution for a nonlocal equation driven by the integrodifferential operator of fractional Laplacian type. We argue for the optimality of some assumptions. Some Lyapunov-type inequalities are given. We also study the continuous dependence of the solution on parameters. In proofs we use monotonicity and variational methods.

A finite difference method for the very weak solution to a Cauchy problem for an elliptic equation

2018 ◽  
Vol 26 (6) ◽  
pp. 835-857 ◽  
Author(s):  
Dinh Nho Hào ◽  
Le Thi Thu Giang ◽  
Sergey Kabanikhin ◽  
Maxim Shishlenin
Keyword(s):  
Cauchy Problem ◽  
Weak Solution ◽  
Finite Difference ◽  
Boundary Value Problem ◽  
Boundary Value ◽  
Weak Sense ◽  
Very Weak Solution ◽  
Local Boundary ◽  
The Cauchy Problem ◽  
Non Local

Abstract We introduce the concept of very weak solution to a Cauchy problem for elliptic equations. The Cauchy problem is regularized by a well-posed non-local boundary value problem whose solution is also understood in a very weak sense. A stable finite difference scheme is suggested for solving the non-local boundary value problem and then applied to stabilizing the Cauchy problem. Some numerical examples are presented for showing the efficiency of the method.

scholarly journals Existence results for nonlinear elliptic problems on fractal domains

2016 ◽  
Vol 5 (1) ◽  
Author(s):  
Massimiliano Ferrara ◽  
Giovanni Molica Bisci ◽  
Dušan Repovš
Keyword(s):  
Weak Solution ◽  
Dirichlet Problem ◽  
Critical Point ◽  
Elliptic Problems ◽  
Open Interval ◽  
Existence Results ◽  
Nonlinear Elliptic Problems ◽  
Nonlinear Elliptic ◽  
Fractal Domains ◽  
Critical Point Result

AbstractSome existence results for a parametric Dirichlet problem defined on the Sierpiński fractal are proved. More precisely, a critical point result for differentiable functionals is exploited in order to prove the existence of a well-determined open interval of positive eigenvalues for which the problem admits at least one non-trivial weak solution.

On Sensitive Vector Poisson and Stokes Problems

1997 ◽  
Vol 07 (05) ◽  
pp. 681-698 ◽  
Author(s):  
J.-L. Guermond ◽  
L. Quartapelle
Keyword(s):  
Boundary Value Problems ◽  
Unique Solution ◽  
Poisson Equation ◽  
Bilinear Form ◽  
Variational Formulation ◽  
Biharmonic Equation ◽  
Stokes Problem ◽  
Boundary Value ◽  
Natural Conditions ◽  
Stokes Problems

Lions/Sanchez-Palencia's theory of sensitive boundary value problems is extended from the scalar biharmonic equation to the vector Poisson equation and the Stokes problem associated with the bilinear form (∇ × u, ∇ × v) + (∇ · u, ∇ · v). For both problems the specification of completely natural conditions for the vector unknown on a part of the boundary leads to a variational formulation admitting a unique solution which is however sensitive to abitrarily small smooth perturbations of the data, as shown in the present paper.

scholarly journals On the existence of a weak solution for some singular p ⁢ ( x ) p(x) -biharmonic equation with Navier boundary conditions

2018 ◽  
Vol 8 (1) ◽  
pp. 1171-1183 ◽  
Author(s):  
Khaled Kefi ◽  
Kamel Saoudi
Keyword(s):  
Weak Solution ◽  
Continuous Function ◽  
Sobolev Spaces ◽  
Biharmonic Equation ◽  
Existence Of Solutions ◽  
Singular Equation ◽  
Positive Parameter ◽  
Biharmonic Operator ◽  
Navier Boundary Conditions ◽  
Variational Techniques

Abstract In the present paper, we investigate the existence of solutions for the following inhomogeneous singular equation involving the {p(x)} -biharmonic operator: \left\{\begin{aligned} &\displaystyle\Delta(\lvert\Delta u\rvert^{p(x)-2}% \Delta u)=g(x)u^{-\gamma(x)}\mp\lambda f(x,u)&&\displaystyle\phantom{}\text{in% }\Omega,\\ &\displaystyle\Delta u=u=0&&\displaystyle\phantom{}\text{on }\partial\Omega,% \end{aligned}\right. where {\Omega\subset\mathbb{R}^{N}} ( {N\geq 3} ) is a bounded domain with {C^{2}} boundary, λ is a positive parameter, {\gamma:\overline{\Omega}\rightarrow(0,1)} is a continuous function, {p\kern-1.0pt\in\kern-1.0ptC(\overline{\Omega})} with {1\kern-1.0pt<\kern-1.0ptp^{-}\kern-1.0pt:=\kern-1.0pt\inf_{x\in\Omega}p(x)% \kern-1.0pt\leq\kern-1.0ptp^{+}\kern-1.0pt:=\kern-1.0pt\sup_{x\in\Omega}p(x)% \kern-1.0pt<\kern-1.0pt\frac{N}{2}} , as usual, {p^{*}(x)\kern-1.0pt=\kern-1.0pt\frac{Np(x)}{N-2p(x)}} , g\in L^{\frac{p^{*}(x)}{p^{*}(x)+\gamma(x)-1}}(\Omega), and {f(x,u)} is assumed to satisfy assumptions (f1)–(f6) in Section 3. In the proofs of our results, we use variational techniques and monotonicity arguments combined with the theory of the generalized Lebesgue Sobolev spaces. In addition, an example to illustrate our result is given.

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