Dirichlet Problem For Laplacian Research Articles

On variational competing [formula omitted]Laplacian Dirichlet problem with gradient depending weight

We investigate the existence of variational generalized solutions to coercive competitive equations driven by a competing p,q−Laplacian type operator with weigth depending on the gradient. An abstract existence result based on a Galerkin scheme is therefore obtained and subsequently applied.

Constructive Approximation on Graded Meshes for the Integral Fractional Laplacian

We consider the homogeneous Dirichlet problem for the integral fractional Laplacian $$(-\Delta )^s$$ . We prove optimal Sobolev regularity estimates in Lipschitz domains provided the solution is $$C^s$$ up to the boundary. We present the construction of graded bisection meshes by a greedy algorithm and derive quasi-optimal convergence rates for approximations to the solution of such a problem by continuous piecewise linear functions. The nonlinear Sobolev scale dictates the relation between regularity and approximability.

Shooting from singularity to singularity and a quasilinear <i>p</i>-Laplace-Beltrami equation with indefinite weight

For surfaces of revolution we prove the existence of infinitely many sign-changing rotationally symmetric solutions to a wide class of $ p $-Laplace-Beltrami equations with polynomial like nonlinearities. We combine the methods in [10] where the semilinear case $ (p = 2) $ with positive weight was studied with those in [5] and [6] where radial solutions to a $ p $-Laplacian Dirichlet problem in a ball was considered. Our equations lead to ordinary differential equations with two singularities and a sign changing weight function which leads to an initial value problem that may blow up in the region of definition of the equation.

Semilinear Dirichlet problem for subordinate spectral Laplacian

We study semilinear problems in bounded $ C^{1, 1} $ domains for non-local operators with a boundary condition. The operators cover and extend the case of the spectral fractional Laplacian. We also study harmonic functions with respect to the non-local operator and boundary behaviour of Green and Poisson potentials.

Existence and multiplicity results for fractional p(x)-Laplacian Dirichlet problem

Abstract In this paper, we study a class of fractional p(x)-Laplacian Dirichlet problems in a bounded domain with Lipschitz boundary. Using variational methods, we prove in different situations the existence and multiplicity of solutions.

Du Bois-Reymond Type Lemma and Its Application to Dirichlet Problem with the p(t)-Laplacian on a Bounded Time Scale.

This paper is devoted to study the existence of solutions and their regularity in the –Laplacian Dirichlet problem on a bounded time scale. First, we prove a lemma of du Bois–Reymond type in time-scale settings. Then, using direct variational methods and the mountain pass methodology, we present several sufficient conditions for the existence of solutions to the Dirichlet problem.

Existence of radial solutions for a p(x)-Laplacian Dirichlet problem

In this paper, using variational methods, we prove the existence of at least one positive radial solution for the generalized p(x)-Laplacian problem −Δp(x)u+R(x)up(x)−2u=a(x)|u|q(x)−2u−b(x)|u|r(x)−2u\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ -\\Delta _{p(x)} u + R(x) u^{p(x)-2}u=a (x) \\vert u \\vert ^{q(x)-2} u- b(x) \\vert u \\vert ^{r(x)-2} u $$\\end{document} with Dirichlet boundary condition in the unit ball in mathbb{R}^{N} (for N geq 3), where a, b, R are radial functions.

A New Approach to the Hyers–Ulam–Rassias Stability of Differential Equations

Based on the lower and upper solutions method, we propose a new approach to the Hyers–Ulam and Hyers–Ulam–Rassias stability of first-order ordinary differential equations $$u'=f(t,u)$$ , in the lack of Lipschitz continuity assumption. Apart from extending and improving the literature by dropping some assumptions, our result provides an estimate for the difference between the solutions of the exact and perturbed models better than from that one obtained by fixed point approach which is commonly used method in this topic. Some examples are also given to illustrate the improvement. Particularly, we examine our approach to the Hyers–Ulam stability problem of second-order elliptic differential equations $$-\Delta u =g(x,u)$$ with homogeneous Dirichlet boundary condition which arise in different applications such as population dynamics and population genetics. This investigation is not only of interest in its own right, but also it supports the usability of our approach to other types of boundary value problems such as p(x)-Laplacian Dirichlet problems, Kirchhoff type problems, fractional differential equations and etc.

Infinitely many radial solutions for a $ p $-Laplacian problem with indefinite weight

<p style='text-indent:20px;'>We prove the existence of infinitely many sign changing radial solutions for a <inline-formula><tex-math id="M2">\begin{document}$ p $\end{document}</tex-math></inline-formula>-Laplacian Dirichlet problem in a ball. Our problem involves a weight function that is positive at the center of the unit ball and negative in its boundary. Standard initial value problems-phase plane analysis arguments do not apply here because solutions to the corresponding initial value problem may blow up near the boundary due to the fact that our weight function is negative at the boundary. We overcome this difficulty by connecting the solutions to a singular initial value problem with those of a regular initial value problem that vanishes at the boundary.

Weak solutions for a (p(z),q(z))-Laplacian Dirichlet problem

We establish the existence of a nontrivial and nonnegative solution for a double phase Dirichlet problem driven by a (p(z); q(z))-Laplacian operator plus a potential term. Our approach is variational, but the reaction term f need not satisfy the usual in such cases Ambrosetti-Rabinowitz condition.

Three solutions for fractional $p(x,.)$-Laplacian Dirichlet problems with weight

Three solutions for fractional $p(x,.)$-Laplacian Dirichlet problems with weight

On the sub-supersolution principle for p(x)-Laplacian equations

This paper deals with the sub-supersolution method for the p(x) -Laplacian Dirichlet problem. A sub-supersolution principle for the Dirichlet problems involving the p(x)-Laplacian is established by using induction method.

Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition

We consider the existence of solutions of the following $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition: \begin{equation*} \begin{cases} -\mbox{div}\hspace{.07em}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) &amp;\text{ in }\Omega , \\ u=0 &amp;\text{ on }\partial \Omega .% \end{cases} \end{equation*} We give a new growth condition and we point out its importance for checking the Cerami compactness condition. We prove the existence of solutions of the above problem via the critical point theory, and also provide some multiplicity properties. The present paper extend previous results of Q. Zhang and C. Zhao (Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, \textit{Computers and Mathematics with Applications}, 2015) and we establish the existence of solutions under weaker hypotheses on the nonlinear term.

An integro-differential inequality related to the smallest positive eigenvalue of p(x)-Laplacian Dirichlet problem

AbstractWe consider the eigenvalue problem for the p(x)-Laplace-Beltrami operator on the unit sphere. We prove same integro-differential inequalities related to the smallest positive eigenvalue of this problem.

Existence of strong solutions of a [formula omitted]-Laplacian Dirichlet problem without the Ambrosetti–Rabinowitz condition

In this paper, we consider the existence of strong solutions of the following p(x)-Laplacian Dirichlet problem via critical point theory: {−div(∣∇u∣p(x)−2∇u)=f(x,u), in Ω,u=0, on ∂Ω. We give a new growth condition, under which, we use a new method to check the Cerami compactness condition. Hence, we prove the existence of strong solutions of the problem as above without the growth condition of the well-known Ambrosetti–Rabinowitz type and also give some results about multiplicity of the solutions.

Nontrivial solution for asymmetric ( p , 2 ) -Laplacian Dirichlet problem

We consider a class of particular -Laplacian Dirichlet problems with a right-hand side nonlinearity which exhibits an asymmetric growth at +∞ and −∞. Namely, it is linear at −∞ and superlinear at +∞. However, it need not satisfy the Ambrosetti-Rabinowitz condition on the positive semi-axis. Some existence results for a nontrivial solution are established by the mountain pass theorem and a variant version of the mountain pass theorem in the general case . Similar results are also established by combining the mountain pass theorem and a variant version of the mountain pass theorem with the Moser-Trudinger inequality in the case of .

On a Helmholtz Style Decomposition for an Exterior Domain

In this paper we prove a variant of the usual Helmholtz decomposition for an exterior domain $\Omega$. The main difference is that in the decomposition $h=\nabla f+v$ for $h\in L^p(\Omega)$, with $\nabla f,v\in L^p(\Omega)^n$, div$v=0$, we assume that $f$, not $v$, vanishes on the boundary of the domain. Generally speaking, we are also more precise about which function spaces $f$ belongs to. It should be mentioned that our results, though not our proofs, have some strong overlap with some results in [C. G. Simader and H. Sohr, The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains, Longman, Harlow, UK, 1996], where a somewhat different approach is used to obtain a similar decomposition. We needed the type of decomposition we construct here for our study of the exterior two-dimensional quasi-geostrophic equation (cf. [L. Kosloff and T. Schonbek, Adv. Differential Equations, 17 (2012), pp. 173--200], [L. Kosloff and T. Schonbek, Discrete Contin. Dyn. Syst. Ser. S, 7 (2014), pp. 1025--1043]...

Multiplicity of solutions for a general p(x)-Laplacian Dirichlet problem

We establish some results on the existence of multiple nontrivial solutions for a class of p(x)-Laplacian elliptic equations. Our approach relies on the variable exponent theory of generalized Lebesgue–Sobolev spaces, combined with adequate variational methods and a variant of the Mountain Pass lemma.

Existence results of infinitely many solutions for p(x)-Laplacian elliptic Dirichlet problems

The existence of infinitely many solutions for a Dirichlet problem involving the p(x)-Laplacian is established. In our main result, under an appropriate oscillating behaviour of the nonlinearity and suitable assumptions on the variable exponent, a sequence of pairwise distinct solutions is obtained. Further, some applications and examples are pointed out. In particular, an existence result of infinitely many solutions for a two-point boundary value problem involving the variable exponent is presented. The approach is based on variational methods.

A Caffarelli–Kohn–Nirenberg-type inequality with variable exponent and applications to PDEs

Given Ω ⊂ ℝ N (N ≥ 2) a bounded smooth domain we establish a Caffarelli–Kohn–Nirenberg type inequality on Ω in the case when a variable exponent p(x), of class C 1, is involved. Our main result is proved under the assumption that there exists a smooth vector function , satisfying and for any x ∈ Ω. Particularly, we supplement a result by Fan et al. [X. Fan, Q. Zhang, and D. Zhao, Eigenvalues of p(x)-Laplacian Dirichlet problem, J. Math. Anal. Appl. 302 (2005), pp. 306–317] regarding the positivity of the first eigenvalue of the p(x)-Laplace operator. Moreover, we provide an application of our result to the study of degenerate PDEs involving variable exponent growth conditions.