A new approach to solving a quasilinear boundary value problem with $p$-Laplacian using optimization

Michaela Bailová; Jiří Bouchala

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Displaying similar documents to “A new approach to solving a quasilinear boundary value problem with $p$ -Laplacian using optimization”

The general methods of finding the sum $S_{n}$ for all kinds of series

K. Orlov (1981)

Matematički Vesnik

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The adaptation of the $k$ -means algorithm to solving the multiple ellipses detection problem by using an initial approximation obtained by the DIRECT global optimization algorithm

Rudolf Scitovski, Kristian Sabo (2019)

Applications of Mathematics

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We consider the multiple ellipses detection problem on the basis of a data points set coming from a number of ellipses in the plane not known in advance, whereby an ellipse $E$ is viewed as a Mahalanobis circle with center $S$ , radius $r$ , and some positive definite matrix $Σ$ . A very efficient method for solving this problem is proposed. The method uses a modification of the $k$ -means algorithm for Mahalanobis-circle centers. The initial approximation consists of the set of circles whose centers...

On a class of $(p, q)$ -Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

M.S. Shahrokhi-Dehkordi (2017)

Communications in Mathematics

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Let $Ω \subset ℝ^{n}$ be a bounded starshaped domain and consider the $(p, q)$ -Laplacian problem $\begin{matrix} - Δ_{p} u - Δ_{q} u = λ (𝐱) {| u |}^{p^{☆} - 2} u + μ {| u |}^{r - 2} u \end{matrix}$ where $μ$ is a positive parameter, $1 < q \leq p < n$ , $r \geq p^{☆}$ and $p^{☆} : = \frac{n p}{n - p}$ is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the $(p, q)$ -Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

Perturbed nonlinear degenerate problems in $ℝ^{N}$

A. El Khalil, S. El Manouni, M. Ouanan (2009)

Applicationes Mathematicae

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Via critical point theory we establish the existence and regularity of solutions for the quasilinear elliptic problem ⎧ $d i v (x, \nabla u) + {a (x) | u |}^{p - 2} u = {g (x) | u |}^{p - 2} u + h (x) {| u |}^{s - 1} u$ in $ℝ^{N}$ ⎨ ⎩ u > 0, $l i m_{| x | \to \infty} u (x) = 0$ , where 1 < p < N; a(x) is assumed to satisfy a coercivity condition; h(x) and g(x) are not necessarily bounded but satisfy some integrability restrictions.

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In this paper we consider a nonlinear elliptic equation with critical growth for the operator $\Delta^{2}$ in a bounded domain $\Omega\subset\mathbb{R}^{n}$ . We state some existence results when $n\geq 8$ . Moreover, we consider $5\leq n\leq 7$ , expecially when $\Omega$ is a ball in $\mathbb{R}^{n}$ .

An adaptive $s$ -step conjugate gradient algorithm with dynamic basis updating

Erin Claire Carson (2020)

Applications of Mathematics

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The adaptive $s$ -step CG algorithm is a solver for sparse symmetric positive definite linear systems designed to reduce the synchronization cost per iteration while still achieving a user-specified accuracy requirement. In this work, we improve the adaptive $s$ -step conjugate gradient algorithm by the use of iteratively updated estimates of the largest and smallest Ritz values, which give approximations of the largest and smallest eigenvalues of $A$ , using a technique due to G. Meurant and...

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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Positive solution for a quasilinear equation with critical growth in $ℝ^{N}$

Lin Chen, Caisheng Chen, Zonghu Xiu (2016)

Annales Polonici Mathematici

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We study the existence of positive solutions of the quasilinear problem ⎧ $- Δ_{N} u + {V (x) | u |}^{N - 2} u = {f (u, | \nabla u |}^{N - 2} \nabla u)$ , $x \in ℝ^{N}$ , ⎨ ⎩ u(x) > 0, $x \in ℝ^{N}$ , where $Δ_{N} u = {d i v (| \nabla u |}^{N - 2} \nabla u)$ is the N-Laplacian operator, $V : ℝ^{N} \to ℝ$ is a continuous potential, $f : ℝ \times ℝ^{N} \to ℝ$ is a continuous function. The main result follows from an iterative method based on Mountain Pass techniques.

Multiplicity results for a family of semilinear elliptic problems under local superlinearity and sublinearity

Djairo Guedes de Figueiredo, Jean-Pierre Gossez, Pedro Ubilla (2006)

Journal of the European Mathematical Society

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We study the existence, nonexistence and multiplicity of positive solutions for the family of problems $- Δ u = f_{λ} (x, u)$ , $u \in H_{0}^{1} (Ω)$ , where $Ω$ is a bounded domain in $ℝ^{N}$ , $N \geq 3$ and $λ > 0$ is a parameter. The results include the well-known nonlinearities of the Ambrosetti–Brezis–Cerami type in a more general form, namely $λ a (x) u^{q} + b (x) u^{p}$ , where $0 \leq q < 1 < p \leq 2^{*} - 1$ . The coefficient $a (x)$ is assumed to be nonnegative but $b (x)$ is allowed to change sign, even in the critical case. The notions of local superlinearity and local sublinearity introduced in [9] are essential...

Arbitrary number of positive solutions for an elliptic problem with critical nonlinearity

Olivier Rey, Juncheng Wei (2005)

Journal of the European Mathematical Society

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We show that the critical nonlinear elliptic Neumann problem $Δ u - μ u + u^{7 / 3} = 0$ in $Ω$ , $u > 0$ in $Ω$ , $\frac{\partial u}{\partial ν} = 0$ on $\partial Ω$ , where $Ω$ is a bounded and smooth domain in $ℝ^{5}$ , has arbitrarily many solutions, provided that $μ > 0$ is small enough. More precisely, for any positive integer $K$ , there exists $μ_{K} > 0$ such that for $0 < μ < μ_{K}$ , the above problem has a nontrivial solution which blows up at $K$ interior points in $Ω$ , as $μ \to 0$ . The location of the blow-up points is related to the domain geometry. The solutions are obtained as critical points of some finite-dimensional...

Existence and nonexistence of solutions for a quasilinear elliptic system

Qin Li, Zuodong Yang (2015)

Annales Polonici Mathematici

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By a sub-super solution argument, we study the existence of positive solutions for the system ⎧ $- Δ_{p} u = a ₁ (x) F ₁ (x, u, v)$ in Ω, ⎪ $- Δ_{q} v = a ₂ (x) F ₂ (x, u, v)$ in Ω, ⎨u,v > 0 in Ω, ⎩u = v = 0 on ∂Ω, where Ω is a bounded domain in $ℝ^{N}$ with smooth boundary or $Ω = ℝ^{N}$ . A nonexistence result is obtained for radially symmetric solutions.

Numerical approximation of the non-linear fourth-order boundary-value problem

Svobodová, Ivona

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We consider functionals of a potential energy $ℙ_{ψ} (u)$ corresponding to $𝑎𝑛 𝑎𝑥𝑖𝑠𝑦𝑚𝑚𝑒𝑡𝑟𝑖𝑐 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 - 𝑣𝑎𝑙𝑢𝑒 𝑝𝑟𝑜𝑏𝑙𝑒𝑚$ . We are dealing with $𝑎 𝑑𝑒𝑓𝑙𝑒𝑐𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑡ℎ𝑖𝑛 𝑎𝑛𝑛𝑢𝑙𝑎𝑟 𝑝𝑙𝑎𝑡𝑒$ with $𝑁𝑒𝑢𝑚𝑎𝑛𝑛 𝑏𝑜𝑢𝑛𝑑𝑎𝑟𝑦 𝑐𝑜𝑛𝑑𝑖𝑡𝑖𝑜𝑛𝑠$ . Various types of the subsoil of the plate are described by various types of the $𝑛𝑜𝑛𝑑𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑏𝑙𝑒$ nonlinear term $ψ (u)$ . The aim of the paper is to find a suitable computational algorithm.

Uniform convergence of the greedy algorithm with respect to the Walsh system

Martin Grigoryan (2010)

Studia Mathematica

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For any 0 < ϵ < 1, p ≥ 1 and each function $f \in L^{p} [0, 1]$ one can find a function $g \in L^{\infty} [0, 1)$ with mesx ∈ [0,1): g ≠ f < ϵ such that its greedy algorithm with respect to the Walsh system converges uniformly on [0,1) and the sequence $| c_{k} (g) | : k \in s p e c (g)$ is decreasing, where $c_{k} (g)$ is the sequence of Fourier coefficients of g with respect to the Walsh system.

Boundary blow-up solutions for a cooperative system involving the p-Laplacian

Li Chen, Yujuan Chen, Dang Luo (2013)

Annales Polonici Mathematici

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We study necessary and sufficient conditions for the existence of nonnegative boundary blow-up solutions to the cooperative system $Δ_{p} u = g (u - α v), Δ_{p} v = f (v - β u)$ in a smooth bounded domain of $ℝ^{N}$ , where $Δ_{p}$ is the p-Laplacian operator defined by $Δ_{p} u = {d i v (| \nabla u |}^{p - 2} \nabla u)$ with p > 1, f and g are nondecreasing, nonnegative C¹ functions, and α and β are two positive parameters. The asymptotic behavior of solutions near the boundary is obtained and we get a uniqueness result for p = 2.

Bubbling along boundary geodesics near the second critical exponent

Manuel del Pino, Monica Musso, Frank Pacard (2010)

Journal of the European Mathematical Society

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The role of the second critical exponent $p = (n + 1) / (n - 3)$ , the Sobolev critical exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem $Δ u + u^{p} = 0$ , $u > 0$ under zero Dirichlet boundary conditions, in a domain $Ω$ in $ℝ^{n}$ with bounded, smooth boundary. Given $Γ$ , a geodesic of the boundary with negative inner normal curvature we find that for $p = (n + 1) / (n - 3 - ε)$ , there exists a solution $u_{ε}$ such that $| \nabla u_{ε} |^{2}$ converges weakly to a Dirac measure on $Γ$ as $ε \to 0^{+}$ , provided that $Γ$ is nondegenerate in the sense of second...

Energy and Morse index of solutions of Yamabe type problems on thin annuli

Mohammed Ben Ayed, Khalil El Mehdi, Mohameden Ould Ahmedou, Filomena Pacella (2005)

Journal of the European Mathematical Society

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We consider the Yamabe type family of problems $(P_{ε}) : - Δ u_{ε} = u_{ε}^{(n + 2) / (n - 2)}$ , $u_{ε} > 0$ in $A_{ε}$ , $u_{ε} = 0$ on $\partial A_{ε}$ , where $A_{ε}$ is an annulus-shaped domain of $ℝ^{n}$ , $n \geq 3$ , which becomes thinner as $ε \to 0$ . We show that for every solution $u_{ε}$ , the energy $\int_{A_{ε}} | \nabla u_{|}^{2}$ as well as the Morse index tend to infinity as $ε \to 0$ . This is proved through a fine blow up analysis of appropriate scalings of solutions whose limiting profiles are regular, as well as of singular solutions of some elliptic problem on $ℝ^{n}$ , a half-space or an infinite strip. Our argument also involves a Liouville...

Critical points of the Moser-Trudinger functional on a disk

Andrea Malchiodi, Luca Martinazzi (2014)

Journal of the European Mathematical Society

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On the unit disk $B_{1} \subset ℝ^{2}$ we study the Moser-Trudinger functional $E (u) = \int_{B_{1}} (e^{u^{2}} - 1) d x, u \in H_{0}^{1} (B_{1})$ and its restrictions ${E |}_{M_{Λ}}$ , where $M_{Λ} : = {u \in H_{0}^{1} (B_{1}) {: ∥ u ∥}_{H_{0}^{1}}^{2} = Λ}$ for $Λ > 0$ . We prove that if a sequence $u_{k}$ of positive critical points of ${E |}_{M_{Λ_{k}}}$ (for some $Λ_{k} > 0$ ) blows up as $k \to \infty$ , then $Λ_{k} \to 4 π$ , and $u_{k} \to 0$ weakly in $H_{0}^{1} (B_{1})$ and strongly in $C_{loc}^{1} ({\bar{B}}_{1} ∖ {0})$ . Using this fact we also prove that when $Λ$ is large enough, then ${E |}_{M_{Λ}}$ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.

Separable solutions of quasilinear Lane–Emden equations

Alessio Porretta, Laurent Véron (2013)

Journal of the European Mathematical Society

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For $0 < p - 1 < q$ and either $ϵ = 1$ or $ϵ = - 1$ , we prove the existence of solutions of $- Δ_{p} u = ϵ u^{q}$ in a cone $C_{S}$ , with vertex 0 and opening $S$ , vanishing on $\partial C_{S}$ , of the form $u (x) = {|x|}^{- β} ω (x / |x|)$ . The problem reduces to a quasilinear elliptic equation on $S$ and the existence proof is based upon degree theory and homotopy methods. We also obtain a nonexistence result in some critical case by making use of an integral type identity.

Displaying similar documents to “A new approach to solving a quasilinear boundary value problem with p -Laplacian using optimization”

Displaying similar documents to “A new approach to solving a quasilinear boundary value problem with $p$ -Laplacian using optimization”