Existence and nonexistence results for quasilinear elliptic equations involving the $p$-Laplacian

Boumediene Abdellaoui; Veronica Felli; Ireneo Peral

Displaying similar documents to “Existence and nonexistence results for quasilinear elliptic equations involving the $p$ -Laplacian”

Existence of two positive solutions for a class of semilinear elliptic equations with singularity and critical exponent

Jia-Feng Liao, Jiu Liu, Peng Zhang, Chun-Lei Tang (2016)

Annales Polonici Mathematici

Similarity:

We study the following singular elliptic equation with critical exponent ⎧ $- Δ u = Q (x) u^{2 * - 1} + λ u^{- γ}$ in Ω, ⎨u > 0 in Ω, ⎩u = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N≥3) is a smooth bounded domain, and λ > 0, γ ∈ (0,1) are real parameters. Under appropriate assumptions on Q, by the constrained minimizer and perturbation methods, we obtain two positive solutions for all λ > 0 small enough.

On Kirchhoff type problems involving critical and singular nonlinearities

Chun-Yu Lei, Chang-Mu Chu, Hong-Min Suo, Chun-Lei Tang (2015)

Annales Polonici Mathematici

Similarity:

In this paper, we are interested in multiple positive solutions for the Kirchhoff type problem ⎧ $- (a + b \int_{Ω} | \nabla u | ² d x) Δ u = u ⁵ + λ u^{q - 1} / {| x |}^{β}$ in Ω ⎨ ⎩ u = 0 on ∂Ω, where Ω ⊂ ℝ³ is a smooth bounded domain, 0∈Ω, 1 < q < 2, λ is a positive parameter and β satisfies some inequalities. We obtain the existence of a positive ground state solution and multiple positive solutions via the Nehari manifold method.

Positive solutions for elliptic problems with critical nonlinearity and combined singularity

Jianqing Chen, Eugénio M. Rocha (2010)

Mathematica Bohemica

Similarity:

Consider a class of elliptic equation of the form $- Δ u - \frac{λ}{{| x |}^{2}} u = u^{2^{*} - 1} + μ u^{- q} in Ω ∖ {0}$ with homogeneous Dirichlet boundary conditions, where $0 \in Ω \subset ℝ^{N}$ ( $N \geq 3$ ), $0 < q < 1$ , $0 < λ < {(N - 2)}^{2} / 4$ and $2^{*} = 2 N / (N - 2)$ . We use variational methods to prove that for suitable $μ$ , the problem has at least two positive weak solutions.

On the spectrum of the p-biharmonic operator involving p-Hardy's inequality

Abdelouahed El Khalil, My Driss Morchid Alaoui, Abdelfattah Touzani (2014)

Applicationes Mathematicae

Similarity:

In this paper, we study the spectrum for the following eigenvalue problem with the p-biharmonic operator involving the Hardy term: ${Δ (| Δ u |}^{p - 2} {Δ u) = λ (| u |}^{p - 2} u) / (δ {(x)}^{2 p})$ in Ω, $u \in W ₀^{2, p} (Ω)$ . By using the variational technique and the Hardy-Rellich inequality, we prove that the above problem has at least one increasing sequence of positive eigenvalues.

Existence of a positive ground state solution for a Kirchhoff type problem involving a critical exponent

Lan Zeng, Chun Lei Tang (2016)

Annales Polonici Mathematici

Similarity:

We consider the following Kirchhoff type problem involving a critical nonlinearity: ⎧ $- [a + b (\int_{Ω} {| \nabla u | ² d x)}^{m} {] Δ u = f (x, u) + | u |}^{2 * - 2} u$ in Ω, ⎨ ⎩ u = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N ≥ 3) is a smooth bounded domain with smooth boundary ∂Ω, a > 0, b ≥ 0, and 0 < m < 2/(N-2). Under appropriate assumptions on f, we show the existence of a positive ground state solution via the variational method.

Stationary states for a two-dimensional singular Schrödinger equation

Paolo Caldiroli, Roberta Musina (2001)

Bollettino dell'Unione Matematica Italiana

Similarity:

In questo articolo studiamo problemi di Dirichlet singolari, lineari e semilineari, della forma ${|x|}^{2} Δ u = f (u)$ in $Ω$ , $u = 0$ su $\partial Ω$ , dove $Ω$ è un dominio in $R^{2}$ e $f (u) = λ u$ o $f (u) = λ u + {|u|}^{p - 2} u$ con $p > 2$ (o nonlinearità più generali). In tali problemi bidimensionali emergono alcune difficoltà a causa della non validità della disuguaglianza di Hardy in $R^{2}$ e a causa delle invarianze dell'equazione $- {|x|}^{2} Δ u = f (u)$ . Pertanto opportune condizioni su $λ$ e $Ω$ sono necessarie al fine di garantire l'esistenza di una soluzione positiva. Per esempio, se $Γ_{0}$ è una curva...

Existence of a renormalized solution of nonlinear degenerate elliptic problems

Youssef Akdim, Chakir Allalou (2014)

Applicationes Mathematicae

Similarity:

We study a general class of nonlinear elliptic problems associated with the differential inclusion $β (u) - d i v (a (x, D u) + F (u)) ∋ f$ in Ω where $f \in L^{\infty} (Ω)$ . The vector field a(·,·) is a Carathéodory function. Using truncation techniques and the generalized monotonicity method in function spaces we prove existence of renormalized solutions for general $L^{\infty}$ -data.

Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition

Zonghu Xiu, Caisheng Chen (2013)

Annales Polonici Mathematici

Similarity:

We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ $- {d i v (| x |}^{- a p} {| \nabla u |}^{p - 2} {\nabla u) + h (x) | u |}^{p - 2} u = g (x) {| u |}^{r - 2} u$ , x ∈ Ω, ⎨ ⎩ ${| x |}^{- a p} {| \nabla u |}^{p - 2} \partial u / \partial ν = λ f (x) {| u |}^{q - 2} u$ , x ∈ ∂Ω, where Ω is an exterior domain in $ℝ^{N}$ , that is, $Ω = ℝ^{N} ∖ D$ , where D is a bounded domain in $ℝ^{N}$ with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function...

Infinitely many solutions for a semilinear elliptic equation in $ℝ^{N}$ via a perturbation method

Marino Badiale (2002)

Annales Polonici Mathematici

Similarity:

We introduce a method to treat a semilinear elliptic equation in $ℝ^{N}$ (see equation (1) below). This method is of a perturbative nature. It permits us to skip the problem of lack of compactness of $ℝ^{N}$ but requires an oscillatory behavior of the potential b.

Singular $φ$ -Laplacian third-order BVPs with derivative dependance

Smaïl Djebali, Ouiza Saifi (2016)

Archivum Mathematicum

Similarity:

This work is devoted to the existence of solutions for a class of singular third-order boundary value problem associated with a $φ$ -Laplacian operator and posed on the positive half-line; the nonlinearity also depends on the first derivative. The upper and lower solution method combined with the fixed point theory guarantee the existence of positive solutions when the nonlinearity is monotonic with respect to its arguments and may have a space singularity; however no Nagumo type condition...

Existence of positive radial solutions for the elliptic equations on an exterior domain

Yongxiang Li, Huanhuan Zhang (2016)

Annales Polonici Mathematici

Similarity:

We discuss the existence of positive radial solutions of the semilinear elliptic equation ⎧-Δu = K(|x|)f(u), x ∈ Ω ⎨αu + β ∂u/∂n = 0, x ∈ ∂Ω, ⎩ $l i m_{| x | \to \infty} u (x) = 0$ , where $Ω = x \in ℝ^{N} : | x | > r ₀$ , N ≥ 3, K: [r₀,∞) → ℝ⁺ is continuous and $0 < \int_{r ₀}^{\infty} r K (r) d r < \infty$ , f ∈ C(ℝ⁺,ℝ⁺), f(0) = 0. Under the conditions related to the asymptotic behaviour of f(u)/u at 0 and infinity, the existence of positive radial solutions is obtained. Our conditions are more precise and weaker than the superlinear or sublinear growth conditions. Our discussion is based on the...

A note on a critical problem with natural growth in the gradient

Boumediene Abdellaoui, Ireneo Peral (2006)

Journal of the European Mathematical Society

Similarity:

The paper analyzes the influence on the meaning of natural growth in the gradient of a perturbation by a Hardy potential in some elliptic equations. Indeed, in the case of the Laplacian the natural problem becomes $- Δ u - Λ_{N} \frac{u}{{| x |}^{2}} = {|\nabla u + \frac{N - 2}{2} \frac{u}{{| x |}^{2}} x|}^{2} {| x |}^{(N - 2) / 2} + λ f (x)$ in $Ω$ , $u = 0$ on $\partial Ω$ , $Λ_{N} = {((N - 2) / 2)}^{2}$ . This problem is a particular case of problem (2). Notice that $(N - 2) / 2$ is optimal as coefficient and exponent on the right hand side.

On a semilinear elliptic eigenvalue problem

Mario Michele Coclite (1997)

Annales Polonici Mathematici

Similarity:

We obtain a description of the spectrum and estimates for generalized positive solutions of -Δu = λ(f(x) + h(u)) in Ω, ${u |}_{\partial Ω} = 0$ , where f(x) and h(u) satisfy minimal regularity assumptions.

Convex integration and the $L^{p}$ theory of elliptic equations

Kari Astala, Daniel Faraco, László Székelyhidi Jr. (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

This paper deals with the $L^{p}$ theory of linear elliptic partial differential equations with bounded measurable coefficients. We construct in two dimensions examples of weak and so-called very weak solutions, with critical integrability properties, both to isotropic equations and to equations in non-divergence form. These examples show that the general $L^{p}$ theory, developed in [1, 24] and [2], cannot be extended under any restriction on the essential range of the coefficients. Our constructions...

An inclusion operator in Hardy spaces on the unit ball in $C^{N}$

M. Jevtić (1988)

Matematički Vesnik

Similarity:

Extending Hardy fields by non- $^{\infty}$ -germs

Krzysztof Grelowski (2008)

Annales Polonici Mathematici

Similarity:

For a large class of Hardy fields their extensions containing non- $^{\infty}$ -germs are constructed. Hardy fields composed of only non- $^{\infty}$ -germs, apart from constants, are also considered.

Solutions to a class of singular quasilinear elliptic equations

Lin Wei, Zuodong Yang (2010)

Annales Polonici Mathematici

Similarity:

We study the existence of positive solutions to ⎧ ${d i v (| \nabla u |}^{p - 2} \nabla u) + q (x) u^{- γ} = 0$ on Ω, ⎨ ⎩ u = 0 on ∂Ω, where Ω is $ℝ^{N}$ or an unbounded domain, q(x) is locally Hölder continuous on Ω and p > 1, γ > -(p-1).

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

Similarity:

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 $\mathbb{R}^{n}$ .

Optimal $L^{p}$ -properties of Green’s functions for non-divergence elliptic equations in two dimensions

Gioconda Moscariello, Carlo Sbordone (2005)

Studia Mathematica

Similarity:

A sharp integrability result for non-negative adjoint solutions to planar non-divergence elliptic equations is proved. A uniform estimate is also given for the Green's function.

Wiener criterion for degenerate elliptic obstacle problem

Marco Biroli, Umberto Mosco (1989)

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

Similarity:

We give a Wiener criterion for the continuity of an obstacle problem relative to an elliptic degenerate problem with a weight in the $A_{2}$ class.

Displaying similar documents to “Existence and nonexistence results for quasilinear elliptic equations involving the p -Laplacian”

Displaying similar documents to “Existence and nonexistence results for quasilinear elliptic equations involving the $p$ -Laplacian”