On Neumann boundary value problems for elliptic equations

Dimitrios A. Kandilakis

Displaying similar documents to “On Neumann boundary value problems for elliptic equations”

Three solutions for a nonlinear Neumann boundary value problem

Najib Tsouli, Omar Chakrone, Omar Darhouche, Mostafa Rahmani (2014)

Applicationes Mathematicae

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The aim of this paper is to establish the existence of at least three solutions for the nonlinear Neumann boundary-value problem involving the p(x)-Laplacian of the form $- Δ_{p (x)} u + a (x) | u^{| p (x) - 2} u = μ g (x, u)$ in Ω, ${| \nabla u |}^{p (x) - 2} \partial u / \partial ν = λ f (x, u)$ on ∂Ω. Our technical approach is based on the three critical points theorem due to Ricceri.

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

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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.

Nonhomogeneous boundary value problem for a semilinear hyperbolic equation

Andrzej Nowakowski (2008)

Applicationes Mathematicae

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We discuss the solvability of a nonhomogeneous boundary value problem for the semilinear equation of the vibrating string $x_{t t} (t, y) - Δ x (t, y) + f (t, y, x (t, y)) = 0$ in a bounded domain and with a certain type of superlinear nonlinearity. To this end we derive a new dual variational method.

A Short Proof of the Variational Principle for a $ℤ_{+}^{N}$ Action on a Compact Space

Michal Misiurewicz (1975)

Publications mathématiques et informatique de Rennes

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

Zonghu Xiu, Caisheng Chen (2013)

Annales Polonici Mathematici

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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...

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 Lincei. Matematica e Applicazioni

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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.

On a semilinear variational problem

Bernd Schmidt (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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We provide a detailed analysis of the minimizers of the functional $u \mapsto \int_{ℝ^{n}} {| \nabla u |}^{2} + D \int_{ℝ^{n}} {| u |}^{γ}$ , $γ \in (0, 2)$ , subject to the constraint ${∥ u ∥}_{L^{2}} = 1$ . This problem,, describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative...

Anti-periodic solutions to a parabolic hemivariational inequality

Jong Yeoul Park, Hyun Min Kim, Sun Hye Park (2004)

Kybernetika

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In this paper we deal with the anti-periodic boundary value problems with nonlinearity of the form $b (u)$ , where $b \in L_{loc}^{\infty} (R) .$ Extending $b$ to be multivalued we obtain the existence of solutions to hemivariational inequality and variational-hemivariational inequality.

Analytic semigroups generated on a functional extrapolation space by variational elliptic equations

Vincenzo Vespri (1988)

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

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We prove that any elliptic operator of second order in variational form is the infinitesimal generator of an analytic semigroup in the functional space $C^{-1,\alpha}(\Omega)$ consinsting of all derivatives of hölder-continuous functions in $\Omega$ where $\Omega$ is a domain in $\mathbb{R}^{n}$ not necessarily bounded. We characterize, moreover the domain of the operator and the interpolation spaces between this and the space $C^{-1,\alpha}(\Omega)$ . We prove also that the spaces $C^{-1,\alpha}(\Omega)$ can be considered as extrapolation spaces relative to suitable non-variational...

The Dirichlet problem with sublinear nonlinearities

Aleksandra Orpel (2002)

Annales Polonici Mathematici

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We investigate the existence of solutions of the Dirichlet problem for the differential inclusion $0 \in Δ x (y) + \partial_{x} G (y, x (y))$ for a.e. y ∈ Ω, which is a generalized Euler-Lagrange equation for the functional $J (x) = \int_{Ω} 1 / 2 | \nabla x (y) | ² - G (y, x (y)) d y$ . We develop a duality theory and formulate the variational principle for this problem. As a consequence of duality, we derive the variational principle for minimizing sequences of J. We consider the case when G is subquadratic at infinity.

On the nonlinear Neumann problem at resonance with critical Sobolev nonlinearity

J. Chabrowski, Shusen Yan (2002)

Colloquium Mathematicae

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We consider the Neumann problem for the equation $- Δ u - λ u = {Q (x) | u |}^{2 * - 2} u$ , u ∈ H¹(Ω), where Q is a positive and continuous coefficient on Ω̅ and λ is a parameter between two consecutive eigenvalues $λ_{k - 1}$ and $λ_{k}$ . Applying a min-max principle based on topological linking we prove the existence of a solution.

Existence of a renormalized solution of nonlinear degenerate elliptic problems

Youssef Akdim, Chakir Allalou (2014)

Applicationes Mathematicae

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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.

Analysis of approximate solutions of coupled dynamical thermoelasticity and related problems

Jozef Kačur, Alexander Ženíšek (1986)

Aplikace matematiky

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The authors study problems of existence and uniqueness of solutions of various variational formulations of the coupled problem of dynamical thermoelasticity and of the convergence of approximate solutions of these problems. First, the semidiscrete approximate solutions is defined, which is obtained by time discretization of the original variational problem by Euler’s backward formula. Under certain smoothness assumptions on the date authors prove existence and uniqueness of the solution...

A Variational Inequality for a Degenerate Elliptic Operator Under Minimal Assumptions on the Coefficients

Carmela Vitanza, Pietro Zamboni (2007)

Bollettino dell'Unione Matematica Italiana

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In this note we obtain the existence and the uniqueness of the solution of a variational inequality associated to the degenerate operator ${}Lu=-\sum^{n}_{i,j=1}(a_{ij}(x)u_{x_{i}}+d_{j}u)_{x_{j}}+\sum^{n}_{i=1}b_{i}u% _{x_{i}}+cu$ assuming the coefficients of the lower terms and the known term belonging to a suitable degenerate Stummel-Kato class. The weight $w$ , which gives the degeneration, belongs to the Muckenoupt class $A^{2}$ .

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

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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.

On discontinuous quasi-variational inequalities

Liang-Ju Chu, Ching-Yang Lin (2007)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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In this paper, we derive a general theorem concerning the quasi-variational inequality problem: find x̅ ∈ C and y̅ ∈ T(x̅) such that x̅ ∈ S(x̅) and ⟨y̅,z-x̅⟩ ≥ 0, ∀ z ∈ S(x̅), where C,D are two closed convex subsets of a normed linear space X with dual X*, and $T : X \to 2^{X *}$ and $S : C \to 2^{D}$ are multifunctions. In fact, we extend the above to an existence result proposed by Ricceri [12] for the case where the multifunction T is required only to satisfy some general assumption without any continuity. Under a...

Uniqueness of solutions for some degenerate nonlinear elliptic equations

Albo Carlos Cavalheiro (2014)

Applicationes Mathematicae

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We investigate the existence and uniqueness of solutions to the Dirichlet problem for a degenerate nonlinear elliptic equation $- \sum_{i, j = 1}^{n} D_{j} (a_{i j} (x) D_{i} u (x)) + b (x) u (x) + d i v (Φ (u (x))) = g (x) - \sum_{j = 1}^{n} f_{j} (x)$ on Ω in the setting of the space H₀(Ω).

T-p(x)-solutions for nonlinear elliptic equations with an L¹-dual datum

El Houssine Azroul, Abdelkrim Barbara, Meryem El Lekhlifi, Mohamed Rhoudaf (2012)

Applicationes Mathematicae

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We establish the existence of a T-p(x)-solution for the p(x)-elliptic problem $- d i v (a (x, u, \nabla u)) + g (x, u) = f - d i v F$ in Ω, where Ω is a bounded open domain of $ℝ^{N}$ , N ≥ 2 and $a : Ω \times ℝ \times ℝ^{N} \to ℝ^{N}$ is a Carathéodory function satisfying the natural growth condition and the coercivity condition, but with only a weak monotonicity condition. The right hand side f lies in L¹(Ω) and F belongs to $\prod_{i = 1}^{N} L^{p^{'} (\cdot)} (Ω)$ .

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...

A variational characterization of condenser capacities in $C^{n}$ within a class of plurisubharmonic functions

Jerzy Kalina (1983)

Annales Polonici Mathematici

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On the maximum of $a_{4} - 3 a_{2} a_{3} + μ a_{2}$ and some related functionals for bounded real univalent functions

Z. J. Jakubowski, H. Siejka, O. Tammi (1985)

Annales Polonici Mathematici

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A nonlocal elliptic equation in a bounded domain

Piotr Fijałkowski, Bogdan Przeradzki, Robert Stańczy (2004)

Banach Center Publications

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The existence of a positive solution to the Dirichlet boundary value problem for the second order elliptic equation in divergence form $- \sum_{i, j = 1}^{n} D_{i} (a_{i j} D_{j} u) = f (u, \int_{Ω} g (u^{p}))$ , in a bounded domain Ω in ℝⁿ with some growth assumptions on the nonlinear terms f and g is proved. The method based on the Krasnosel’skiĭ Fixed Point Theorem enables us to find many solutions as well.

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

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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.

Analytic semigroups generated on a functional extrapolation space by variational elliptic equations

Vincenzo Vespri (1988)

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

Similarity: