The multiplicity of solutions and geometry of a nonlinear elliptic equation

Q. Choi; Sungki Chun; Tacksun Jung

Displaying similar documents to “The multiplicity of solutions and geometry of a nonlinear elliptic equation”

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.

On a semilinear elliptic eigenvalue problem

Mario Michele Coclite (1997)

Annales Polonici Mathematici

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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.

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

Yongxiang Li, Huanhuan Zhang (2016)

Annales Polonici Mathematici

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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...

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).

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.

Solvability for semilinear PDE with multiple characteristics

Alessandro Oliaro, Luigi Rodino (2003)

Banach Center Publications

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We prove local solvability in Gevrey spaces for a class of semilinear partial differential equations. The linear part admits characteristics of multiplicity k ≥ 2 and data are fixed in $G^{σ}$ , 1 < σ < k/(k-1). The nonlinearity, containing derivatives of lower order, is assumed of class $G^{σ}$ with respect to all variables.

On the principal eigencurve of the p-Laplacian related to the Sobolev trace embedding

Abdelouahed El Khalil, Mohammed Ouanan (2005)

Applicationes Mathematicae

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We prove that for any λ ∈ ℝ, there is an increasing sequence of eigenvalues μₙ(λ) for the nonlinear boundary value problem ⎧ $Δ ₚ u = {| u |}^{p - 2} u$ in Ω, ⎨ ⎩ ${| \nabla u |}^{p - 2} \partial u / \partial ν = {λ ϱ (x) | u |}^{p - 2} u + μ {| u |}^{p - 2} u$ on crtial ∂Ω and we show that the first one μ₁(λ) is simple and isolated; we also prove some results about variations of the density ϱ and the continuity with respect to the parameter λ.

Asymptotics for quasilinear elliptic non-positone problems

Zuodong Yang, Qishao Lu (2002)

Annales Polonici Mathematici

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In the recent years, many results have been established on positive solutions for boundary value problems of the form $- d i v (| \nabla u (x) |^{p - 2} \nabla u (x)) = λ f (u (x))$ in Ω, u(x)=0 on ∂Ω, where λ > 0, Ω is a bounded smooth domain and f(s) ≥ 0 for s ≥ 0. In this paper, a priori estimates of positive radial solutions are presented when N > p > 1, Ω is an N-ball or an annulus and f ∈ C¹(0,∞) ∪ C⁰([0,∞)) with f(0) < 0 (non-positone).

On positive solutions of quasilinear elliptic systems

Yuanji Cheng (1997)

Czechoslovak Mathematical Journal

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In this paper, we consider the existence and nonexistence of positive solutions of degenerate elliptic systems $\}\begin{matrix} - Δ_{p} u = f (x, u, v), & in Ω, - Δ_{p} v = g (x, u, v), & in Ω, u = v = 0, & on \partial Ω, \end{matrix}$ where $- Δ_{p}$ is the $p$ -Laplace operator, $p > 1$ and $Ω$ is a $C^{1, α}$ -domain in $ℝ^{n}$ . We prove an analogue of [7, 16] for the eigenvalue problem with $f (x, u, v) = λ_{1} v^{p - 1}$ , $g (x, u, v) = λ_{2} u^{p - 1}$ and obtain a non-existence result of positive solutions for the general systems.

A nonlinear elliptic equation with singular potential and applications to nonlinear field equations

Marino Badiale, Vieri Benci, Sergio Rolando (2007)

Journal of the European Mathematical Society

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We prove the existence of cylindrical solutions to the semilinear elliptic problem $- Δ u + \frac{u}{{| y |}^{2}} = f (u)$ , $u \in H^{1} (ℝ^{N})$ , $u \geq 0$ , where $(y, z) \in ℝ^{k} \times ℝ^{N - k}$ , $N > k \geq 2$ and $f$ has a double-power behaviour, subcritical at infinity and supercritical near the origin. This result also implies the existence of solitary waves with nonvanishing angular momentum for nonlinear Schr¨odinger and Klein–Gordon equations.

On solutions of a fourth-order Lidstone boundary value problem at resonance

Mariusz Jurkiewicz (2009)

Annales Polonici Mathematici

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We consider a Lidstone boundary value problem in $ℝ^{k}$ at resonance. We prove the existence of a solution under the assumption that the nonlinear part is a Carathéodory map and conditions similar to those of Landesman-Lazer are satisfied.

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 $\prod_{i = 1}^{N} L^{p^{'} (\cdot)} (Ω)$ .

Weighted inequalities for gradients on non-smooth domains

Caroline Sweezy, J. Michael Wilson

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We prove sufficiency of conditions on pairs of measures μ and ν, defined respectively on the interior and the boundary of a bounded Lipschitz domain Ω in d-dimensional Euclidean space, which ensure that, if u is the solution of the Dirichlet problem. Δu = 0 in Ω, ${u |}_{\partial Ω} = f$ , with f belonging to a reasonable test class, then $(\int_{Ω} {| \nabla u |}^{q} {d μ)}^{1 / q} \leq (\int_{\partial Ω} {| f |}^{p} {d ν)}^{1 / p}$ , where 1 < p ≤ q < ∞ and q ≥ 2. Our sufficiency conditions resemble those found by Wheeden and Wilson for the Dirichlet problem on $ℝ ₊^{d + 1}$ . As in that case we attack the...

Fonctions biharmoniques adjointes

Emmanuel P. Smyrnelis (2010)

Annales Polonici Mathematici

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The study of the equation (L₂L₁)*h = 0 or of the equivalent system L*₂h₂ = -h₁, L*₁h₁ = 0, where $L_{j} (j = 1, 2)$ is a second order elliptic differential operator, leads us to the following general framework: Starting from a biharmonic space, for example the space of solutions (u₁,u₂) of the system L₁u₁ = -u₂, L₂u₂ = 0, $L_{j} (j = 1, 2)$ being elliptic or parabolic, and by means of its Green pairs, we construct the associated adjoint biharmonic space which is in duality with the initial one.

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

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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...

Linear independence of boundary traces of eigenfunctions of elliptic and Stokes operators and applications

Roberto Triggiani (2008)

Applicationes Mathematicae

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This paper is divided into two parts and focuses on the linear independence of boundary traces of eigenfunctions of boundary value problems. Part I deals with second-order elliptic operators, and Part II with Stokes (and Oseen) operators. Part I: Let $λ_{i}$ be an eigenvalue of a second-order elliptic operator defined on an open, sufficiently smooth, bounded domain Ω in ℝⁿ, with Neumann homogeneous boundary conditions on Γ = tial Ω. Let ${φ_{i j}}_{j = 1}^{ℓ_{i}}$ be the corresponding linearly independent (normalized)...

Existence of positive solutions for a nonlinear fourth order boundary value problem

Ruyun Ma (2003)

Annales Polonici Mathematici

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We study the existence of positive solutions of the nonlinear fourth order problem $u^{(4)} (x) = λ a (x) f (u (x))$ , u(0) = u’(0) = u”(1) = u”’(1) = 0, where a: [0,1] → ℝ may change sign, f(0) < 0, and λ < 0 is sufficiently small. Our approach is based on the Leray-Schauder fixed point theorem.

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...