Multiplicity of positive solutions for a nonlinear fourth order equation

D. R. Dunninger

Displaying similar documents to “Multiplicity of positive solutions for a nonlinear fourth order equation”

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.

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.

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.

Some Remarks on Nonlinear Composition Operators in Spaces of Differentiable Functions

J. Appell, Z. Jesús, O. Mejía (2011)

Bollettino dell'Unione Matematica Italiana

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In this note we study the nonlinear composition operator $f\mapsto g\circ f$ in various spaces of differentiable functions over an interval. It turns out that this operator is always bounded in the corresponding norm, whenever it maps such a space into itself, but continuous only in exceptional cases.

On the equation x”(t) = F(t, x(t)) in the Sobolev space $H^{1} (R)$

Piotr Fijałkowski (1991)

Annales Polonici Mathematici

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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.

Blow-up of the solution to the initial-value problem in nonlinear three-dimensional hyperelasticity

J. A. Gawinecki, P. Kacprzyk (2008)

Applicationes Mathematicae

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We consider the initial value problem for the nonlinear partial differential equations describing the motion of an inhomogeneous and anisotropic hyperelastic medium. We assume that the stored energy function of the hyperelastic material is a function of the point x and the nonlinear Green-St. Venant strain tensor $e_{j k}$ . Moreover, we assume that the stored energy function is $C^{\infty}$ with respect to x and $e_{j k}$ . In our description we assume that Piola-Kirchhoff’s stress tensor $p_{j k}$ depends on the tensor...

Existence Theorems for a Fourth Order Boundary Value Problem

A. El-Haffaf (2009)

Bulletin of the Polish Academy of Sciences. Mathematics

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This paper treats the question of the existence of solutions of a fourth order boundary value problem having the following form: $x^{(4)} (t) + f (t, x (t), x^{''} (t)) = 0$ , 0 < t < 1, x(0) = x’(0) = 0, x”(1) = 0, $x^{(3)} (1) = 0$ . Boundary value problems of very similar type are also considered. It is assumed that f is a function from the space C([0,1]×ℝ²,ℝ). The main tool used in the proof is the Leray-Schauder nonlinear alternative.

Abstract inclusions in Banach spaces with boundary conditions of periodic type

Lahcene Guedda, Ahmed Hallouz (2014)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

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We study in the space of continuous functions defined on [0,T] with values in a real Banach space E the periodic boundary value problem for abstract inclusions of the form ⎧ $x \in S (x (0), s e l_{F} (x))$ ⎨ ⎩ x (T) = x(0), where, $F : [0, T] \times \to 2^{E}$ is a multivalued map with convex compact values, ⊂ E, $s e l_{F}$ is the superposition operator generated by F, and S: × L¹([0,T];E) → C([0,T]; ) an abstract operator. As an application, some results are given to the periodic boundary value problem for nonlinear differential inclusions governed...

Twists and resonance of $L$ -functions, I

Jerzy Kaczorowski, Alberto Perelli (2016)

Journal of the European Mathematical Society

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We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents $\leq 1 / d$ of the $L$ -functions of any degree $d \geq 1$ in the extended Selberg class. In particular, this solves the resonance problem in all such cases.

The restriction theorem for fully nonlinear subequations

F. Reese Harvey, H. Blaine Lawson (2014)

Annales de l’institut Fourier

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Let $X$ be a submanifold of a manifold $Z$ . We address the question: When do viscosity subsolutions of a fully nonlinear PDE on $Z$ , restrict to be viscosity subsolutions of the restricted subequation on $X$ ? This is not always true, and conditions are required. We first prove a basic result which, in theory, can be applied to any subequation. Then two definitive results are obtained. The first applies to any “geometrically defined” subequation, and the second to any subequation which can be...

On the number of positive solutions of singularly perturbed 1D nonlinear Schrödinger equations

Patricio Felmer, Salomé Martínez, Kazunaga Tanaka (2006)

Journal of the European Mathematical Society

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We study singularly perturbed 1D nonlinear Schrödinger equations (1.1). When $V (x)$ has multiple critical points, (1.1) has a wide variety of positive solutions for small $ε$ and the number of positive solutions increases to $\infty$ as $ε \to 0$ . We give an estimate of the number of positive solutions whose growth order depends on the number of local maxima of $V (x)$ . Envelope functions or equivalently adiabatic profiles of high frequency solutions play an important role in the proof.

Positive solutions to a class of elastic beam equations with semipositone nonlinearity

Qingliu Yao (2010)

Annales Polonici Mathematici

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Let h ∈ L¹[0,1] ∩ C(0,1) be nonnegative and f(t,u,v) + h(t) ≥ 0. We study the existence and multiplicity of positive solutions for the nonlinear fourth-order two-point boundary value problem $u^{(4)} (t) = f (t, u (t), u^{'} (t))$ , 0 < t < 1, u(0) = u’(0) = u’(1) =u”’(1) =0, where the nonlinear term f(t,u,v) may be singular at t=0 and t=1. By constructing a suitable cone and integrating certain height functions of f(t,u,v) on some bounded sets, several new results are obtained. In mechanics, the problem models the...

Existence results for a class of nonlinear parabolic equations with two lower order terms

Ahmed Aberqi, Jaouad Bennouna, M. Hammoumi, Mounir Mekkour, Ahmed Youssfi (2014)

Applicationes Mathematicae

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We investigate the existence of renormalized solutions for some nonlinear parabolic problems associated to equations of the form ⎧ $\partial (e^{β u} - 1) / \partial t - {d i v (| \nabla u |}^{p - 2} \nabla u) + d i v (c (x, t) {| u |}^{s - 1} u) + b (x, t) {| \nabla u |}^{r} = f$ in Q = Ω×(0,T), ⎨ u(x,t) = 0 on ∂Ω ×(0,T), ⎩ $(e^{β u} - 1) (x, 0) = (e^{β u ₀} - 1) (x)$ in Ω. with s = (N+2)/(N+p) (p-1), $c (x, t) \in {(L^{τ} (Q T))}^{N}$ , τ = (N+p)/(p-1), r = (N(p-1) + p)/(N+2), $b (x, t) \in L^{N + 2, 1} (Q T)$ and f ∈ L¹(Q).

Finite $p$ -groups with exactly two nonlinear non-faithful irreducible characters

Yali Li, Xiaoyou Chen, Huimin Li (2019)

Czechoslovak Mathematical Journal

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Let $G$ be a finite group with exactly two nonlinear non-faithful irreducible characters. We discuss the properties of $G$ and classify finite $p$ -groups with exactly two nonlinear non-faithful irreducible characters.

On spaces $L^{p (x)}$ and $W^{k, p (x)}$

Ondrej Kováčik, Jiří Rákosník (1991)

Czechoslovak Mathematical Journal

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On the radius of spatial analyticity for the higher order nonlinear dispersive equation

Aissa Boukarou, Kaddour Guerbati, Khaled Zennir (2022)

Mathematica Bohemica

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In this work, using bilinear estimates in Bourgain type spaces, we prove the local existence of a solution to a higher order nonlinear dispersive equation on the line for analytic initial data $u_{0}$ . The analytic initial data can be extended as holomorphic functions in a strip around the $x$ -axis. By Gevrey approximate conservation law, we prove the existence of the global solutions, which improve earlier results of Z. Zhang, Z. Liu, M. Sun, S. Li, (2019).

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 boundary value problems for systems of nonlinear generalized ordinary differential equations

Malkhaz Ashordia (2017)

Czechoslovak Mathematical Journal

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A general theorem (principle of a priori boundedness) on solvability of the boundary value problem $d x = d A (t) \cdot f (t, x), h (x) = 0$ is established, where $f : [a, b] \times ℝ^{n} \to ℝ^{n}$ is a vector-function belonging to the Carathéodory class corresponding to the matrix-function $A : [a, b] \to ℝ^{n \times n}$ with bounded total variation components, and $h : {BV}_{s} ([a, b], ℝ^{n}) \to ℝ^{n}$ is a continuous operator. Basing on the mentioned principle of a priori boundedness, effective criteria are obtained for the solvability of the system under the condition $x (t_{1} (x)) = ℬ (x) \cdot x (t_{2} (x)) + c_{0},$ where $t_{i} : {BV}_{s} ([a, b], ℝ^{n}) \to [a, b]$ $(i = 1, 2)$ and $ℬ : {BV}_{s} ([a, b], ℝ^{n}) \to ℝ^{n}$ are continuous...

Existence and multiplicity results for a nonlinear stationary Schrödinger equation

Danila Sandra Moschetto (2010)

Annales Polonici Mathematici

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We revisit Kristály’s result on the existence of weak solutions of the Schrödinger equation of the form -Δu + a(x)u = λb(x)f(u), $x \in ℝ^{N}$ , $u \in H ¹ (ℝ^{N})$ , where λ is a positive parameter, a and b are positive functions, while $f : ℝ \to ℝ$ is sublinear at infinity and superlinear at the origin. In particular, by using Ricceri’s recent three critical points theorem, we show that, under the same hypotheses, a much more precise conclusion can be obtained.