scholarly journals Nonlinear eigenvalue problems in Sobolev spaces with variable exponent

2006 ◽  
Vol 4 (3) ◽  
pp. 225-242 ◽  
Author(s):  
Teodora-Liliana Dinu
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Eigenvalue Problems ◽  
Mountain Pass Lemma ◽  
Smooth Bounded Domain ◽  
Nonlinear Eigenvalue Problems ◽  
First Case ◽  
Symmetric Version

We study the boundary value problem-div⁡((|∇u|p1(x)-2+|∇u|p2(x)-2)∇u)=f(x,u)inΩ,u=0on∂Ω, whereΩis a smooth bounded domain inℝN. We focus on the cases whenf±(x,  u)=±(-λ|u|m(x)-2u+|u|q(x)-2u), wherem(x)≔max⁡⁡{p1(x),p2(x)}<q(x)<N⋅m(x)N-m(x)for anyx∈Ω̅. In the first case we show the existence of infinitely many weak solutions for anyλ>0. In the second case we prove that ifλis large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with aℤ2-symmetric version for even functionals of the Mountain Pass Lemma and some adequate variational methods.

scholarly journals A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids

2006 ◽  
Vol 462 (2073) ◽  
pp. 2625-2641 ◽  
Author(s):  
Mihai Mihăilescu ◽  
Vicenţiu Rădulescu
Keyword(s):  
Boundary Value Problem ◽  
Sobolev Spaces ◽  
Variable Exponent ◽  
Electrorheological Fluids ◽  
Mountain Pass Lemma ◽  
Multiplicity Result ◽  
Smooth Bounded Domain ◽  
Degenerate Problem ◽  
Laplace Type ◽  
Nonlinear Degenerate

We study the boundary value problem in , u =0 on , where is a smooth bounded domain in and is a -Laplace type operator, with . We prove that if λ is large enough then there exist at least two non-negative weak solutions. Our approach relies on the variable exponent theory of generalized Lebesgue–Sobolev spaces, combined with adequate variational methods and a variant of the Mountain Pass lemma.

EIGENVALUE PROBLEMS ASSOCIATED WITH NONHOMOGENEOUS DIFFERENTIAL OPERATORS, IN ORLICZ–SOBOLEV SPACES

2008 ◽  
Vol 06 (01) ◽  
pp. 83-98 ◽  
Author(s):  
MIHAI MIHĂILESCU ◽  
VICENŢIU RĂDULESCU
Keyword(s):  
Continuous Function ◽  
Boundary Value Problem ◽  
Real Number ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Positive Real Number ◽  
Differential Operators ◽  
Eigenvalue Problems ◽  
Smooth Boundary ◽  
Positive Real

We study the boundary value problem - div ((a1(|∇ u|) + a2(|∇ u|))∇ u) = λ|u|q(x)-2u in Ω, u = 0 on ∂Ω, where Ω is a bounded domain in ℝN (N ≥ 3) with smooth boundary, λ is a positive real number, q is a continuous function and a1, a2 are two mappings such that a1(|t|)t, a2(|t|)t are increasing homeomorphisms from ℝ to ℝ. We establish the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that any λ ∈ [λ1, ∞) is an eigenvalue, while any λ ∈ (0, λ0) is not an eigenvalue of the above problem.

scholarly journals Existence Theorems for Some Classes of Boundary Value Problems Involving the P(X)-Laplacian

2008 ◽  
Vol 13 (2) ◽  
pp. 145-158
Author(s):  
Ionica Andrei
Keyword(s):  
Eigenvalue Problem ◽  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Theoretical Approach ◽  
Eigenvalue Problems ◽  
Nonlinear Eigenvalue Problem ◽  
Existence Theorems ◽  
Boundary Value ◽  
Mountain Pass ◽  
Mountain Pass Lemma

We prove an alternative for a nonlinear eigenvalue problem involving the p(x)-Laplacian and study a subcritical boundary value problem for the same operator. The theoretical approach is the Mountain Pass Lemma and one of its variants, which is very useful in the study of eigenvalue problems.

scholarly journals Existence of Three Positive Solutions for a Nonlocal Singular Dirichlet Boundary Problem

2019 ◽  
Vol 19 (2) ◽  
pp. 333-352 ◽  
Author(s):  
Jacques Giacomoni ◽  
Tuhina Mukherjee ◽  
Konijeti Sreenadh
Keyword(s):  
Positive Solutions ◽  
Eigenvalue Problems ◽  
Nonlocal Problem ◽  
Singular Problem ◽  
Smooth Bounded Domain ◽  
Critical Point Theorem ◽  
Nonlinear Eigenvalue Problems ◽  
Nondecreasing Map ◽  
Dirichlet Boundary Problem ◽  
Ordered Banach Spaces

AbstractIn this article, we prove the existence of at least three positive solutions for the following nonlocal singular problem:\left\{\begin{aligned} \displaystyle(-\Delta)^{s}u&\displaystyle=\lambda\frac{% f(u)}{u^{q}},&&\displaystyle u>0\text{ in }\Omega,\\ \displaystyle u&\displaystyle=0&&\displaystyle\phantom{}\text{in }\mathbb{R}^{% n}\setminus\Omega,\end{aligned}\right.where {(-\Delta)^{s}} denotes the fractional Laplace operator for {s\in(0,1)}, {n>2s}, {q\in(0,1)}, {\lambda>0} and Ω is a smooth bounded domain in {\mathbb{R}^{n}}. Here {f:[0,\infty)\to[0,\infty)} is a continuous nondecreasing map satisfying\lim_{u\to\infty}\frac{f(u)}{u^{q+1}}=0.We show that under certain additional assumptions on f, the above problem possesses at least three distinct solutions for a certain range of λ. We use the method of sub-supersolutions and a critical point theorem by Amann [H. Amann, Fixed point equations and nonlinear eigenvalue problems in ordered Banach spaces, SIAM Rev. 18 1976, 4, 620–709] to prove our results. Moreover, we prove a new existence result for a suitable infinite semipositone nonlocal problem which played a crucial role to obtain our main result and is of independent interest.

GEOMETRY OF SOBOLEV SPACES WITH VARIABLE EXPONENT AND A GENERALIZATION OF THE p-LAPLACIAN

2009 ◽  
Vol 07 (04) ◽  
pp. 373-390 ◽  
Author(s):  
GEORGE DINCA ◽  
PAVEL MATEI
Keyword(s):  
Sobolev Space ◽  
Bounded Domain ◽  
Sobolev Spaces ◽  
Lebesgue Space ◽  
Variable Exponent ◽  
Uniformly Convex ◽  
Smooth Bounded Domain ◽  
Generalized Lebesgue Space

Let Ω ⊂ ℝN, N ≥ 2, be a smooth bounded domain. It is shown that: (a) if [Formula: see text] and ess inf x ∈ y p(x) > 1, then the generalized Lebesgue space (Lp (·)(Ω), ‖·‖p(·)) is smooth; (b) if [Formula: see text] and p(x) > 1, [Formula: see text], then the generalized Sobolev space [Formula: see text] is smooth. In both situations, the formulae for the Gâteaux gradient of the norm corresponding to each of the above spaces are given; (c) if [Formula: see text] and p(x) ≥ 2, [Formula: see text], then [Formula: see text] is uniformly convex and smooth.

scholarly journals Nonlinear eigenvalue problems

1975 ◽  
Vol 12 (3) ◽  
pp. 467-472 ◽  
Author(s):  
L.Ju. Fradkin ◽  
G.C. Wake
Keyword(s):  
Eigenvalue Problems ◽  
Sufficient Conditions ◽  
Nonlinear Eigenvalue Problems ◽  
Nonlinear Elliptic ◽  
First Case ◽  
Elliptic Eigenvalue Problems ◽  
Nonlinear Eigenvalue

The object of this paper is to prove two new results on the nature of the spectrum of a class of nonlinear elliptic eigenvalue problems. In the first case, sufficient conditions are given for which the spectrum is bounded and, in the second case, conditions are given for which the spectrum is open.

scholarly journals Initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Liming Xiao ◽  
Mingkun Li
Keyword(s):  
Weak Solution ◽  
Boundary Value Problem ◽  
Parabolic Equations ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Higher Order ◽  
Evolution Problem ◽  
Positive Energy ◽  
Initial Boundary Value

AbstractIn this paper, we study the initial boundary value problem for a class of higher-order n-dimensional nonlinear pseudo-parabolic equations which do not have positive energy and come from the soil mechanics, the heat conduction, and the nonlinear optics. By the mountain pass theorem we first prove the existence of nonzero weak solution to the static problem, which is the important basis of evolution problem, then based on the method of potential well we prove the existence of global weak solution to the evolution problem.

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