74 research outputs found
Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations
AbstractIn connection with the maximizing problem for the functional R(u) = â„uâ„Lqâ„âœuâ„Lp in W01,p(Ω)ÎČ{0}, we consider the equation âdiv(ŠâœuŠp â 2 âœu(x)) = ŠuŠq â 2 u(x), x Ï” Ω, 1 < p, q < â, p â q, (E) u(x) = 0, x Ï” âΩ. It is shown that for the case q < pâ (pâ = â if p ⧠N, and pâ = Np(N â p) if p < N), (E) has always a nonnegative nontrivial solution belonging to W01,p(Ω) â© Lâ(Ω), and for the case p < N and q > pâ (resp. q = pâ), (E) has no nontrivial (resp. nonnegative nontrivial) solution belonging to the class P = {u Ï” W01,p(Ω) â© Lq(Ω); xiŠuŠq â 2u Ï” Lp(p â 1)(Ω), i = 1, 2, âŠ, N} â W01,p(Ω) â© â(Ω), provided that Ω is star shaped. The crucial point of the proof of our result is to obtain an Lâ-estimate of weak solutions and to verify a certain âPohozaev-type inequalityâ for weak solutions belonging to P
Nonmonotone perturbations for nonlinear parabolic equations associated with subdifferential operators, Cauchy problems
GLOBAL SOLVABILITY FOR DOUBLE-DIFFUSIVE CONVECTION SYSTEM BASED ON BRINKMANâFORCHHEIMER EQUATION IN GENERAL DOMAINS
Infinite-dimensional attractors for parabolic equations with p-Laplacian in heterogeneous medium
Generalized elliptic functions and their application to a nonlinear eigenvalue problem with -Laplacian
The Jacobian elliptic functions are generalized and applied to a nonlinear
eigenvalue problem with -Laplacian. The eigenvalue and the corresponding
eigenfunction are represented in terms of common parameters, and a complete
description of the spectra and a closed form representation of the
corresponding eigenfunctions are obtained. As a by-product of the
representation, it turns out that a kind of solution is also a solution of
another eigenvalue problem with -Laplacian.Comment: 17 page
Approximation properties of the -sine bases
For the eigenfunctions of the non-linear eigenvalue problem
associated to the one-dimensional -Laplacian are known to form a Riesz basis
of . We examine in this paper the approximation properties of this
family of functions and its dual, in order to establish non-orthogonal spectral
methods for the -Poisson boundary value problem and its corresponding
parabolic time evolution initial value problem. The principal objective of our
analysis is the determination of optimal values of for which the best
approximation is achieved for a given problem.Comment: 20 pages, 11 figures and 2 tables. We have fixed a number of typos
and added references. Changed the title to better reflect the conten
Lagrange multiplier and singular limit of double obstacle problems for Allen--Cahn equation with constraint
We consider an Allen--Cahn equation with a constraint of double obstacle-type. This constraint is a subdifferential of an indicator function on the closed interval, which is a multivalued function. In this paper we study the properties of the Lagrange multiplier to our equation. Also, we consider the singular limit of our system and clarify the limit of the solution and the Lagrange multiplier to our double obstacle problem. Moreover, we give some numerical experiments of our problem by using the Lagrange multiplier
Solvability of complex Ginzburg-Landau equations with non-dissipative terms (Theory of evolution equations and applications to nonlinear problems)
In this paper, we consider the following complex Ginzburg-Landau equation. (CGL) left{begin{aray}{l} u{t}-(+i ) u-( +i )|u|^{mathrm{q}-2}u- u=f & (t, x)in[0, T] times , u(t, x)=0 & (t, x)in[0, T]timespartial , mathrm{u}(0, x)=u{0}(x) & xin , end{aray}right. where subset mathbb{R}^{N} is a smooth bounded domain. Parameters , are positive, while , , in mathbb{R} are real parameters and i=sqrt{-1} is the imaginary unit. We assume that q is Sobolev sub-critical, i.e., 2 < q < +infty when N= 1, 2 and 2 < q < displaystyle frac{2N}{N-2} when N geq 3. We study the local well-posedness of (CGL) and the global continuation of local solutions for small data
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