Existence and nonexistence of nontrivial solutions of some nonlinear degenerate elliptic equations

AbstractIn connection with the maximizing problem for the functional R(u) = ∥u∥Lq∥▽u∥Lp in W01,p(Ω)β{0}, we consider the equation −div(¦▽u¦p − 2 ▽u(x)) = ¦u¦q − 2 u(x), x ϵ Ω, 1 < p, q < ∞, p ≠ q, (E) u(x) = 0, x ϵ ∂Ω. It is shown that for the case q < p∗ (p∗ = ∞ if p ≧ N, and p∗ = Np(N − p) if p < N), (E) has always a nonnegative nontrivial solution belonging to W01,p(Ω) ∩ L∞(Ω), and for the case p < N and q > p∗ (resp. q = p∗), (E) has no nontrivial (resp. nonnegative nontrivial) solution belonging to the class P = {u ϵ W01,p(Ω) ∩ Lq(Ω); xi¦u¦q − 2u ϵ Lp(p − 1)(Ω), i = 1, 2, …, N} ⊂ W01,p(Ω) ∩ ∞(Ω), provided that Ω is star shaped. The crucial point of the proof of our result is to obtain an L∞-estimate of weak solutions and to verify a certain “Pohozaev-type inequality” for weak solutions belonging to P

Generalized elliptic functions and their application to a nonlinear eigenvalue problem with pp-Laplacian

The Jacobian elliptic functions are generalized and applied to a nonlinear eigenvalue problem with $p$-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 $p/2$-Laplacian.Comment: 17 page

Approximation properties of the qq-sine bases

For $q>12/11$ the eigenfunctions of the non-linear eigenvalue problem associated to the one-dimensional $q$-Laplacian are known to form a Riesz basis of $L^2(0,1)$. 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 $p$-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 $q$ for which the best approximation is achieved for a given $p$ 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}-($lambda$+i $alpha$) $Delta$ u-( $kapa$+i $beta$)|u|^{mathrm{q}-2}u- $gamma$ u=f & (t, x)in[0, T] times $Omega$, u(t, x)=0 & (t, x)in[0, T]timespartial $Omega$, mathrm{u}(0, x)=u{0}(x) & xin $Omega$, end{aray}right. where $Omega$ subset mathbb{R}^{N} is a smooth bounded domain. Parameters $lambda$, $kappa$ are positive, while $alpha$, $beta$, $gamma$ 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