768 research outputs found
Self-adjoint boundary-value problems on time-scales
Full text linkNOTE: THE MATHEMATICAL SYMBOLS IN THIS ABSTRACT CANNOT BE DISPLAYED CORRECTLY ON THIS PAGE. PLEASE REFER TO THE ABSTRACT IN THE ATTACHED FILE OR THE PUBLISHERS WEBSITE FOR AN ACCURATE DISPLAY. In this paper we consider a second order, Sturm-Liouville-type boundary-value operator of the form Lu := -[pur] + qu,on an arbitrary, bounded time-scale T, for suitable functions p, q, together with suitable boundary conditions. We show that, with a suitable choice of domain, this operator can be formulated in the Hilbert space L2(T ), in such a way that the resulting operator is self-adjoint, with compact resolvent (here,‘self-adjoint’ means in the standard functional analytic meaning of this term). Previous discussions of operators of this, and similar, form have described them as ‘self-adjoint’, but have not demonstrated self-adjointness in the standard functional analytic sense
p-Laplacian problems with jumping nonlinearities
No full textWe consider the p-Laplacian boundary value problem{A formula is presented}{A formula is presented} where p > 1 is a fixed number, f{symbol}p ( s ) = | s |p - 2 s, s ? R, and for each j = 0, 1, | cj 0 | + | cj 1 | > 0. The function f : [ 0, 1 ] × R2 ? R is a Carathéodory function satisfying, for ( x, s, t ) ? [ 0, 1 ] × R2,{A formula is presented} where ?±, ?± ? L1 ( 0, 1 ), and E has the form E ( x, s, t ) = ? ( x ) e ( | s | + | t | ), with ? ? L1 ( 0, 1 ), ? {greater than or slanted equal to} 0, e {greater than or slanted equal to} 0 and limr ? 8 e ( r ) r1 - p = 0. This condition allows the nonlinearity in (1) to behave differently as u ? ± 8. Such a nonlinearity is often termed jumping. Related to (1), (2) is the problem{A formula is presented} together with (2), where a, b ? L1 ( 0, 1 ), ? ? R, and u± ( x ) = max { ± u ( x ), 0 } for x ? [ 0, 1 ]. This problem is 'positively-homogeneous' and jumping. Values of ? for which (2), (3) has a nontrivial solution u will be called half-eigenvalues, while the corresponding solutions u will be called half-eigenfunctions. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having certain nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to existence and nonexistence results for the problem (1), (2). We also consider a related bifurcation problem, and obtain a global bifurcation result similar to the well-known Rabinowitz global bifurcation theorem. This then leads to a multiplicity result for (1), (2). When the functions a and b are constant the set of half-eigenvalues is closely related to the 'Fucík spectrum' of the problem, and equivalent solvability results are obtained using the two approaches. However, when a and b are not constant the half-eigenvalue approach yields stronger results. © 2005 Elsevier Inc. All rights reserved.</p
Spectral properties of second-order, multi-point, p-Laplacian boundary value problems
No full textWe consider the multi-point boundary value problem - f{symbol}p (u')' = ? f{symbol}p (u), on (- 1, 1),u (± 1) = underover(?, i = 1, m±) ai± u (?i±), where p > 1, f{symbol}p (s) {colon equals} | s |p - 1 sgn s for s ? R, ? ? R, m± = 1 are integers, ?i± ? (- 1, 1), 1 = i = m±, and the coefficients ai± satisfy underover(?, i = 1, m±) | ai± | < 1 . A number ? ? R is said to be an eigenvalue of the above problem if there exists a non-trivial solution u. The spectrum is the set of eigenvalues. In this paper we obtain some basic spectral and degree-theoretic properties of this eigenvalue problem. These results have numerous applications to more general problems. As an example, a Rabinowitz-type, global bifurcation theorem is briefly described. © 2010 Elsevier Ltd. All rights reserved.</p
A global curve of stable, positive solutions for a p-Laplacian problem
No full textWe consider the boundary-value problem where (), , , , and the function is and satisfies These assumptions on imply that the trivial solution is the only solution with or , and if then any solution is {em positive}, that is, on
Spectral properties of p-Laplacian problems with Neumann and mixed-type multi-point boundary conditions
No full textWe consider the boundary value problem consisting of the p-Laplacian equation -fp(u')'=?fp(u),on (-1,1), where p>1, fp(s):=|s|p-1sgns for s?R, ??R, together with the multi-point boundary conditions fp(u'(±1))=?i=1m±ai±f p(u'(?i±)), or u(±1)=?i=1m±ai±u(?i±), or a mixed pair of these conditions (with one condition holding at each of x=-1 and x=1). In (2), (3), m± =1 are integers, ?i±?(-1,1), 1 =i=m±, and the coefficients ai± satisfy ? i=1m±|ai±|<1. We term the conditions (2) and (3), respectively, Neumann-type and Dirichlet-type boundary conditions, since they reduce to the standard Neumann and Dirichlet boundary conditions when a±=0. Given a suitable pair of boundary conditions, a number ? is an eigenvalue of the corresponding boundary value problem if there exists a non-trivial solution u (an eigenfunction). The spectrum of the problem is the set of eigenvalues. In this paper we obtain various spectral properties of these eigenvalue problems. We then use these properties to prove Rabinowitz-type, global bifurcation theorems for related bifurcation problems, and to obtain nonresonance conditions (in terms of the eigenvalues) for the solvability of related inhomogeneous problems. © 2010 Elsevier Ltd. All rights reserved.</p
The spectrum of the periodic p-Laplacian
No full textWe consider one-dimensional p-Laplacian eigenvalue problems of the form- ?p u = (? - q) | u |p - 1 sgn u, on (0, b), together with periodic or separated boundary conditions, where p > 1, ?p is the p-Laplacian, q ? C1 [0, b], and b > 0, ? ? R. It will be shown that when p ? 2, the structure of the spectrum in the general periodic case (that is, with q ? 0 and periodic boundary conditions), can be completely different from those of the following known cases: (i) the general periodic case with p = 2, (ii) the periodic case with p ? 2 and q = 0, and (iii) the general separated case with any p > 1. © 2006.</p
Global stability, or instability, of positive equilibria of p-Laplacian boundary value problems with p-convex nonlinearities
Full text linkWe consider the parabolic, initial value problem
vt = Δp(v) + λg(x, v)φp(v), in Ω x (0,∞), v = 0, in ∂Ω x (0,∞), (IVP) v = v0 > 0, in Ω x {0}, where Ω is a bounded domain in RN , for some integer N > 1, with smooth boundary ∂Ω, φp(s) := |s|p−1 sgn s , s ∈ R , and Δp denotes the p -Laplacian, with p > max{2,N} , v0 ∈ C0(Ω) , and λ > 0 . The function g : Ω x [0,∞) → (0,∞) is C0 and, for each x ∈ Ω , the function g(x, ·) : [0,∞) → (0,∞) is Lipschitz continuous and strictly increasing.
Clearly, (IVP) has the trivial solution v ≡ 0 , for all λ > 0 . In addition, there exists 0 < λmin(g) < λmax(g) such that:
• if λ ∈/ (λmin(g),λmax(g)) then (IVP) has no non-trivial, positive
equilibrium;
• there exists a closed, connected set of positive equilibria bifurcating
from (λmax(g), 0) and ‘meeting infinity’ at λ = λmin(g) .
We prove the following results on the positive solutions of (IVP):
• if 0 < λ < λmin(g) then the trivial solution is globally asymptotically
stable;
• if λmin(g) < λ < λmax(g) then the trivial solution is locally asymptotically stable and all non-trivial, positive equilibria are unstable;
• if λmax(g) < λ then any non-trivial solution blows up in finite
time
Eigenvalue criteria for existence of positive solutions Of second-order, multi-point, p-laplacian boundary value problems
No full textIn this paper we consider the existence and uniqueness of positive solutions of the multi-point boundary value problem (1) - (Øp(u?)?+(a+g(x,u,u?)Øp(u) =0, a.e. on (-1,1), (2) u(±1)=Si=1 ±ai±u(? i±) where p>1, Øp(s) p-2s, se R,m± = 1 are integer, and ?i±?(-1,1), ai± > 0, i=1 ,?,m±, Si=1m±ai± Also a ? L1(-1,1) and g: [-1,1]× R 2?R is Carath´eodory, with (3) g(x,0,0)=0, x ?[-1,1]. Our criteria for existence of positive solutions of (1), (2) will be expressed in terms of the asymptotic behaviour of g(x, s, t), as s ? 8, and the principal eigenvalues of the multi-point boundary value problem consisting of the equation (4) -Ø(u?)?+aØp(u)= ?Øp(u), on (-1,1) Copyright © 2010 Juliusz Schauder Center for Nonlinear Studies.</p
The formulation of second-order boundary value problems on time scales
No full textWe reconsider the basic formulation of second-order, two-point, Sturm-Liouville-type boundary value problems on time scales. Although this topic has received extensive atten-tion in recent years, we present some simple examples which show that there are certain difficulties with the formulation of the problem as usually used in the literature. These difficulties can be avoided by some additional conditions on the structure of the time scale, but we show that these conditions are unnecessary, since in fact, a simple, amended formulation of the problem avoids the difficulties. Copyright © 2006 F. A. Davidson and B. P. Rynne. This is an open access article distrib-uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1
Variational and non-variational eigenvalues of the p-Laplacian
No full textIt is well known that all the eigenvalues of the linear eigenvalue problem? u = (q - ? r) u, in O ? RN, can (under appropriate conditions on q, r and O) be characterized by minimax principles, but it has been a long-standing question whether that remains true for analogous equations involving the p-Laplacian ?p. It will be shown that there are corresponding nonlinear eigenvalue problems?p u = (q - ? r) | u |p - 1 sgn u, in O ? RN, with 1 < p ? 2 and q, r ? C1 (over(O, -)), r > 0 on over(O, -), for which not all eigenvalues are of variational type. As far as we know, this is the first observation of such a phenomenon, and examples will be given for one- and higher-dimensional equations. The question of exactly which eigenvalues are variational is also discussed when N = 1. © 2007 Elsevier Inc. All rights reserved.</p
- …
