Global asymptotic stability of bifurcating, positive equilibria of p-Laplacian boundary value problems with p-concave nonlinearities

We consider the parabolic, initial value problem $$v_t =\Delta_p(v)+\lambda g(x,v)\phi_p(v), \quad \text{in $\Omega \times (0,\infty),$}$$ $$v =0, \text{in $\partial\Omega \times (0,\infty),$}\tag{IVP} v =v_0\ge0, \text{in $\Omega \times \{0\},$}$$ where $\Omega$ is a bounded domain in ${\mathbb R}^N$, for some integer $N\ge1$, with smooth boundary $\partial\Omega$, $\phi_p(s):=|s|^{p-1} {\rm sgn}s$, $s\in{\mathbb R}$, $\Delta_p$ denotes the $p$-Laplacian, with $p>\max\{2,N\}$, $v_0\in C^0(\overline{\Omega})$, and $\lambda>0$. The function $g:\overline{\Omega } \times [0,\infty)\to(0,\infty)$ is $C^0$ and, for each $x\in\overline{\Omega }$, the function $g(x,\cdot):[0,\infty)\to(0,\infty)$ is Lipschitz continuous and strictly decreasing. Clearly, (IVP) has the trivial solution $v\equiv0$, for all $\lambda>0$. In addition, there exists $0<\lambda_{\rm min}(g)<\lambda_{\rm max}(g)$ ($\lambda_{\rm max}(g)$ may be $\infty$) such that: $(a)$ if $\lambda\not\in(\lambda_{\rm min}(g),\lambda_{\rm max}(g))$ then (IVP) has no non-trivial, positive equilibrium; $(b)$ if $\lambda\in(\lambda_{\rm min}(g),\lambda_{\rm max}(g))$ then (IVP) has a unique, non-trivial, positive equilibrium $e_\lambda\in W_0^{1,p}(\Omega)$. We prove the following results on the positive solutions of (IVP): $(a)$ if $0<\lambda<\lambda_{\rm min}(g)$ then the trivial solution is globally asymptotically stable; $(b)$ if $\lambda_{\rm min}(g)<\lambda<\lambda_{\rm max}(g)$ then $e_\lambda$ is globally asymptotically stable; $(c)$ if $\lambda_{\rm max}(g)<\lambda$ then any non-trivial solution blows up in finite time

Landesman-Lazer conditions at half-eigenvalues of the p-Laplacian

We study the existence of solutions of the Dirichlet problem {gather} -\phi_p(u')' -a_+ \phi_p(u^+) + a_- \phi_p(u^-) -\lambda \phi_p(u) = f(x,u), \quad x \in (0,1), \label{pb.eq} \tag{1} u(0)=u(1)=0,\label{pb_bc.eq} \tag{2} {gather} where $p>1$, \phi_p(s):=|s|^{p-1}\sgn s for $s \in \mathbb{R}$, the coefficients $a_\pm \in C^0[0,1]$, $\lambda \in \mathbb{R}$, and $u^\pm := \max\{\pm u,0\}$. We suppose that $f\in C^1([0,1]\times\mathbb{R})$ and that there exists $f_\pm \in C^0[0,1]$ such that $\lim_{\xi\to\pm\infty} f(x,\xi) = f_\pm(x)$, for all $x \in [0,1]$. With these conditions the problem \eqref{pb.eq}-\eqref{pb_bc.eq} is said to have a `jumping nonlinearity'. We also suppose that the problem {gather} -\phi_p(u')' = a_+ \phi_p(u^+) - a_- \phi_p(u^-) + \lambda \phi_p(u) \quad\text{on} \ (0,1), \tag{3} \label{heval_pb.eq} {gather} together with \eqref{pb_bc.eq}, has a non-trivial solution $u$. That is, $\lambda$ is a `half-eigenvalue' of \eqref{pb_bc.eq}-\eqref{heval_pb.eq}, and the problem \eqref{pb.eq}-\eqref{pb_bc.eq} is said to be `resonant'. Combining a shooting method with so called `Landesman-Lazer' conditions, we show that the problem \eqref{pb.eq}-\eqref{pb_bc.eq} has a solution. Most previous existence results for jumping nonlinearity problems at resonance have considered the case where the coefficients $a_\pm$ are constants, and the resonance has been at a point in the `Fucik spectrum'. Even in this constant coefficient case our result extends previous results. In particular, previous variational approaches have required strong conditions on the location of the resonant point, whereas our result applies to any point in the Fucik spectrum.Comment: 14 page

Half eigenvalues and the Fucik spectrum of multi-point, boundary value problems

We consider the nonlinear boundary value problem consisting of the equation \tag{1} -u" = f(u) + h, \quad \text{a.e. on $(-1,1)$,} where $h \in L^1(-1,1)$, together with the multi-point, Dirichlet-type boundary conditions \tag{2} u(\pm 1) = \sum^{m^\pm}_{i=1}\alpha^\pm_i u(\eta^\pm_i) where $m^\pm \ge 1$ are integers, $\alpha^\pm = (\alpha_1^\pm, ...,\alpha_m^\pm) \in [0,1)^{m^\pm}$, $\eta^\pm \in (-1,1)^{m^\pm}$, and we suppose that $$\sum_{i=1}^{m^\pm} \alpha_i^\pm < 1 .$$ We also suppose that $f : \mathbb{R} \to \mathbb{R}$ is continuous, and $$0 < f_{\pm\infty}:=\lim_{s \to \pm\infty} \frac{f(s)}{s} < \infty.$$ We allow $f_{\infty} \ne f_{-\infty}$ --- such a nonlinearity $f$ is {\em jumping}. Related to (1) is the equation \tag{3} -u" = \lambda(a u^+ - b u^-), \quad \text{on $(-1,1)$,} where $\lambda,\,a,\,b > 0$, and $u^{\pm}(x) =\max\{\pm u(x),0\}$ for $x \in [-1,1]$. The problem (2)-(3) is `positively-homogeneous' and jumping. Regarding $a,\,b$ as fixed, values of $\lambda = \lambda(a,b)$ for which (2)-(3) has a non-trivial solution $u$ will be called {\em half-eigenvalues}, while the corresponding solutions $u$ will be called {\em half-eigenfunctions}. We show that a sequence of half-eigenvalues exists, the corresponding half-eigenfunctions having specified nodal properties, and we obtain certain spectral and degree theoretic properties of the set of half-eigenvalues. These properties lead to solvability and non-solvability results for the problem (1)-(2). The set of half-eigenvalues is closely related to the `Fucik spectrum' of the problem, which we briefly describe. Equivalent solvability and non-solvability results for (1)-(2) are obtained from either the half-eigenvalue or the Fucik spectrum approach

Self-adjoint boundary-value problems on time-scales

In this paper we consider a second order, Sturm-Liouville-type boundary-value operator of the form $$L u := -[p u^{ abla}]^{Delta} + qu,$$ on an arbitrary, bounded time-scale $mathbb{T}$, 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 $L^2(mathbb{T}_kappa)$, 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

Bifurcation from Zero or Infinity in Sturm–Liouville Problems Which Are Not Linearizable

AbstractWe consider the nonlinear Sturm–Liouville problem[formula][formula]whereai,biare real numbers with |ai|+|bi|>0,i=0,1, λ is a real parameter, and the functionspandaare strictly positive on [0,π]. Suppose that the nonlinearityhsatisfies a condition of the form[formula]as either |(ξ,η)|→0 or |(ξ,η)|→∞, for some constantsM0,M1. Then we show that there exist global continua of nontrivial solutions (λ,u) bifurcating fromu=0 or “u=∞,” respectively. These global continua have properties similar to those of the continua found in Rabonowitz' well-known global bifurcation theorem

Linear, second-order problems with Sturm-Liouville-type multi-point boundary conditions

We consider the linear eigenvalue problem \tag{1} -u" = \lambda u, \quad \text{on $(-1,1)$}, where $\lambda \in \mathbb{R}$, together with the general multi-point boundary conditions \tag{2} \alpha_0^\pm u(\pm 1) + \beta_0^\pm u'(\pm 1) = \sum^{m^\pm}_{i=1} \alpha^\pm_i u(\eta^\pm_i) + \sum_{i=1}^{m^\pm} \beta^\pm_i u'(\eta^\pm_i). We also suppose that: \alpha_0^\pm \ge 0, \quad \alpha_0^\pm + |\beta_0^\pm| > 0, \tag{3} \pm \beta_0^\pm \ge 0, \tag{4} (\frac{\sum_{i=1}^{m^\pm} |\alpha_i^\pm|}{\alpha_0^\pm})^2 + (\frac{\sum_{i=1}^{m^\pm} |\beta_i^\pm|}{\beta_0^\pm})^2 < 1, \tag{5} with the convention that if any denominator in (5) is zero then the corresponding numerator must also be zero, and the corresponding fraction is omitted from (5) (by (3), at least one denominator is nonzero in each condition). In this paper we show that the basic spectral properties of this problem are similar to those of the standard Sturm-Liouville problem with separated boundary conditions. Similar multi-point problems have been considered before under more restrictive hypotheses. For instance, the cases where $\beta_i^\pm = 0$, or $\alpha_i^\pm = 0$, $i = 0,..., m^\pm$ (such conditions have been termed Dirichlet-type or Neumann-type respectively), or the case of a single-point condition at one end point and a Dirichlet-type or Neumann-type multi-point condition at the other end. Different oscillation counting methods have been used in each of these cases, and the results here unify and extend all these previous results to the above general Sturm-Liouville-type boundary conditions

Second order, multi-point problems with variable coefficients

In this paper we consider the eigenvalue problem consisting of the equation -u" = \la r u, \quad \text{on $(-1,1)$}, where $r \in C^1[-1,1], \ r>0$ and \la \in \R, together with the multi-point boundary conditions u(\pm 1) = \sum^{m^\pm}_{i=1} \al^\pm_i u(\eta^\pm_i), where $m^\pm \ge 1$ are integers, and, for $i = 1,...,m^\pm$, \al_i^\pm \in \R, $\eta_i^\pm \in [-1,1]$, with $\eta_i^+ \ne 1$, $\eta_i^- \ne -1$. We show that if the coefficients \al_i^\pm \in \R are sufficiently small (depending on $r$) then the spectral properties of this problem are similar to those of the usual separated problem, but if the coefficients \al_i^\pm are not sufficiently small then these standard spectral properties need not hold. The spectral properties of such multi-point problems have been obtained before for the constant coefficient case ($r \equiv 1$), but the variable coefficient case has not been considered previously (apart from the existence of `principal' eigenvalues). Some nonlinear multi-point problems are also considered. We obtain a (partial) Rabinowitz-type result on global bifurcation from the eigenvalues, and various nonresonance conditions for existence of general solutions and also of nodal solutions --- these results rely on the spectral properties of the linear problem

BHPR research: qualitative1. Complex reasoning determines patients' perception of outcome following foot surgery in rheumatoid arhtritis

Background: Foot surgery is common in patients with RA but research into surgical outcomes is limited and conceptually flawed as current outcome measures lack face validity: to date no one has asked patients what is important to them. This study aimed to determine which factors are important to patients when evaluating the success of foot surgery in RA Methods: Semi structured interviews of RA patients who had undergone foot surgery were conducted and transcribed verbatim. Thematic analysis of interviews was conducted to explore issues that were important to patients. Results: 11 RA patients (9 ♂, mean age 59, dis dur = 22yrs, mean of 3 yrs post op) with mixed experiences of foot surgery were interviewed. Patients interpreted outcome in respect to a multitude of factors, frequently positive change in one aspect contrasted with negative opinions about another. Overall, four major themes emerged. Function: Functional ability & participation in valued activities were very important to patients. Walking ability was a key concern but patients interpreted levels of activity in light of other aspects of their disease, reflecting on change in functional ability more than overall level. Positive feelings of improved mobility were often moderated by negative self perception ("I mean, I still walk like a waddling duck”). Appearance: Appearance was important to almost all patients but perhaps the most complex theme of all. Physical appearance, foot shape, and footwear were closely interlinked, yet patients saw these as distinct separate concepts. Patients need to legitimize these feelings was clear and they frequently entered into a defensive repertoire ("it's not cosmetic surgery; it's something that's more important than that, you know?”). Clinician opinion: Surgeons' post operative evaluation of the procedure was very influential. The impact of this appraisal continued to affect patients' lasting impression irrespective of how the outcome compared to their initial goals ("when he'd done it ... he said that hasn't worked as good as he'd wanted to ... but the pain has gone”). Pain: Whilst pain was important to almost all patients, it appeared to be less important than the other themes. Pain was predominately raised when it influenced other themes, such as function; many still felt the need to legitimize their foot pain in order for health professionals to take it seriously ("in the end I went to my GP because it had happened a few times and I went to an orthopaedic surgeon who was quite dismissive of it, it was like what are you complaining about”). Conclusions: Patients interpret the outcome of foot surgery using a multitude of interrelated factors, particularly functional ability, appearance and surgeons' appraisal of the procedure. While pain was often noted, this appeared less important than other factors in the overall outcome of the surgery. Future research into foot surgery should incorporate the complexity of how patients determine their outcome Disclosure statement: All authors have declared no conflicts of interes