Variational eigencurve and bifurcation for two-parameter nonlinear Sturm-Liouville equations (Q1381009)

The author considers the two-parameter nonlinear boundary value problem \[ u''+ \mu u^p =\lambda u^q,\;u(x)>0 \quad \text{when }x\in I \] while \[ u(0)= u(1)=0 \] satisfied by the function \(u=u(x)\) defined on the interval \(I=(0,1)\) and proves the existence of a unique variational eigenvalue \(\lambda= \lambda (\mu)\). Here \(p\) and \(q\) are given constants \((1<q< p<q+2)\) while \(\mu>0\) and \(\lambda>0\) stand for eigenvalue parameters. The variational eigenvalue and the associated eigenfunction \(u_\mu\) satisfy the following minimization problem: \[ \mu>0,\;\lambda (\mu)>0,\;u_\mu\in N_{\mu, \alpha} \] \[ {1\over q+1} \| u_\mu \|^{q+1}_{q+1}: =\inf_{u\in N_{\mu,\alpha}} {1\over q+1} \| u\|^{q+1}_{q+1}. \] Here \(N_{\mu, \alpha}\) denotes the so-called level set defined by \[ N_{\mu,\alpha}: =\left\{u\in X: {1 \over 2} \| u\|^2_X- {\mu\over p+1} \| u\|^{p+1}_{p+1}= -\alpha\right\}, \] with a certain (fixed) normalizing parameter \(\alpha>0\). The space \(X\) is the closure of \(C^\infty_0 (I)\) equipped with the norm \(\| u\|^2_X= \int_I| u'(x) |^2dx\). The main results are stated in two theorems which are based on four and nine lemmas, respectively. The first theorem shows the existence and uniqueness of \(\lambda= \lambda(\mu)\) while the second theorem gives the asymptotic relation \(\lambda \sim C\mu^{(q-1)/(p-1)}\) as \(\mu\to 0\).