Motivated by [3], we define the “Ambrosetti–Hess problem” to be the problem of bifurcation from infinity and of the local behavior of continua of solutions of nonlinear elliptic eigenvalue problems. Although the works in this direction underline the asymptotic properties of the nonlinearity, here we point out that this local behavior is determined by the global shape of the nonlinearity.

We consider linear differential equations of the form $${\left(p\left(t\right)x^{\text{'}}\right)}^{\text{'}}+\lambda q\left(t\right)x=0(p\left(t\right)>0,q\left(t\right)>0)\left(\mathrm{A}\right)$$ on an infinite interval $[a,\infty )$ and study the problem of finding those values of $\lambda$ for which () has principal solutions $x_0(t;\lambda )$ vanishing at $t=a$. This problem may well be called a singular eigenvalue problem, since requiring $x_0(t;\lambda )$ to be a principal solution can be considered as a boundary condition at $t=\infty$. Similarly to the regular eigenvalue problems for () on compact intervals, we can prove a theorem asserting that there exists a sequence $\left\{{\lambda }_n\right\}$ of eigenvalues such...

We give some results about the topological structure of solution sets of multivalued Sturm-Liouville problems in Banach spaces.

Let ${\lambda }_1\left(Q\right)$ be the first eigenvalue of the Sturm-Liouville problem $$y^{\text{'}\text{'}}-Q\left(x\right)y+\lambda y=0,y\left(0\right)=y\left(1\right)=0,0<x<1.$$ We give some estimates for $m_{\alpha ,\beta ,\gamma }={inf}_{Q\in T_{\alpha ,\beta ,\gamma }}{\lambda }_1\left(Q\right)$ and $M_{\alpha ,\beta ,\gamma }={sup}_{Q\in T_{\alpha ,\beta ,\gamma }}{\lambda }_1\left(Q\right)$, where $T_{\alpha ,\beta ,\gamma }$ is the set of real-valued measurable on $\left[0,1\right]$$x^{\alpha }{...}^{}$

We consider the Sturm-Liouville problem with symmetric boundary conditions and an integral condition. We estimate the first eigenvalue ${\lambda }_1$ of this problem for different values of the parameters.

In this paper, we consider the nonlinear fourth order eigenvalue problem. We show the existence of family of unbounded continua of nontrivial solutions bifurcating from the line of trivial solutions. These global continua have properties similar to those found in Rabinowitz and Berestycki well-known global bifurcation theorems.

We consider nonlinear Sturm-Liouville problems with spectral parameter in the boundary condition. We investigate the structure of the set of bifurcation points, and study the behavior of two families of continua of nontrivial solutions of this problem contained in the classes of functions having oscillation properties of the eigenfunctions of the corresponding linear problem, and bifurcating from the points and intervals of the line of trivial solutions.