Eigenvalues and the one-dimensional \(p\)-Laplacian. (English) Zbl 1002.34019

Agarwal, Ravi P.; Lü, Haishen; O’Regan, Donal

J. Math. Anal. Appl. 266, No. 2, 383-400 (2002).

The authors are concerned with determining values of \(\lambda\), for which there exist positive solutions to the boundary value problem \[ (\phi_p(u'))'+ \lambda F(t,u)= 0\quad\text{in }(0,1),\quad u(0)= u(1)= 0,\tag{P} \] with \(\phi_p(s)=|s|^{p-2}s\) and \(p> 1\). They provide conditions to guarantee that the set \(E= \{\lambda> 0\mid\text{(P)}\) has positive solutions} is a bounded interval or an unbounded interval. They give the explicit eigenvalue interval in terms of \[ f_0= \lim_{x\to 0^+} {f(x)\over x^{p- 1}}\quad\text{and}\quad f_\infty= \lim_{x\to\infty} {f(x)\over x^{p-1}}. \] Also, they show the existence of two positive solutions when \(\lambda\) in an appropriate interval. The proofs are based on the Guo-Krasnosel’skii fixed-point theorem in cones.

Reviewer: Ruyun Ma (Lanzhou)

MSC:

34B24	Sturm-Liouville theory
34B18	Positive solutions to nonlinear boundary value problems for ordinary differential equations
34B15	Nonlinear boundary value problems for ordinary differential equations
47H10	Fixed-point theorems
34L05	General spectral theory of ordinary differential operators
47J10	Nonlinear spectral theory, nonlinear eigenvalue problems

Keywords:

one-dimensional \(p\)-Laplacian; positive solutions; cone; fixed-point; eigenvalue

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