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

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)


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
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