The existence of solutions to a class of semilinear differential equations

Abstract

We will study the existence of solutions of $$ -(\alpha^{p-1}(t)|u'|^{p-2}u')'= f(t,u)+h(t) \quad \text{in $[0,1]$}, $$ subject to various boundary conditions and $p>1$. In addition, we will give a detailed characterization of the eigenvalues and the Fucik spectrum of the corresponding differential operators. We use the Sturm comparison theorem and degree theory.