Solvability of a nonlinear two-point boundary value problem at resonance II. (English) Zbl 1054.34031

Summary: The author applies the Leray-Schauder continuation method to obtain existence theorems for solutions to \[ u''+ u+ g(x,u)= h\quad\text{in }(0,\pi),\;u(0)= u(\pi)= 0,\tag{i} \] in which the nonlinearity \(g\) grows superlinearly in \(u\) in one of directions \(u\to\infty\) and \(u\to-\infty\), and may grow sublinearly in the other, and to \[ -u''- u+ g(x,u)= h\quad\text{in }(0,\pi),\;u(0)= u(\pi)= 0,\tag{ii} \] in which the nonlinearity \(g\) has no growth restriction in \(u\) as \(| u|\to\infty\). Let the \(L^1(0,\pi)\) function \(h\) satisfy \[ \int^\pi_0 g^-_\beta(x)\sin x\,dx< \int^\pi_0 h(x)\sin x\,dx= 0<\int^\pi_0 g^+_\alpha(x)\sin x\,dx, \] with \(\alpha,\beta\geq 0\), \(g^+_\alpha(x)= \liminf_{u\to\infty}\, g(x,u)| u|^\alpha\), and \(g^-_\beta= \limsup_{u\to-\infty}\, g(x,u)| u|^\beta\).
For part I see [J. Differ. Equations 140, 1–9 (1997; Zbl 0887.34016)].


34B15 Nonlinear boundary value problems for ordinary differential equations


Zbl 0887.34016
Full Text: DOI


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