We consider a strongly nonlinear differential equation of the following general type: (Φ(a(t,x(t))x′(t)))′=f(t,x(t),x′(t)),a.e. on[0,T],where f is a Carathédory function, Φ is a strictly increasing homeomorphism (the Φ -Laplacian operator), and the function a is continuous and non-negative. We assume that a(t, x) is bounded from below by a non-negative function h(t), independent of x and such that 1 / h∈ Lp(0 , T) for some p> 1 , and we require a weak growth condition of Wintner–Nagumo type. Under these assumptions, we prove existence results for the Dirichlet problem associated with the above equation, as well as for different boundary conditions. Our approach combines fixed point techniques and the upper/lower solution method.
Existence results for boundary value problems associated with singular strongly nonlinear equations
Biagi S.
;Calamai A.;Papalini F.
2020-01-01
Abstract
We consider a strongly nonlinear differential equation of the following general type: (Φ(a(t,x(t))x′(t)))′=f(t,x(t),x′(t)),a.e. on[0,T],where f is a Carathédory function, Φ is a strictly increasing homeomorphism (the Φ -Laplacian operator), and the function a is continuous and non-negative. We assume that a(t, x) is bounded from below by a non-negative function h(t), independent of x and such that 1 / h∈ Lp(0 , T) for some p> 1 , and we require a weak growth condition of Wintner–Nagumo type. Under these assumptions, we prove existence results for the Dirichlet problem associated with the above equation, as well as for different boundary conditions. Our approach combines fixed point techniques and the upper/lower solution method.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
