Existence results for boundary value problems associated with singular strongly nonlinear equations
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Journal of Fixed Point Theory and Applications
Existence results for boundary value problems associated with singular strongly nonlinear equations Stefano Biagi , Alessandro Calamai
and Francesca Papalini
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 Carath´edory 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 nonnegative 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. Mathematics Subject Classification. Primary 34B15; Secondary 34L30, 34B24, 34C25. Keywords. Boundary value problems, singular φ-Laplacian, lower/upper solutions, fixed-point, Winter–Nagumo condition.
1. Introduction Recently, many papers have been devoted to the study of boundary value problems (BVPs for short) associated with nonlinear ODEs involving the so-called Φ-Laplace operator (see, e.g., [3–5,9]). Namely, ODEs of the type: Φ(x ) = f (t, x, x ), where f is a Carath´edory function and Φ : R → R is a strictly increasing homeomorphism such that Φ(0) = 0. The class of Φ-Laplacian operators includes as a special case the classical r-Laplacian Φ(y) := y|y|r−2 , with r > 1. Such operators arise in some models, e.g., in non-Newtonian fluid theory, diffusion of flows in porous media, 0123456789().: V,-vol
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nonlinear elasticity, and theory of capillary surfaces. Other models (for example reaction-diffusion equations with non-constant diffusivity and porous media equations) lead to consider mixed differential operators, that is, differential equations of the type: (1.1) a(x) Φ(x ) = f (t, x, x ), where a is a continuous positive function (see, e.g., [8]). Furthermore, several papers have been devoted to the case of singular or non-surjective operators (see [1,6,10]). Usually, these existence results stem from a combination of fixed point techniques with the upper and lower solution method. In this context, an important tool to get a priori bounds for the derivatives of the solutions is a Nagumo-type growth condition on the function f . Let us observe that, when in the differential operator is present the nonlinear term a, some assumptions are required on the differential operator Φ, which in general is assumed to be homogeneous, or having at most linear growth at infinity. More recently, in collaboration with Cristina Marcelli, we considered two different generalizations of Eq. (1.1). In the paper [15], we investigated the case in which the function a may depend also on t. More precisely, we obtained existence results for genera
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