Existence of positive solutions for nonlinear Dirichlet problems with gradient dependence and arbitrary growth

We consider a nonlinear elliptic problem driven by the Dirichlet p-Laplacian and a reaction term which depends also on the gradient (convection). No growth condition is imposed on the reaction term f (z, ·, y). Using topological tools and the asymptotic analysis of a family of perturbed problems, we prove the existence of a positive smooth solution.


Introduction
Let Ω ⊆ R N be a bounded domain with a C 2 -boundary ∂Ω.We study the following nonlinear Dirichlet problem with gradient dependence In this problem ∆ p denotes the p-Laplace differential operator defined by The dependence of the reaction term f (z, x, y) on the gradient ∇u of the unknown function u, precludes the use of variational methods in the study of problem (1.1).Instead our approach is topological and uses the theory of nonlinear operators of monotone type and the asymptotic analysis of a perturbation of the original problem.We prove the existence of a positive smooth solution, without imposing any growth condition on f (z, •, y).Instead, we employ a Hartman-type condition on f (z, •, y) which leads to an a priori bound for the positive solutions.This absence of any growth condition on f (z, •, y) distinguishes our work from previous ones on elliptic equations with convection.We refer to the papers of de Figueiredo-Girardi-Matzeu [3], Girardi-Matzeu [6] (semilinear problems driven by the Laplacian) and Faraci-Motreanu-Puglisi [2], Ruiz [9] (nonlinear problems driven by the p-Laplacian).We mention also the recent work of Gasi ński-Papageorgiou [5] on Neumann problems driven by a differential operator of the form div(a(u)∇u).In all the aforementioned works f (z, •, y) exhibits the usual subcritical polynomial growth.

Mathematical background -hypotheses
Let X be a reflexive Banach space and X * its topological dual.By •, • we denote the duality brackets for the pair (X * , X).A map V : X → X * is said to be "pseudomonotone", if it has the following property: A maximal monotone, everywhere defined operator is pseudomonotone.Moreover, if V = A + K with A maximal monotone and everywhere defined and Pseudomonotone operators exhibit remarkable surjectivity properties.More precisely we have (see Gasi ński-Papageorgiou [4], p. 336).Proposition 2.1.If V : X → X * is pseudomonotone and strongly coercive, that is, The following two spaces will be used in the analysis of problem (1.1): • the Sobolev space W 1,p 0 (Ω); . By • we denote the norm of W 1,p 0 (Ω).On account of the Poincaré inequality, we can take u = ∇u p for all u ∈ W 1,p 0 (Ω).The Banach space C 1 0 (Ω) is an ordered Banach space with positive (order) cone This cone has a nonempty interior given by int where ∂u ∂n = (∇u, n) R N with n(•) being the outward unit normal on ∂Ω.
We know that (2.1) has a smallest eigenvalue λ 1 which has the following properties: • λ 1 > 0 and λ 1 is isolated (that is, if σ(p) denotes the spectrum of (2.1), then we can find The infimum in (2.2) is realized on the one-dimensional eigenspace corresponding to λ 1 .The above properties imply that the elements of this eigenspace, have fixed sign.By u 1 we denote the L p -normalized (that is, u 1 p = 1) positive eigenfunction corresponding to λ 1 .The nonlinear regularity theory and the nonlinear maximum principle (see, for example, Gasi ński-Papageorgiou [4], pp.737-738) imply that • for almost all z ∈ Ω, (x, y) → f (z, x, y) is continuous.
Such a function is necessarily jointly measurable (see Hu-Papageorgiou [8], p. 142).The hypotheses on the reaction term f (z, x, y) are the following: (iii) for every c > 0, there exists Remark 2.2.Since we are looking for positive solutions and all the above hypotheses concern the positive semiaxis R + = [0, +∞), without any loss of generality we assume that f (z, x, y) = 0 for a.a.z ∈ Ω, all x ≤ 0, all y ∈ R N .
Hypothesis H( f ) (i) is essentially a condition due to Hartman [7] (p.433).It was used by Hartman [7] for ordinary Dirichlet differential systems.
Example 2.3.The following function satisfies hypotheses H( f ).For the sake of simplicity we drop the z-dependence.
Let M > 0 be as in hypothesis H( f ) (i) and consider the nonexpansive function (that is, Lipschitz continuous with Lipschitz constant 1) p M : R → R defined by Clearly |p M (x)| ≤ |x| for all x ∈ R. We introduce the following function This is a Carathéodory function.Let e ∈ int C + and ε > 0. We consider the following auxiliary Dirichlet problem: Proposition 2.4.If hypotheses H( f ) hold and ε > 0 is small, then problem (2.4) admits a solution u ε ∈ int C + and 0 ≤ u ε (z) ≤ M for all z ∈ Ω. Proof.

be the nonlinear map defined by
It is well-known (see, for example Gasi ński-Papageorgiou [4]), that the map A(•) is bounded (that is, maps bounded sets to bounded ones), continuous, strictly monotone, hence maximal monotone too.Also let This map too is bounded, continuous and maximal monotone (recall that L p (Ω) → W −1,p (Ω)).
Let N f denote the Nemitsky map corresponding to the Carathéodory function f , that is, Hypothesis H( f ) (ii) and the Krasnoselskii theorem (see Gasi ński-Papageorgiou [4], p. 407) imply that The compact embeddings of (use the Sobolev embedding theorem and Lemma 2.2.27, p. 141 of Gasi ński-Papageorgiou [4]), imply that ψ p and N f are both completely continuous maps.
Evidently V(•) is bounded, continuous and pseudomonotone.Also, for all u ∈ W 1,p 0 (Ω) we have Then we can use Proposition 2.1 and find for all h ∈ W 1,p 0 (Ω).