Robin problems for the p-Laplacian with gradient dependence

We consider a nonlinear elliptic equation with Robin boundary condition driven by the p-Laplacian and with a reaction term which depends also on the gradient. By using a topological approach based on the Leray-Schauder alternative principle, we show the existence of a smooth solution.


Introduction.
Let Ω ⊆ R N be a bounded domain with a C 2 −boundary ∂Ω. In this paper, we study the following nonlinear Robin problem with dependence on the gradient    −∆ p u(z) = f (z, u(z), Du(z)), in Ω, ∂u ∂n p + β(z)|u| p−2 u = 0, on ∂Ω.
The reaction term f (z, x, y) is a Carathéodory function (that is, for all (x, y) ∈ R × R N the function z ∈ R → f (z, x, y) is measurable, and for a.a. z ∈ Ω the map (x, y) → f (z, x, y) is continuous). Note that the reaction term depends also on the gradient of u (convection term). This fact precludes the use of variational methods directly on problem (1). Finally, in the boundary condition ∂u ∂n p denotes the conormal derivative defined by extension of the map u ∈ C 1 (Ω) → |Du| p−2 ∂u ∂n , n(·) being the outward unit normal on ∂Ω.
Using topological methods and, more precisely, the Leray-Schauder alternative principle, we show the existence of a smooth solution for problem (1), under general conditions on the convection term. The main hypothesis is that, asymptotically as x → ±∞, the quotient f (z, x, y) |x| p−2 x stays below the principal eigenvalueλ 1 uniformly in y ∈ R N in the measure sense, in the sense that we allow only partial interaction withλ 1 (nonuniform nonresonance), see condition H(f ).ii) below. Elliptic problems with a reaction term depending on the gradient have been studied using a variety of methods. Most of the works deal with the Dirichlet problem. In this setting, we mention the papers of de Figueiredo -Girardi -Matzeu [2], Girardi -Matzeu [5] (semilineat equations) and Ruiz [13], Faraci -Motreanu -Puglisi [1], Huy -Quan -Khanh [6] (nonlinear equations driven by the p−Laplacian). For the Neumann problem, we have the recent works of Gasinski -Papageorgiou [4] and Papageorgiou -Radulescu -Repovs [12]. We remark that in these works the differential operator depends strongly on both u and Du.

2.
Mathematical background -hypotheses. In this section we recall the main mathematical tools which we will use in the analysis of problem (1) and we state our hypotheses on the reaction term f (z, x, y).
Let X be a reflexive Banach space and X * its topological dual. By ·, · we denote the duality brackets for the pair (X, X * ). Given a map A : X → 2 X * , the graph of A is the set The domain of A is the set We say that A(·) is monotone if The map A(·) is maximal monotone, if it is monotone and its graph is maximal with respect to inclusion among the graphs of monotone maps, that is, From Gasinski -Papageorgiou [3], p. 319, we have: is continuous and maps bounded subsets of V into relatively compact subsets of Y .
The analysis of problem (1) uses the Sobolev space W 1,p (Ω), the Banach space C 1 (Ω) and the boundary Lebesgue spaces L q (∂Ω), 1 ≤ q ≤ ∞. By · we denote the norm of W 1,p (Ω) defined by Recall that the Banach space Using this measure, we can define in the usual way the boundary Lebesgue spaces L q (∂Ω), 1 ≤ q ≤ ∞. The theory of Sobolev spaces says that there exists a unique continuous linear map γ 0 : W 1,p (Ω) → L p (∂Ω) known as the trace map such that So, the trace map assigns boundary values to every Sobolev function. We know that Our hypotheses on the boundary coefficient β(·) are the following: H(β) : β ∈ C 0,α (∂Ω) with α ∈ (0, 1) and β(z) ≥ 0 for all z ∈ ∂Ω. Remark 1. Of course, the case β ≡ 0 corresponds to the Neumann problem.
• if ξ(u) = Du p p + ∂Ω β(z)|u| p dσ for all u ∈ W 1,p (Ω), then Hence, in (3) the infimum is realized on the one-dimensional eigenspace corresponding toλ 1 . The above properties imply that the elements of this eigenspace do not change sign and lead to the following lemma (see [9,Lemma 4.11]).
H(f ) f : Ω×R×R N → R is an L ∞ −locally Hölder continuous function such that 3. Existence of solutions. In this section we shall prove the existence of one solution to problem (1). For this, let g ∈ L ∞ (Ω) andξ > Θ ∞ . We start by considering the following auxiliary Robin problem Proposition 2. If Hypothesis H(β) holds, then for every g ∈ L ∞ (Ω) problem (4) admits a unique solution E(g) ∈ C 1 (Ω), and the map E : Proof. First we show the existence and uniqueness of a solution for problem (4).