Constant Sign and Nodal Solutions for Nonlinear Robin Equations with Locally Defined Source Term

We consider a parametric Robin problem driven by a nonlinear, nonhomogeneous differential operator which includes as special cases the p-Laplacian and the (p, q)-Laplacian. The source term is parametric and only locally defined (that is, in a neighborhood of zero). Using suitable cut-off techniques together with variational tools and comparison principles, we show that for all big values of the parameter, the problem has at least three nontrivial smooth solutions, all with sign information (positive, negative and nodal).

In this problem, the map a : R N → R N involved in the differential operator of (P λ ), is strictly monotone and continuous, hence maximal monotone too. It satisfies certain other growth and regularity conditions, listed in hypotheses H(a) below. These conditions are not restrictive and provide a broad framework in which we can fit many differential operators of interest such as the p-Laplacian, the (p, q)-Laplacian (that is, the sum of a p-Laplacian and of a q-Laplacian) and the modified capillary differential operator. We point out that the differential operator is not homogeneous and this is a source of difficulties in the analysis of problem (P λ ). The potential function ξ(·) ∈ L ∞ (Ω) and ξ(z) ≥ 0 for a.a. z ∈ Ω. The source term λf (z, x) is parametric with λ > 0 being the parameter and f (z, x) a Carathéodory function (that is, for all x ∈ R, z → f (z, x) is measurable and for a.a. z ∈ Ω, x → f (z, x) is continuous). The interesting and distinguishing feature of our work is that this source term f (z, ·) is only locally defined, that is, we only fix the properties of f (z, ·) near zero. Away from that neighborhood of zero, f (z, ·) can be arbitrary.
In the boundary condition, ∂u ∂na denotes the conormal derivative of u corresponding to the map a(·). This derivative is understood via the nonlinear Green's identity (see Papageorgiou-Rǎdulescu-Repovš [16], Corollary 1.5.16, p. 34) and when u ∈ C 1 (Ω), then ∂u ∂n a = (a(∇u), n) R N with n(·) being the outward unit normal on ∂Ω. The boundary coefficient β(·) is nonnegative. The case β ≡ 0 is also included and corresponds to the Neumann problem. Using suitable cut-off techniques together with variational tools based in the critical point theory, we show that for all λ > 0 big problem (P λ ) has at least three nontrivial smooth solutions all with sign information, a positive, a negative and a nodal (sign changing) solutions.
The first to examine parametric elliptic equations with a source term defined only locally, was Wang [20], who studied a semilinear equation driven by the Dirichlet Laplacian. Imposing a local symmetry condition on the source term (it is assumed that it is odd), Wang [20] proves that for every parameter value λ > 0, the problem has a whole sequence {u n } n≥1 ⊆ H 1 0 (Ω) ∩ L ∞ (Ω) of weak solutions such that u n ∞ → 0 as n → +∞. The work of Wang [20] was extended by Li-Wang [7] to semilinear Schrödinger equations. Extensions of these results were obtained by Papageorgiou-Vetro-Vetro [18] (semilinear Robin problems), Papageorgiou-Rǎdulescu-Repovš [15] (nonlinear Robin problems) and Papageorgiou-Rǎdulescu-Repovš [14] (Dirichlet (p, 2)-equations). In all the aforementioned works the source term is symmetric near zero and this leads to an application of a version of the symmetric mountain pass theorem, which generates the desired sequence of distinct nodal solutions. In contrast here we do not impose any symmetry condition on f (z, ·). Finally we should mention the recent paper of Guarnotta-Marano-Papageorgiou [5]. There the authors also deal with a nonlinear Robin problem driven by a nonhomogeneous differential operator plus an indefinite potential term. However, in [5] the reaction term is nonparametric and this changes the hypotheses on the reaction and consequently the geometry of the problem and the approach used. In [5] it is assumed that the reaction has constant sign near zero (see hypothesis (f 1 )) and this is crucial in the analysis since is the reason for which the reaction is only locally defined. In contrast here we rely on cut-off techniques. Moreover, the asymptotic condition as x → 0 on the reaction is in our case different and do not require the presence of a concave term near zero (compare H(f ) (i), (ii) of this paper with (f 3 ), (f 4 ) of [5]). All these facts distinguish our work here from that of [5] and for that reason the tools and techniques are different.

Remark 1.
Hypotheses H(a) (i), (ii), (iii) come from the nonlinear regularity theory of Lieberman [8] and the nonlinear maximum principle of Pucci-Serrin [19], while hypothesis H(a) (iv) is motivated by the particular needs of our problem. However, the condition is mild and it is satisfied in all cases of interest.
Therefore G(·) is the primitive of a(·) and from the convexity of G(·) and since G(0) = 0, we have Hypotheses H(a) lead to the following properties of the map a(·) (see Papageorgiou-Rǎdulescu [11]).
(a) a(·) is continuous and strictly monotone (thus maximal monotone too); This lemma and (2.1) lead to the following bilateral growth estimates for the primitive G(·).
The examples that follow show that the framework provided by these conditions on a(·), is broad an includes as special cases many differential operators of interest (see Papageorgiou-Rǎdulescu [11]).
This map corresponds to the p-Laplace differential operator defined by ∆ p u = div (|∇u| p−2 ∇u) for all u ∈ W 1,p (Ω).
(b) a(y) = |y| p−2 y + |y| q−2 y with 1 < q < p < +∞. This map corresponds to the (p, q)-Laplace differential operator defined by Such operators arise in many models of physical phenomena which involve the combination of two operators of different nature. We mention the works of Benci-D'Avenia-Fortunato-Pisani [2] (quantum physics -soliton solutions), Cherfils-Il yasov [3] (reaction-diffusion systems) and Zhikov [21,22] (elasticity theory). A survey of some recent results concerning such equations, can be found in the paper of Marano-Mosconi [9]. Also, we mention the work of Papageorgiou-Vetro [17] on (p, q)-equations with variable exponents.
The differential operator corresponding to this map is and appears in problems of plasticity theory (see [11]).
Remark 2. Evidently the case β ≡ 0 is included and corresponds to the Neumann problem.
The next lemma can be found in Mugnai-Papageorgiou [10] (Lemma 4.11). In the sequel by · we denote the norm of the Sobolev space W 1,p (Ω) defined by Using this measure, we can define in the usual way the boundary Lebesgue spaces L r (∂Ω) with 1 ≤ r ≤ +∞. We know that there exists a unique continuous linear map γ 0 : W 1,p (Ω) → L p (∂Ω), known as the "trace map", such that This map is compact, im γ 0 = W 1 p ,p (∂Ω) 1 p + 1 p = 1 and ker γ 0 = W 1,p 0 (Ω). In the sequel, for the sake of notational simplicity, we drop the use of the trace map γ 0 (·). All restrictions of Sobolev functions are understood in the sense of traces.
Consider the C 1 -functional γ : From Corollary 1 and Lemmata 2 and 3, we infer that there exists c 7 > 0 such that Now we introduce the conditions on the source term f (z, x). Note that by p * > 1 we denote the critical Sobolev exponent corresponding to p and defined by Carathéodory function such that f (z, 0) = 0 for a.a. z ∈ Ω and (i) there exist c 1 , c 2 > 0 and r ∈ (p, p * ) such that uniformly for a.a.
Remark 3. We emphasize that conditions (i) and (ii) concern only the behavior of f (z, ·) near zero. There are no conditions concerning the behavior of x → f (z, x) for |x| big. Also we point out that no local sign condition is assumed on f (z, ·).

From (2.3)-(2.5) it follows that
| f (z, x)| ≤ c 9 |x| r−1 for a.a. z ∈ Ω, all x ∈ R, some c 9 > 0, Finally we recall the definition of critical groups for a C 1 -functional at an isolated critical point. We will use critical groups to distinguish between critical points. So, let X be a Banach space and ϕ ∈ C 1 (X). We introduce the following sets For every topological pair (Y 1 , Y 2 ) with Y 2 ⊆ Y 1 ⊆ X and every integer k ≥ 0, by H k (Y 1 , Y 2 ) we denote the k th -singular homology group with integer coefficients. Then for u ∈ K ϕ isolated and c = ϕ(u), the critical groups of ϕ at u are defined by with U a neighborhood of u such that K ϕ ∩ ϕ c ∩ U = {u}. The excision property of singular homology guarantees that this definition is independent of the choice of the isolating neighborhood.
For this problem when λ > 0 is big, we can have constant sign smooth solutions. Recall that the Banach space C 1 (Ω) is ordered with positive (order) cone This cone has a nonempty interior given by Proof. We do the proof for the positive solution, the proof for the negative solution being similar.
Here by |Ω| N we denote the Lebesgue measure of Ω.
Similarly, if we work with the C 1 -functional ϕ − λ : for all u ∈ W 1,p (Ω), via the mountain pass theorem, we produce a negative Proof. From the proof of Proposition 2 we know that is of mountain pass type). From (2.3) and (2.5) we see that F (z, x) ≥ c 15 |x| τ for a.a. z ∈ Ω, all |x| ≤ δ 0 , some c 15 > 0. (3.13) Consider the function k λ : R + → R defined by This function is differentiable, unbounded below (recall that τ > p) and attains its supremum on R + at t 0 > 0. Let On account of (3.12) we have Then the nonlinear regularity theory of Lieberman [8] implies that there exist α ∈ (0, 1) and c 21 > 0 such that u n ∈ C 1,α (Ω) and u n C 1,α (Ω) ≤ c 21 for all n ∈ N. (3.19) On account of (3.19) and the compact embedding of C 1,α (Ω) into C 1 (Ω), we have, at least for a subsequence, that u n → u in C 1 (Ω). We must have u ≡ 0 or otherwise from (3.18) and hypothesis H(f ) (iii), we have a contradiction.
Suppose that u * λ = 0. Then u n → 0 in C 1 (Ω) (see (3.22)). (3.24) Let y n = u n / u n , n ∈ N. Then y n = 1 for all n ∈ N and so we may assume that y n w − → y in W 1,p (Ω) and y n → y in L p (Ω) and in L p (∂Ω).
On account of (2.5), (3.24) and hypothesis H(f ) (i), we have From (3.23) and via Fatou's lemma, we have a contradiction. So, u * λ = 0. Passing to the limit as n → +∞ in (3.21) we conclude that Note that in this case, since S − λ is upward directed, we can find an increasing sequence {v n } n≥1 ⊆ S − λ such that sup n≥1 v n = sup S − λ .
From Proposition 3, we have at once the following corollary.

Nodal solutions
In this section, using the extremal constant sign solutions produced in Proposition 4, we will produce a nodal solution.