Existence and uniqueness of positive solutions for the Neumann p-Laplacian

We consider a nonlinear Neumann problem driven by the p-Laplacian and with a Carathéodory reaction which satisfies only a unilateral growth restriction. Using the principal eigenvalue of an eigenvalue problem involving the Neumann p-Laplacian plus an indefinite potential, we produce necessary and sufficient conditions for the existence and uniqueness of positive smooth solutions.

We are interested in the existence and uniqueness of positive solutions when the nonlinearity f (z, ·) is only unilaterally restricted (only from above). Problems like this were studied primarily in the context of semilinear (i.e., p = 2) equations with Dirichlet boundary conditions. We mention the works of Amann [2], Brézis and Oswald [4], Dancer [6], de Figueiredo [7], Hess [16], Krasnoselskii [19], Laetsch [20], and Simpson and Cohen [24]. Extensions to the Dirichlet p-Laplacian can be found in the works of Guo [14], Guo and Webb [15] and Kamin and Veron [18], but for special classes of equations, such as logistic equations. To the best of our knowledge, there are no such results for the Neumann p-Laplacian. Some other existence results for Neumann p-Laplacian problems, but with no information on the sign of solutions can be found in Gasiński and Papageorgiou [9][10][11] and with some sign information on the solution (but without uniqueness) can be found in Gasiński and Papageorgiou [12,13].
As it is remarked in de Figueiredo [7], the problem of uniqueness for elliptic equations, is in general a difficult one and requires special structure on the reaction term. Our work here is closely related to that of Brézis and Oswald [4]. In fact our result is a twofold generalization of that in [4]. First, we pass from the Laplacian (semilinear equation; i.e., p = 2) to the p-Laplacian (nonlinear equation; i.e., p ∈ (1, +∞)). Second, we pass from the Dirichlet to the Neumann boundary condition. We should mention that sufficient conditions for the uniqueness of the positive solutions of the Dirichlet p-Laplacian were obtained by Belloni and Kawohl [3], were the authors exploited in a direct way the convexity of the energy functional u −→ ϕ(u) in u p .

An eigenvalue problem
In this section we discuss the first eigenvalue of the nonlinear eigenvalue problem involving the negative Neumann p-Laplacian plus an indefinite potential. This quantity plays a central role in our subsequential considerations, but it is also of independent interest.
The eigenvalue problem under consideration is the following: (2.1)
Proof Let ξ : W 1, p ( ) −→ R be the C 1 -functional, defined by We set Thus λ 1 − β ∞ . We will show that the infimum in (2.2) is realized at a u 1 ∈ W 1, p ( ), with u 1 p = 1. To this end, let {u n } n 1 ⊆ M be a minimizing sequence, i.e., Clearly the sequence {u n } n 1 ⊆ W 1, p ( ) is bounded and so by passing to a suitable subsequence if necessary, we may assume that It is clear from (2.3) that u 1 p = 1, i.e., u 1 ∈ M. Hence ξ( u 1 ) = λ 1 . The Lagrange multiplier rule (see, e.g., Papageorgiou and Kyritsi [23, p. 76]) implies that λ 1 is an eigenvalue of problem (2.1), with the corresponding eigenfunction u 1 ∈ W 1, p ( ). Using the Moser iteration technique, we show that u 1 ∈ L ∞ ( ) (see, e.g., Hu and Papageorgiou [17]) and the nonlinear regularity theorem of Lieberman [21], implies that u 1 ∈ C 1,α ( ) for some α ∈ (0, 1). Moreover, since we infer that u 1 does not change sign and we may assume that u 1 0. Invoking the nonlinear maximum principle of Vázquez [25], we conclude that Next, we show the simplicity of λ 1 . So, let v 1 ∈ W 1, p ( ) be another eigenfunction corresponding to λ 1 . As above, we show that v 1 ∈ C 1 ( ) and v 1 (z) > 0 for all z ∈ . We introduce From the generalized Picone identity of Allegretto and Huang [1] and the nonlinear Green's identity (see Casas and Fernández [5]), we have so finally u 1 = k v 1 for some k > 0 (see Allegretto and Huang [1]). This proves that λ 1 is simple (i.e., it is a principal eigenvalue).
From the above proof, we have Note that in the second infimum in (2.6), the integral {u =0} β|u| p dz makes sense even when β is only a measurable function and there exists c > 0, such that In the first case λ 1 (β) ∈ [−∞, +∞) and in the second case In what follows by A : This map is continuous and maximal monotone (see [8] or [23]).

Existence of positive solutions
In this section we prove the existence of a positive smooth solution. The hypotheses on the reaction f are the following: Remark 3.1 Since we are looking for positive solutions and hypotheses H f concern only the positive semiaxis R + = [0, +∞), by truncating if necessary, we may (and will) assume that Note that H f (i) is a unilateral growth condition. Hypothesis H f (ii) implies that both functions η and η 0 are measurable. Moreover, we have and thus Similarly, we have If η, η 0 ∈ L ∞ ( ), then λ 1 (−η), λ(−η 0 ) ∈ R and are the principal eigenvalues of (2.1) when β = −η and β = −η 0 respectively. If f (z, ζ ) = f (ζ ) (autonomous case), then hypotheses H f (iii) and H f (iv) are equivalent to saying that η < λ 1 = 0 < η 0 (recall that the first eigenvalue of the negative Neumann p-Laplacian (i.e., problem (2.1) with β ≡ 0) is zero).

Example 3.2 Let
Then f satisfies hypotheses H f . This function corresponds to the equidiffusive p-logistic equation and η 0 = λ > 0, η = −∞. More generally, let Note that this f has no polynomial growth restriction from below.
We introduce the following truncation-perturbation of f : This is a Carathéodory function. We set Note that hypothesis H f (i) and (3.1) imply that and some c 1 > 0. Because of (3.2), we see that we can introduce the functional ϕ : So, passing to a subsequence if necessary, we may assume that  .1)). Note that So, it follows that f (z, 0) 0 for almost all z ∈ .
Since y + n −→ y + in L p ( ) (see (3.7)), by passing to a further subsequence if necessary, we may also assume that (3.14) From (3.11), we have (see hypotheses H f (ii) and (iii)). For a given ε > 0, we can find M 2 = M 2 (ε, z) > 0, such that Since ε > 0 was arbitrary, we let ε 0 to conclude that then for every ξ > 0, we can find Since ξ > 0 was arbitrary, we let ξ → +∞ to conclude that Therefore, finally we have proved (3.18).

Proposition 3.4 If hypotheses H f hold, then ϕ is sequentially weakly lower semicontinuous.
Proof From the expression of ϕ and since the norm in a Banach space is sequentially weakly lower semicontinuous, it suffices to show that the integral functional ψ : is sequentially weakly lower semicontinuous. To this end, we need to show that for every λ ∈ R, the sublevel set is sequentially weakly closed. To this end, let {u n } n 1 ⊆ L λ and assume that Then u n −→ u in L p ( ) (by the Sobolev embedding theorem) and since L p ( ) is a Banach lattice, we also have that We may also assume that and (see (3.22)). Also, from (3.23) and Fatou's lemma, we have (3.27) Then, from (3.24) and using (3.25) and (3.27), in the limit as n → +∞, we have so u ∈ L λ and so ψ is sequentially weakly lower semicontinuous. Now we are ready to establish the existence of positive solutions.

Proposition 3.5
It hypotheses H f hold, then problem (1.1) has a positive solution u 0 ∈ C 1 ( ) with u 0 (z) > 0 for all z ∈ .
Since u 0k is a minimizer of ϕ k , we have We have Since v k ∈ L ∞ ( ), we conclude that Claim 2 holds. Next, let h ∈ C 1 ( ) and t ∈ (−1, 1). We set where N g (u)(·) = g(·, u(·)) for all u ∈ W 1, p ( ). Here we have used Fubini's theorem and Hölder's inequality. Hypotheses H f (i), (ii) and the fact that h ∈ C 1 ( ), imply that g z, u 0 (z) + th(z) a(z) for almost all z ∈ , all t ∈ (−1, 1), with a ∈ L ∞ ( ). From this it follows that through nonlinear regularity theory and the nonlinear maximum principle of Vázquez for some k > 0 (see Allegretto and Huang [1]). Hypothesis H f (ii) implies that k = 1 and so u = v.
As we already remarked, we are going to show that hypotheses H f (iii) and (iv) are also necessary for the uniqueness of positive solutions for problem (1.1).