Concave-Convex Problems for the Robin p-Laplacian Plus an Indefinite Potential

We consider nonlinear Robin problems driven by the p-Laplacian plus an indefinite potential. In the reaction, we have the competing effects of a parametric concave (that is, ( p − 1 ) -sublinear) term and of a convex (that is, ( p − 1 ) -superlinear) term which need not satisfy the Ambrosetti–Rabinowitz condition. We prove a "bifurcation-type" theorem describing in a precise way the dependence the dependence of the set of positive solutions on the parameter λ > 0 . In addition, we show the existence of a smallest positive solution u λ * and determine the monotonicity and continuity properties of the map λ ↦ u λ * .

The potential function ξ ∈ L ∞ (Ω) and in general it is sign changing. Thus, the left-hand side of (1) (the differential operator plus the potential term) is not coercive. On the right-hand side of the problem (the reaction), we have the competing effects of two nonlinearities. One is the parametric term λg(z, x) with g(·, ·) being a Carathéodory function (that is, for all x ∈ R, z → g(z, x) is measurable and for a.a. z ∈ Ω, x → g(z, x) is continuous). The function g(z, ·) is strictly (p − 1)-sublinear near +∞ and concave near the origin (that is, (p − 1)-superlinear near zero). The second nonlinearity f (z, x) is also a Carathéodory function and f (z, ·) is (p − 1)-superlinear near +∞. The superlinearity of f (z, ·) is not expressed using the well-known Ambrosetti-Rabinowitz condition (the AR-condition for short). Instead, we employ a less restrictive condition which incorporates in our framework a larger class of (p − 1)-superlinear functions. In the boundary condition, ∂u ∂n p denotes the conormal derivative of u corresponding to the p-Laplacian and it is interpreted via the nonlinear Green's identity (see , p. 211).
In particular, ∂u ∂n p = |Du| p−2 (Du, n) R N , for all u ∈ C 1 (Ω), with n(·) being the outward unit normal on ∂Ω. The boundary coefficient β(·) is nonnegative and can be identically zero (Neumann problem). From the above description, we see that problem (1) is a generalized concave-convex nonlinear Robin problem. We look for positive solutions of (1) and our goal is to describe in detail the changes in the set of positive solutions of (1) as the parameter λ > 0 varies in the open positive semiaxis R + = (0, +∞).
Thus, we prove a "bifurcation-type" theorem, according to which there exists a critical parameter λ * > 0 such that • for all λ ∈ (0, λ * ) problem (1) has at least two positive solutions; • for λ = λ * problem (1) has at least one positive solution; • for all λ > λ * there are no positive solutions for problem (1).
In addition, we show that, for every admissible parameter λ ∈ L = (0, λ * ], problem (1) has a smallest positive solution u * λ , and we also examine the monotonicity and continuity properties of the map L λ → u * λ . Our work here extends to nonlinear problems driven by the p-Laplacian the recent semilinear work of Papageorgiou-Rȃdulescu-Repovš [2]. An inspection of their method of proof reveals that it is heavily dependent on the fact that the Sobolev space H 1 (Ω) can be written as the orthogonal direct sum of the eigenspaces of −∆ + ξ(z)I with Robin boundary condition. No such decomposition is available for W 1,p (Ω) in the case of the p-Laplacian. Moreover, in the semilinear case strong comparison principles are an easy consequence of the Hopf boundary point theorem. In contrast, in the nonlinear case, it is much more difficult to come up with strong comparison principles and stronger conditions are needed. We point out that our conditions on the two nonlinearities g(z, x) and f (z, x), are in general less restrictive than those in [2]. Finally, we mention the recent work of Marano-Marino-Papageorgiou [3] on Dirichlet problems with no potential term and a more restrictive reaction.
The study of parametric concave-convex problem (as they are usually called in the literature problems exhibiting the competing effects on sublinear and superlinear nonlinearities), started with the seminal paper of Ambrosetti-Brezis-Cerami [4], which deals with semilinear Dirichlet problems driven by the Laplacian. Their work was extended to equations driven by the Dirichlet p-Laplacian by García Azorero-Manfredi-Peral Alonso [5] and Guo-Zhang [6]. In these works, the reaction has the particular form Here, p * denotes the critical Sobolev exponent corresponding to p, that is, More general reactions can be found in the works of Hu-Papageorgiou [7] and Marano-Papageorgiou [8]. In all these works, ξ ≡ 0 and the differential operator is coercive. Finally, we mention the recent work of Papageorgiou-Scapellato [9] dealing with a similar parametric Robin problem in which the reaction term is (p − 1)-linear and exhibits an asymmetric behavior as x → ±∞. The authors produce nodal solutions (see also [10]).
It is worth pointing out that many other researchers studied differential problems close to those considered in this paper. DiBenedetto-Gianazza-Vespri [11] studied the local boundedness and the local Hölder continuity of local weak solutions to anisotropic p-Laplacian type equations. Drábek-Hernández [12] studied quasilinear eigenvalue problems with singular weights driven by the p-Laplacian. Drábek-Ho-Sarkar considered an eigenvalue problem involving the weighted p-Laplacian in radially symmetric domains [13] and the Fredholm alternative for the p-Laplacian in exterior domains [14]. , using the fibrering method, proved the existence of multiple positive solutions to quasilinear problems of second order driven by the p-Laplacian and also proved nonexistence results. Jebelean-Mawhin-Şerban [16] considered a system of difference equations with periodic nonlinearities and applying a modification argument to a suitable problem with a left-hand member of p-Laplacian type and using Morse theory, obtained multiple periodic solutions. Finally, we mention the work of Manásevich-Mawhin [17] that deals with the spectrum of p-Laplacian-type systems with certain boundary conditions.

Mathematical Background-Hypotheses
In the study of problem (1), there are three main spaces that we will use: the Sobolev space W 1,p (Ω), the Banach space C 1 (Ω), and the boundary Lebesgue spaces L q (∂Ω) ( The symbol · denotes the norm of W 1,p (Ω), defined by The Banach space C 1 (Ω) is ordered by the positive (order) cone This cone has a nonempty interior given by int C + = u ∈ C + : u(z) > 0 for all z ∈ Ω .
We will also use another open cone in C 1 (Ω), namely On ∂Ω, we consider the (N − 1)-dimensional Hausdorff (surface) measure σ(·). Having this measure on ∂Ω, we can define in the usual way the boundary Lebesgue spaces L q (∂Ω) (1 ≤ q ≤ ∞). It is known that there exists a unique continuous linear map γ 0 : W 1,p (Ω) → L p (∂Ω), called the trace map, such that We know that In the sequel for notational economy, we drop the use of γ 0 . All restrictions of Sobolev functions on ∂Ω, are understood in the sense of traces.
In addition, by int C 1 (Ω) [v, u], we denote the interior in the C 1 (Ω)−norm topology of [v, u] ∩ C 1 (Ω). Moreover, for any u ∈ W 1,p (Ω), we define For every x ∈ R, we set x ± = max{±x, 0} and then for any u ∈ W 1,p (Ω), we define u ± (·) = u(·) ± . We know that Given X a Banach space, by X * , we denote its topological dual and with the symbol ·, · we denote the duality brackets for the pair (X * , X). If ϕ ∈ C 1 (X, R), then we say that ϕ(·) satisfies the Cerami condition (the C-condition for short), if the following property holds: admits a strongly convergent subsequence.
By K ϕ , we denote the critical set of ϕ, that is, In addition, a map A : X → X * is said to be an (S) + -map if it has the following property: u n w − → u in X and lim sup n→∞ A(u n ), u n − u ≤ 0 ⇒ u n → u in X (see [18], p. 203) Let A : W 1,p (Ω) → W 1,p (Ω) * be defined by A(u), h = Ω |Du| p−2 (Du, Dh) R N dz for all u, h ∈ W 1,p (Ω). Proposition 1. The nonlinear map A : W 1,p (Ω) → W 1,p (Ω) * defined above is bounded (that is, it maps bounded sets to bounded sets), continuous, monotone (hence maximal monotone too) and of type (S) + . Now, we introduce our hypotheses on the data of problem (1).

Remark 1.
We see that ξ(·) is not nonnegative. This makes the left-hand side of (1) noncoercive, a feature of the problem that makes its analysis more difficult.
Remark 2. When β ≡ 0, we recover the Neumann problem. The regularity requirements on β(·) will be used to have global regularity results for the produced solutions.

Remark 3.
Since we look for positive solutions and the above hypotheses concern the positive semiaxis R + = [0, +∞), without any loss of generality, we may assume that g(z, x) = 0 for a.a. z ∈ Ω, all x ≤ 0.

Remark 4.
As we did for the nonlinearity g(z, ·), we may assume that From hypotheses H( f ) (ii),(iii), we infer that Thus, the perturbation term f (z, ·) is (p − 1)-superlinear. In the literature, equations having a (p − 1)-superlinear reaction are usually treated using the AR-condition, which says that there exist η > p and M > 0 such that (see Ambrosetti-Rabinowitz [19]). Actually, this is a unilateral version of the AR-condition since we have assumed (2) and (3). Integrating (4a) and using (4b), we obtain a weaker condition, namely that Therefore, the AR-condition implies that f (z, ·) has at least (η − 1)-polynomial growth. The AR-condition although convenient in the verification of the C-condition for the energy (Euler) functional of the problem, it is rather restrictive and excludes from consideration superlinear nonlinearities with slower growth near +∞ (see the Examples below). Here, instead, we use the less restrictive condition H( f ) (iii). Note that, if f (z, ·) satisfies the AR-condition, then we may assume that η > max 1, (r − p) N p and we have Thus, the behavior of f (z, ·) both near +∞ and near 0 + is complementary to that of g(z, ·).
The following pairs of function satisfy hypotheses H(g), H( f ), H 0 . For the sake of simplicity, we drop the z-dependence: Note that the pair (g 1 , f 1 ) is the classical "concave-convex" pair with f 1 satisfying the AR-condition. On the other hand, f 2 does not satisfy the AR-condition.

Positive Solutions
We introduce the following sets L = {λ > 0 : problem (1) admits a positive solution} , (set of admissible parameters), S λ = set of positive solutions of (1).
Thus, we have proved that S λ ⊆ intC + for all λ > 0.

Nonemptiness of L
In this subsection, we show that L = ∅; namely, we show that there exist admissible parameters. In what follows by γ p : Proof. Let η > ξ ∞ and consider the C 1 -functional ϕ λ : Note that, by appropriately modifying c 2 > 0 and c 3 > 0 if necessary, we may assume that (recall that ϑ < p < r). Since q < ϑ < p < r, given > 0, we can find c 6 = c 6 ( ) > 0 such that Moreover, hypotheses H(g) imply that given > 0, we can find c 7 = c 7 ( ) > 0 such that Then, for every u ∈ W 1,p (Ω), we have We consider the function Then, we have It follows that Since ∈ 0, c 4 λ and δ > 0 are arbitrary, we can choose both small so that (10)).
Hypotheses H( f ) (i), (iii) imply that we can find c 13 , c 14 > 0 such that In addition, from hypotheses H(g), we see that we can find c 15 , c 16 > 0 such that In (19), we use (20) and (21). Since q < τ, we obtain u n τ τ ≤ c 17 1 + u n q τ for some c 17 > 0, all n ∈ N, From hypothesis H( f ) (iii), it is clear that we can have τ < r < p * . Then, let t ∈ (0, 1) be such that Invoking the interpolation inequality (see Papageorgiou-Winkert [23], p. 116), we have u n r ≤ u n 1−t τ u n t p * ⇒ u n r r ≤ c 18 u n tr for some c 18 > 0, all n ∈ N (see (22)).
Hypotheses H(g), H( f ) (i) imply that Recall that By assuming that r > p is close to p * (see hypothesis H( f ) (i)), we can always have that τ ≥ p. Then, from (22), we infer that Hence, with η > ξ ∞ , we have  (24)).
By passing to a suitable subsequence if necessary, we may assume that u n w − → u in W 1,p (Ω) and u n → u in L r (Ω) and in L p (∂Ω).

Structural Properties of L
In this subsection, we show that L is an interval and establish a kind of monotonicity property for the solution multifunction λ → S λ . Proposition 3. If hypotheses H(ξ), H(β), H(g), H( f ), H 0 hold, λ ∈ L and 0 < µ < λ, then µ ∈ L .
Moreover, using the Sobolev embedding theorem and the compactness of the trace map, we have that ψ µ (·) is sequentially weakly lower semicontinuous.
In addition, from hypothesis H(g) (iii), we have Let λ 1 ∈ R be the first eigenvalue of the operator u → −∆ p u + ξ(z)u with the Robin boundary condition and let u 1 ∈ W 1,p (Ω) be the L p -normalized (that is, u 1 p = 1) eigenfunction corresponding to λ 1 . We know that u 1 has a fixed sign and we can take it to be positive. The nonlinear regularity theory and the nonlinear Hopf boundary point theorem imply that u 1 ∈ int C + . We have γ p ( u 1 ) = λ 1 (for details, we refer to Papageorgiou-Rȃdulescu [24]). Let t ∈ (0, 1) be small such that t u 1 (z) ∈ (0, δ] for a.a. z ∈ Ω. We have (31)).

The Critical Parameter λ *
Here, we show that λ * < +∞, that is, the set of admissible parameters L is a bounded interval and also we show that λ * > 0 is admissible, that is, λ * ∈ L . To this end, first, we prove a weak form of the antimaximum principle for the Robin p-Laplacian plus and indefinite potential (see Godoy-Gossez-Paczka [26]). We start with a lemma which is stated in a more general form than the one that we need in the sequel, but which is of independent interest. Lemma 1. If u, ϑ ∈ W 1,p (Ω) ∩ C 1 (Ω) ∩ L ∞ (Ω), u(z) > 0 and ϑ(z) ≥ 0 for all z ∈ Ω, then we can find c * > 0 independent of u such that Proof. For > 0, let u = u + . We have Using Young's inequality with δ > 0 (see Papageorgiou-Winkert [23], p. 113), we have Returning to (40) and using (41), we obtain Choosing δ ∈ (0, 1), we obtain Note that ϑ p u p−1 ∈ W 1,p (Ω). In addition, u ∈ intC + . Thus, according to Proposition 4.1.22 of Papageorgiou-Rȃdulescu-Repovš [18], we can find µ > 0 such that ϑ ≤ µu . Then, we have In addition, From (42), we have (43)).
Using (56) and (57) and reasoning as in the proof of Proposition 2 (see the proof of the Claim), we show that {u n } n≥1 ⊆ W 1,p (Ω) is bounded.
Thus, we may assume that u n w − → u * in W 1,p (Ω) and u n → u * in L r (Ω) and in L p (∂Ω).
If in the above argument we reverse the roles of v and u, we have and this proves the uniqueness of the solution of (61). Claim: u ≤ u n for all n ∈ N. For n ∈ N, we consider the following Carathéodory function: We set I n (z, x) = x 0 i n (z, s) ds and consider the C 1 -functional From (67) and since η > ξ ∞ , we see that d n (·) is coercive. In addition, it is sequentially weakly lower semicontinuous. Thus, we can find u n ∈ W 1,p (Ω) such that We have d n (u n ) = 0, In (68), we choose h = −u − n ∈ W 1,p (Ω) and then h = (u n − u n ) + ∈ W 1,p (Ω) and we obtain ⇒ u n is a positive solution of (61) (see (67), (69)), Therefore, u ≤ u n for all n ∈ N, and this proves the Claim.
Passing to the limit as n → ∞ in (57) and using (59) and the Claim, we obtain for all h ∈ W 1,p (Ω), Then, (72) and (73) imply From Propositions 3 and 7, we have
We have for all h ∈ W 1,p (Ω), all n ∈ N.
Thus, we may assume that u n w − → u * λ in W 1,p (Ω) and u n → u * λ in L r (Ω) and in L p (∂Ω).
Thus, if in (89) we pass to the limit as n → ∞ and use (92), we obtain Moreover, from the proof of Proposition 7 (see the Claim), we have (in this case, in problem (61) λ 1 is replaced by λ). Then, (93) and (94) imply that u * λ ∈ S λ ⊆ int C + and u * λ = inf S λ .
We examine the properties of the map λ → u * λ .