Multiple solutions for Robin (p, q)-equations plus an indefinite potential and a reaction concave near the origin

We consider a Robin problem driven by the (p, q)-Laplacian plus an indefinite potential term. The reaction is either resonant with respect to the principal eigenvalue or (p-1)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(p-1)$$\end{document}-superlinear but without satisfying the Ambrosetti-Rabinowitz condition. For both cases we show that the problem has at least five nontrivial smooth solutions ordered and with sign information. When q=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=2$$\end{document} (a (p, 2)-equation), we show that we can slightly improve the conclusions of the two multiplicity theorems.


Introduction
Let ⊆ R N be a bounded domain with a C 2 -boundary ∂ . In this paper, we study the following nonlinear, nonhomogeneous Robin problem with 1 < q < p. For every r ∈ (1, ∞), by r we denote the r -Laplace differential operator defined by r u = div |Du| r −2 Du for all u ∈ W 1,r ( ).
In problem (1.1) we have the sum of two such operators. So, the differential operator in (1.1) is not homogeneous and so many of the techniques used in p-Laplacian equations, can not be employed here. Equations driven by the sum of two operators of different nature, arise often in the mathematical models of various physical processes. We mention the works of Bahrouni-Rȃdulescu-Repovš [3] (transonic flow problems), Benci-D'Avenia-Fortunato-Pisani [5] (quantum physics), Cherfils-Il'yasov [7] (reaction-diffusion systems), Zhikov [34] (nonlinear elasticity theory).
In problem (1.1), in addition to the ( p, q)-differential operator there is also a potential term ξ(z)|u| p−2 u, with the potential function ξ ∈ L ∞ ( ) being in general indefinite (that is, sign-changing). This means that the left-hand side of (1.1) is not coercive, an additional difficulty in dealing with problem (1.1).
In the reaction (right-hand side of (1.1)), the function f (z, x) is a Carathéodory function (that is, z → f (z, x) is measurable for all x ∈ R and x → f (z, x) is continuous for a.a. z ∈ ). We consider two different cases concerning the growth of f (z, ·) as x → ±∞. First we assume that f (z, ·) exhibits ( p − 1)-linear growth as x → ±∞ (that is, f (z, ·) is asymptotically ( p − 1)-homogeneous). In this case we permit resonance with respect to the principal eigenvalue of u → − p u+ξ(z)|u| p−2 u with Robin boundary condition. The resonance occurs from the right of the principal eigenvalue λ 1 ( p), in the sense that pF(z, x) − λ 1 ( p)|x| p → +∞ uniformly for a.a. z ∈ , as x → ±∞, with F(z, x) = x 0 f (z, s) ds. This makes the energy (Euler) functional of the problem unbounded from below (hence noncoercive) and so we can not use the direct method of the calculus of variations. In the second case we assume that f (z, ·) is ( p − 1)-superlinear as x → ±∞ but without satisfying the usual in such cases Ambrosetti-Rabinowitz condition (the "AR-condition" for short). In both cases we assume that f (z, ·) is concave near the origin and also has an oscillatory behavior.
Using variational tools based on the critical point theory together with truncation and comparison techniques, we show that in both cases problem (1.1) has at least five nontrivial smooth solutions which are ordered and we provide sign information for all of them.
The starting point of our work here is the recent paper of Papageorgiou-Scapellato [24] where the authors deal with a generalized version of the classical concave-convex problem, for equations driven only by the p-Laplacian. The reaction there is parametric and nonnegative and they prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter λ > 0 varies. Here there is no parameter and for this reason we require that f (z, ·) changes sign. Moreover, in the present work in addition to constant sign solutions, we also produce nodal (that is, sign-changing) solutions.

Mathematical background-hypotheses
In the analysis of problem (1.1), the main spaces are the Sobolev space W 1, p ( ) and the Banach space C 1 ( ). By · we denote the norm of W 1, p ( ) defined by The space C 1 ( ) is an ordered Banach space with positive (order) cone C + = {u ∈ C 1 ( ) : u(z) ≥ 0 for all z ∈ }. This cone has a nonempty interior given by Let r ∈ (1, ∞). By A r : W 1,r ( ) → W 1,r ( ) * we denote the nonlinear operator defined by The next proposition summarizes the main properties of this map (see Gasiński-Papageorgiou [12], p. 279).

Proposition 2.1
The operator A r : W 1,r ( ) → W 1,r ( ) * is bounded (that is, it maps bounded sets to bounded sets), continuous, monotone (hence maximal monotone too) and of type (S) + , that is, it has the following property "If u n w − → u in W 1,r ( ) and lim n→∞ A r (u n ), u n − u ≤ 0, then u n → u in For x ∈ R, we set x ± = max{±x, 0}. Then, for u ∈ W 1, p ( ), we define u ± (z) = u(z) ± for all z ∈ . We know that Also, we define Given a set S ⊆ W 1, p ( ), we say that S is downward (resp. upward) directed, if for every u 1 , u 2 ∈ S we can find u ∈ S such that u ≤ u 1 , u ≤ u 2 (resp. u 1 ≤ u, u 2 ≤ u).
Let X be a Banach space and ϕ ∈ C 1 (X ). By K ϕ we denote the critical set of ϕ, that is, We say that ϕ(·) satisfies the C-condition, if it has the following property "Every sequence {u n } n∈N ⊆ X such that admits a strongly convergent subsequence".
we denote the k th relative singular homology group with integer coefficients for the pair (Y 1 , Y 2 ). Suppose that u ∈ K ϕ is isolated and let c = ϕ(u). Then, the critical groups of ϕ(·) at u, are defined by The excision property of singular homology, implies that this definition is independent of the choice of the isolating neighborhood U.
We mention that λ 1 ( p) is the only eigenvalue with eigenfunctions of fixed sign. All other eigenvalues have nodal (that is, sign-changing) eigenfunctions. Now we can introduce our hypotheses on the data of problem (1.1).

Remarks
We stress thay ξ(·) is in general sign changing. If β ≡ 0, then we have a Neumann problem.
For the resonant case, the hypotheses on the reaction f (z, x) are the following .
(iv) there exist μ ∈ (1, q) and δ > 0 such that (vi) for every ρ > 0, there exists ξ ρ > 0 such that for a.a. z ∈ , the function Remarks If lim x→±∞ uniformly for a.a. z ∈ , then hypothesis H 1 (ii) is satisfied. Therefore we see thay hypothesis H 1 (ii) covers the case of problems which are resonant as x → ±∞ with respect to the principal eigenvalue λ 1 ( p). In the process of the proof, we will show that uniformly for a.a. z ∈ . This means that the resonance occurs from the right of λ 1 ( p) and this makes the energy functional of the problem noncoercive, hence we can not employ the direct method of the calculus of variations. Hypothesis H 1 (iv) implies the presence of a concave term near zero. When there is no potential term (that is, ξ ≡ 0), then hypotheses H 1 (iv), (v) dictate an oscillatory behavior for f (z, ·) near zero. In the general case the oscillatory behavior concerns the function It is satisfied if f (z, ·) is differentiable for a.a. z ∈ and for every ρ > 0, we can find ξ ρ > 0 such that f x (z, x)x 2 ≥ − ξ ρ |x| p for a.a. z ∈ , all |x| ≤ ρ.

Constant sign solutions
In this section we produce multiple constant sign solutions. We start by producing two nontrivial smooth solutions, one positive and the other negative, using only the conditions on f (z, ·) near zero.
Proof First we produce the positive solution.
Clearly we can always assume that ξ ρ > ξ ∞ . We have (3.5) and hypothesis H 1 (vi)) On account of hypothesis H 1 (v) and using Proposition 2.10 of Papageorgiou-Rȃdulescu-Repovš [20], from (3.6) we infer that Suppose that for some z 0 ∈ ∂ we have u 0 (z 0 ) = ϑ + . Then, from [20] we have ∂u 0 ∂n (z 0 ) > 0. On the other hand from the Robin boundary condition, we have a contradiction. Therefore, we conclude that For the negative solution, we introduce the Carathéodory function k − (z, x) defined by Using w − (·), (3.7) and the direct method of the calculus of variations, as before, we produce a negative solution v 0 ∈ W 1, p ( ) for problem (1.1) such that v 0 ∈ −int C + , ϑ − < v 0 (z) < 0 for all z ∈ .
Using these two constant sign solutions and activating the asymptotic hypotheses as x → ±∞ (see H 1 (ii), (iii)), we can produce two more constant sign smooth solutions localized with respect to u 0 and v 0 respectively. So, let u 0 ∈ int C + and v 0 ∈ −int C + be the two solutions from Proposition 3.1. As before, let η > ξ ∞ and consider the Carathéodory functions g + , g − : × R → R defined by Since u 0 (z) < ϑ + and ϑ − < v 0 (z) for all z ∈ (see Proposition 3.1), we can also introduce the following truncations of g ± (z, ·) Both are Carathéodory functions. We set G * Proof We do the proof for ϕ + (·), the proof for ϕ − (·) being similar.
So, we consider a sequence {u n } n∈N ⊆ W 1, p ( ) such that (3.14) From (3.14) we have In (3.15) we choose h = −u − n ∈ W 1, p ( ). Then, from (3.8) we have Next we show that {u + n } ⊆ W 1, p ( ) is bounded too. We argue by contradiction. So, by passing to a subsequence if necessary, we may assume that u + n → +∞ as n → ∞.
for some c 12 > 0, all n ∈ N, (recall that q < τ and see (3.17)). (3.28) Comparing (3.28) and (3.25), we have a contradiction. Hence We may assume that u n w − → u in W 1, p ( ) and u n → u in L p ( ) and in L p (∂ ). (3.29) In (3.15) we choose h = u n − u ∈ W 1, p ( ), pass to the limit as n → ∞ and use (3.29). Then This proves that ϕ + (·) satisfies the C-condition. In a similar fashion, we show that ϕ − (·) satisfies the C-condition.

Proposition 3.3 If hypotheses
Proof For a.a. z ∈ and all x > 0, we have d dx On account of hypothesis H 1 (iii), we can find γ 2 ∈ (0, γ 0 ) and M > 0 such that We return to (3.30) and use (3.31). Then d dx We pass to the limit as x → +∞ and use hypothesis H 1 (ii). Then We infer that (3.32) Then we have Using (3.32) and the fact that τ > q (see hypothesis H 1 (iii)), we infer that In a similar fashion we show that Using (3.8) and (3.9), we can easily check that We may assume that Clearly the functionals ϕ * + and ϕ * − are coercive (see (3.10) and (3.11)) and sequentially weakly lower semicontinuous. Moreover, we can easily check that The Weierstrass-Tonelli theorem, together with (3.34), (3.35) and (3.12) and the fact that Note that we may say that the sets otherwise on account of (3.35), (3.8), (3.9) we already have a sequence of distinct positive solutions bigger than u 0 and a sequence of distinct negative solutions smaller than v 0 and so we are done. From (3.36), (3.37), (3.38) and Theorem 5.7.6, p. 449, of Papageorgiou-Rȃdulescu-Repovš, we see that we can find ρ ∈ (0, 1) small such that Now we are ready to produce two more constant sign solutions for problem (1.1).

Extremal constant sign solutions
In this section we show that problem (1.1) has extremal constant sign solutions, that is, there exist a smallest positive solution and a biggest negative solution. Using these extremal solutions, in the next section, we will be able to produce a nodal (signchanging) solution. Let S + (resp. S − ) be the set of positive (resp. negative) solutions of (1.1). In Sect. 3, we saw that ∅ = S + ⊆ int C + and ∅ = S − ⊆ −int C + . Proof From Papageorgiou-Rȃdulescu-Repovš [19] (see the proof of Proposition 7), we know that S + is downward directed. So, using Lemma 3.10, p. 178, of Hu-Papageorgiou [14], we can find a decreasing sequence {u n } n∈N ⊆ S + such that inf n≥1 u n = inf S + .
We have 0 ≤ u n ≤ u 1 for all n ∈ N. From (4.1) (with h = u n ∈ W 1, p ( )), (4.2) and hypothesis H 1 (i) it follows that {u n } n≥1 ⊆ W 1, p ( ) is bounded. So, we may assume that u n w − → u * in W 1, p ( ) and u n → u * in L p ( ) and in L p (∂ ). (4.3) In (4.1) we choose h = u n − u * ∈ W 1, p ( ), pass to the limit as n → ∞ and use (4.3). Then, as in the proof of Proposition 3.2, we obtain lim sup (4.4) So, passing to the limit as n → ∞ in (4.1) and using (4.4), we have that If we show that u * = 0, then u * ∈ S + and so u * = inf S + . On account of hypotheses H 1 (i), (iv), we have f (z, x) ≥ c 0 |x| μ − c 13 |x| r for a.a. z ∈ , all x ∈ R, with c 13 > 0, r ∈ ( p, p * ). (4.5) The unilateral growth restriction on f (z, ·) leads to the following auxiliary Robin problem From Proposition 12 of Papageorgiou-Vetro-Vetro [27], we know that this problem has a unique positive solution u ∈ int C + and since the equation is odd, v = − u ∈ −int C + is the unique negative solution of (4.6).
Similarly we show that v ≤ v for all v ∈ S − . This proves the Claim.
Similarly, we prove the existence of v * ∈ S − and v ≤ v * for all v ∈ S − . We point out that now S − is upward directed.

Nodal solutions
In this section, using the extremal constant sign solutions u * ∈ int C + and v * ∈ −int C + produced in Proposition 4.1, we show the existence of a nodal (sign-changing) solution. The idea is simple. We focus on the order interval [v * , u * ] and using a combination of tools and techniques we show that the problem has a solution in [v * , u * ] distinct from 0, u * , v * . Then, on account of the extremality of u * , v * , such a solution will be nodal.
So, summarizing the situation for the resonant case, we can state the following multiplicity theorem for problem (1.1).
We argue as in the resonant case. In this case, Proposition 3.2 (the C-condition for the functionals ϕ + , ϕ − ), is proved as in the Claim in the proof of Proposition 2 in Papageorgiou-Scapellato [24]. Moreover, in this case Proposition 3.3, is an immediate consequence of hypothesis H 1 (ii). Since the conditions near zero are the same, the rest of the results (Proposition 3.4 and those in Sect. 4), remain valid and so finally we can state the following multiplicity theorem for the superlinear problem.
Remark We stress that in both Theorems 5.2 and 6.1 the solutions are ordered.

The (p, 2)-equation
In this section we show that when q = 2, we can slightly improve the two multiplicity theorems (Theorem 5.2 and Theorem 6.1). Our work in this section is also related to the recent paper of Vetro [33] on semilinear equations driven by the Robin Laplacian plus an indefinite potential. The author proves a multiplicity result using the reduction technique of Amann [1].
Evidently div a(Du) = p u + u for all u ∈ W 1, p ( ).
In a similar fashion we also have v(z) < v 0 (z) for all z ∈ , v 0 (z) < y 0 (z) < u 0 (z) for all z ∈ . Therefore for problem (7.1) we can have the following slightly improved version of Theorems 5.2 and 6.1.