Nonlinear, Nonhomogeneous Robin Problems with Indefinite Potential and General Reaction

We consider a nonlinear elliptic equation driven by a nonhomogeneous differential operator plus an indefinite potential. On the reaction term we impose conditions only near zero. Using variational methods, together with truncation and perturbation techniques and critical groups, we produce three nontrivial solutions with sign information. In the semilinear case we improve this result by obtaining a second nodal solution for a total of four nontrivial solutions. Finally, under a symmetry condition on the reaction term, we generate a whole sequence of distinct nodal solutions.

In this problem, the map a : R N → R N involved in the differential operator is a continuous, strictly monotone (thus maximal monotone operator, too) map which satisfies certain other regularity and growth conditions listed in hypotheses H (a) below. These conditions are general enough to generate a broad framework that incorporates many differential operators of interest, such as the p-Laplacian and the ( p, q)-Laplacian (that is, the sum of a p-Laplacian and a q-Laplacian, with 1 < q < p < ∞). Note that in general, the differential operator u → div a(Du) is not homogeneous. The potential function ξ(·) ∈ L ∞ ( ) and in general, ξ(·) is nodal (that is, sign changing). So, the left-hand side of problem (1) needs not be coercive. The reaction term f (z, x) is a Carathéodory function (that is, for all x ∈ R, the mapping z → f (z, x) is measurable, while for almost all z ∈ , the mapping x → f (z, x) is continuous). The special feature of our paper is that no global growth condition is imposed on f (z, ·). The only conditions imposed on f (z, ·) concern its behavior near zero and that f (z, ·) must be locally L ∞ -bounded. In the boundary condition, ∂u ∂n a denotes the generalized normal derivative corresponding to the map a(·). It is defined by extension of the map with n(·) being the outward unit normal on ∂ . This kind of conormal derivative is dictated by the nonlinear Green identity (see Gasinski and Papageorgiou [9,p. 210]) and was also used by Lieberman [15] in his nonlinear regularity theory. The boundary coefficient is β ∈ C 0,α (∂ ), with 0 < α < 1 and β(z) 0 for all z ∈ ∂ . The aim of the present paper is to prove a multiplicity theorem for such equations, providing sign information for all solutions produced. Wang [32] was the first to study elliptic problems with a general reaction term of arbitrary growth. The equation studied by Wang [32] was a nonlinear problem driven by Dirichlet p-Laplacian with zero potential. Using cut-off techniques and imposing a symmetry condition on f (z, ·) (that is, assuming that f (z, ·) is odd), Wang [32] produced an infinity of nontrivial solutions. More recently, Li and Wang [14], using similar tools, improved this result by producing an infinity of nodal solutions for semilinear Schrödinger equations. Their result was extended by Papageorgiou and Rȃdulescu [21] who considered nonlinear, nonhomogeneous Robin problems with zero potential (that is, ξ ≡ 0). Assuming that the reaction term f (z, ·) has zeros of constant sign and that it is odd, they produced an infinity of smooth nodal solutions. We also mention the recent work of Papageorgiou and Winkert [23], who considered a reaction term of general growth and with zeros. Under stronger conditions on the map a(·) and with zero potential, they produced constant sign and nodal solutions. We refer to Pucci et al. [2,4] for eigenvalue problems associated to p-Laplacian type operators. Related results in the framework of problems with unbalanced growth are due to Fiscella and Pucci [8], and Papageorgiou et al. [26]. Finally, we also point out the papers of He et al. [11] on nonlinear, nonhomogeneous Neumann problems with nonnegative potential (that is, ξ 0, ξ ≡ 0), Iturriaga et al. [12] on parametric equations driven by Dirichlet p-Laplacian with zero potential and a reaction with zeros, and Tan and Fang [30] on nonlinear, nonhomogeneous Dirichlet problems using the formalism of Orlicz spaces.

Mathematical Background
Let X be a Banach space and X * its topological dual. By ·, · we denote the duality brackets for the pair (X * , X ). Given ϕ ∈ C 1 (X , R), we say that ϕ satisfies the "Cerami condition" (the C-condition for short), if the following property holds: "Every sequence {u n } n 1 ⊆ X such that {ϕ(u n )} n 1 ⊆ R is bounded and (1 + ||u n ||)ϕ (u n ) → 0 in X * as n → ∞, admits a strongly convergent subsequence".
This compactness-type condition on the functional ϕ, leads to a deformation theorem from which one can derive the minimax theory for the critical values of ϕ. A fundamental result in this theory is the so-called "mountain pass theorem". Theorem 1 Let X be a Banach space and assume that ϕ ∈ C 1 (X , R) satisfies the C-condition, u 0 , u 1 ∈ X, ||u 1 − u 0 || > ρ > 0, Then c m ρ and c is a critical value of ϕ.
for 1 s < p We introduce the following conditions on the map α(·) (see also Papageorgiou and Rȃdulescu [20,22]): H (a) : a(y) = a 0 (|y|)y for all y ∈ R with a 0 (t) > 0 for all t > 0 and |y| for all y ∈ R N \{0}, and for some c 3 > 0;

Remark 1 Hypotheses H (a)(i), (ii), (iii)
were dictated by the nonlinear regularity theory of Lieberman [15] and the nonlinear maximum principle of Pucci and Serrin [27, pp. 111, 120]. Hypothesis H (a)(iv) serves the needs of our problem. It is a mild condition and it is satisfied in all cases of interest (see the examples below). Evidently, G 0 (·) is strictly convex and strictly increasing. We set G(y) = G 0 (|y|) for all y ∈ R N .
Hence G(·) is the primitive of a(·) and so by a well-known property of convex functions, we have G(y) (a(y), y) R N for all y ∈ R N .
The following lemma is an easy consequence of hypotheses H (a) and summarizes the main properties of a(·) (see Papageorgiou and Rȃdulescu [20]).
The corresponding differential operator is the p-Laplacian defined by The corresponding differential operator is the ( p, q)-Laplacian defined by Such operators arise in problems of mathematical physics and recently there have been some existence and multiplicity results for equations driven by such operators, see Cingolani and Degiovanni [3], Mugnai and Papageorgiou [17], Papageorgiou and Rȃdulescu [18], Papageorgiou et al. [24], Sun [28], and Sun et al. [29].
The corresponding differential operator is the generalized p-mean curvature differential operator defined by The corresponding differential operator is defined by Such operators arise in problems of plasticity.

Now let
The following spaces will be used in the analysis of problem (1): We denote by || · || the norm of W 1, p ( ) given by The Banach space C 1 ( ) is an ordered Banach space with positive (order) cone given by This cone has a nonempty interior containing the set On ∂ we consider the (N − 1)-dimensional Hausdorff (surface) measure σ (·). Using this measure, we can define in the usual way the "boundary" Lebesgue spaces L q (∂ ) (for 1 q ∞). From the theory of Sobolev spaces we know that there exists a unique continuous linear map γ 0 : W 1, p ( ) → L p (∂ ) known as the "trace map", such that Hence the trace map assigns boundary values to any Sobolev function. The trace map is compact into L q (∂ ) for all q ∈ 1, p(N −1) In what follows, for the sake of notational simplicity, we drop the use of the map γ 0 . All restrictions of Sobolev functions on ∂ are understood in the sense of traces.

Remark 2
If β ≡ 0, then we recover the Neumann problem.
Let γ : W 1, p ( ) → R be the C 1 -functional defined by Also, let f 0 : × R → R be a Carathéodory function such that | f 0 (z, x)| a 0 (z)(1 + |x| r −1 ) for almost all z ∈ , and all x ∈ R, The following result is due to Papageorgiou and Rȃdulescu [22] and is an outgrowth of the nonlinear regularity theory of Lieberman [15].

Proposition 5 Assume that hypotheses H (a)(i), (ii), (iii), H (ξ ), H (β) hold and
Then u 0 ∈ C 1,η ( ) for some η ∈ (0, 1) and u 0 is also a local W 1, p ( )-minimizer of ψ 0 , that is, there exists ρ 1 > 0 such that In the special case of semilinear equations (that is, when a(y) = y for all y ∈ R N ), we will be able to improve the multiplicity theorem and produce additional nodal solutions. In this case we can also relax the requirements on the potential function ξ(z) and make use of the spectrum of u → − u + ξ(z)u with Robin boundary condition.
So, we consider the following linear eigenvalue problem Now we assume that We consider the C 1 -functionalγ : From D'Agui et al. [5], we know that there exists μ > 0 such that γ (u) + μ||u|| 2 2 c 6 ||u|| 2 for all u ∈ H 1 ( ), and some c 6 > 0. (5) Using (5) and the spectral theorem for compact self-adjoint operators on a Hilbert space, we show that the spectrumσ (2) of (4) consists of a sequence {λ k } k 1 of distinct eigenvalues which satisfyλ k → +∞ as k → +∞. By E(λ k ) we denote the corresponding eigenspace. We can say the following about these items: (i)λ 1 is simple (that is, dim E(λ 1 ) = 1) and (ii) For every m 2 we havê and it has the "Unique Continuation Property" ("UCP" for short), that is, if u ∈ E(λ k ) vanishes on a set of positive measure, then u ≡ 0 (see de Figueiredo and Gossez [6]).
Finally, let us recall some basic definitions and facts from Morse theory (critical groups), which we will need in the sequel.
With X being a Banach space, let For ϕ ∈ C 1 (X , R) and c ∈ R we introduce the following sets: Suppose that u ∈ K c ϕ is isolated. Then the critical groups of ϕ at u are defined by Here, U is a neighborhood of u such that K ϕ ∩ ϕ c ∩U = {u}. The excision property of singular homology theory implies that the above definition of critical groups is independent of the choice of the isolating neighborhood U .
Suppose that ϕ ∈ C 1 (X , R) satisfies the C-condition and that inf ϕ(K ϕ ) > −∞. Then the critical groups of ϕ at infinity are defined by This definition is independent of the choice of c < inf ϕ(K ϕ ). To see this, let c < inf ϕ(K ϕ ) and without any loss of generality assume that c < c. Then from Motreanu et al. [16,Theorem 5.34, p. 110], we have that Assume that K ϕ is finite. We introduce the following quantities The Morse relation says that where Let H be a Hilbert space, u ∈ H , and U a neighborhood of u. Suppose that ϕ ∈ C 2 (U ). If u ∈ K ϕ , then the "Morse index" m of u is defined to be the supremum of the dimensions of the vector subspaces of H on which ϕ (u) is negative definite. The "nullity" ν of u is the dimension of ker ϕ (u). We say that u ∈ K ϕ is "nondegenerate" if ϕ (u) is invertible (that is, ν = 0). Suppose that ϕ ∈ C 2 (U ) and u ∈ K ϕ is isolated and nondegenerate with Morse index m. Then Here δ k,m denotes the Kronecker symbol, that is,

Solutions of Constant Sign
In this section, we produce solutions of constant sign for problem (1). We assume the following conditions on the reaction term f (z, x).

Remark 3
We see that no global growth condition is imposed on f (z, ·). All our hypotheses on f (z, ·) concern its behaviour near zero. Note that H ( f ) 1 , (ii), (iii) imply a kind of oscillatory behaviour near zero for Evidently, we can find ϑ 0 > 0 such that Then we define Note that μ(z, x) is a Carathéodory function and for all z ∈ , μ(z, ·) is odd. We consider the following auxiliary Robin problem In what follows, given h 1 ,
Also, using the Sobolev embedding theorem and the compactness of the trace map, we see that ψ + is sequentially weakly lower semicontinuous. Invoking the Weierstrass-Tonelli theorem, we can findũ ∈ W 1, p ( ) such that Hypothesis H (a)(iv) implies that we can find c 7 > 0 and δ ∈ (0, δ 0 ] such that Let u ∈ D + and choose small t ∈ (0, 1) such that tu δ 0 . Then we have for some c 8 , c 9 > 0 (see (12), (14) and hypothesis H (β)).
From (17) and Papageorgiou and Rȃdulescu [22], we havẽ Then from the regularity theory of Lieberman [15] we havẽ Hence by the nonlinear strong maximum principle of Pucci and Serrin [27, pp. 111, 120], we haveũ Next, we show that this positive solution is unique. To this end, we introduce the integral functional j : We have y ∈ W 1, p ( ). Using Lemma 1 of Diaz and Saa [7], we have Also since τ p and β 0 (see hypothesis H (β)), it follows that It follows that the integral functional j(·) is convex and, by Fatou's lemma, it is lower semicontinuous.
This proves the uniqueness of the positive solutioñ is the unique negative solution of (11).
Next, applying the nonlinear regularity theory of Lieberman [15], we have Hypotheses H ( f ) 1 (i), (ii) imply that we can find c 11 > 0 such that f (z, x) + c 11 x p−1 0 for almost all x ∈ , and all 0 x η.
Therefore we have proved that ∅ = S + ⊆ D + . For negative solutions we consider the Carathéodory function 0f − (z, s)ds and consider the C 1 -functional Reasoning as above, using this timeφ − and (25), we produce a negative solution The next result provides a lower bound for the elements of S + and an upper bound for the elements of S − . These bounds will lead to the existence of extremal constant sign solutions. H (ξ ), H (β), H ( f ) 1 hold, thenũ u for all u ∈ S + and v ṽ for all v ∈ S − . Proof Let u ∈ S + and consider the following Carathéodory function
The proof is now complete.
We are now ready to produce extremal constant sign solutions for problem (1), that is, a smallest positive solutionū + and a biggest negative solutionv − . In the next section, usingū + andv − we will produce a nodal (sign-changing) solution.
Proof From Papageorgiou et al. [25] we know that • S + is downward directed (that is, if u 1 , u 2 ∈ S + , then we can find u ∈ S + such that u u 1 , u u 2 ).
Then as in the proof of Proposition 6 of Papageorgiou and Rȃdulescu [22], we can find {u n } n 1 ⊆ S + such that inf S + = inf n 1 u n ,ũ u n for all n ∈ N (see Proposition 8).
Similarly we producev The proof is now complete.

Nodal Solutions
In this section, using the extremal constant sign solutionsv − ∈ −D + andū + ∈ D + , we produce a nodal (sign changing) solution. The idea is to use truncation techniques to focus on the order interval [v − ,ū + ]. Using variational tools we obtain a solution y 0 in this order interval, which is distinct from 0,v − ,ū + . The extremality ofv − and u + means that this solution y 0 is necessarily nodal.

Proposition 10 If hypotheses H (a), H (ξ ), H (β), H ( f ) 1 hold, then problem (1) admits a nodal solutions y
Proof Letv − ∈ −D + andū + ∈ D + be the two extremal constant sign solutions produced in Proposition 9 and let ϑ > ||ξ || ∞ . We introduce the following Carathéodory function We also consider the positive and negative truncations of (z, ·), that is, the Carathéodory functions We set Let u ∈ Kφ. We have In a similar fashion we show that The extremality of solutionsū + ∈ D + andv − ∈ −D + implies that This proves Claim 1.

Semilinear Equations
In this section, we introduce a special case of problem (1) in which a(y) = y for all y ∈ R N (that is, the differential operator is the Laplacian, which corresponds to a semilinear equation). So, the problem under consideration is the following In this case we can also relax the conditions on the potential function ξ(·) and allow it to be unbounded. For problem (41) we were able to improve Theorem 11 and produce a second nodal solution for a local of four nontrivial smooth solutions. Now the hypotheses on the data of (41) are the following: Remark 4 Again we can have β ≡ 0, which corresponds to the Neumann problem.

Theorem 12 If hypotheses H (ξ ) , H (β)
, H ( f ) 2 hold, then problem (41) has at least four nontrivial smooth solutions Proof Since we have relaxed the conditions on the potential function ξ(·) and on the boundary coefficient β(·), we need to show how the solutions of problem (1) exhibit the global (that is, up to the boundary) regularity properties claimed by the theorem. So, let u ∈ [−η, η] be a nontrivial solution of (41). Then u(z)) for almost all z ∈ , ∂u ∂n + β(z)u = 0 on ∂ (see Papageorgiou and Rȃdulescu [19]) Then it follows that g 1 , g 2 ∈ L ∞ ( ). We rewrite (42) as follows: Note that g 1 − ξ ∈ L s ( ) s > N (see hypothesis H (ξ ) ). Invoking Lemma 5.1 of Wang [31], we have u ∈ L ∞ ( ). Then the Calderon-Zygmund estimates (see Wang [31,Lemma 5.2]), imply that u ∈ W 2,s ( ), (by the Sobolev embedding theorem). Now the condition near zero (hypothesis H ( f ) 2 (ii)) is different. Here, f (z, ·) is linear near zero, while hypothesis H ( f ) 1 (ii) implies the presence of a concave nonlinearity near zero. So, we need to verify that Theorem 11 remains valid also in the present setting. Note that now, given r > 2, we can find ϑ 0 = ϑ 0 (r ) > 0 such that λ m x 2 − ϑ 0 |x| r for almost all z ∈ , and all |x| η.
We introduce the following Carathéodory function and then consider the following auxiliary Robin problem (see (10) and (11) for the corresponding items in the previous setting). Reasoning as in the proof of Proposition 6, we obtain a unique positive solutioñ is the unique negative solution of (44).
In fact in this case, due to the semilinearity of the problem, we can have an alternative more direct proof of the uniqueness of the positive solution of problem (44). So, suppose thatũ,û are two positive solutions of (44). We havẽ Let t * > 0 be the biggest real number such that Assume that 0 < t * < 1. Evidently, we can findξ η > 0 such that the function (45) and (43)) This contradicts the maximality of t * . Hence t * 1 and sô u ũ (see (46)).
Interchanging the roles ofũ andû in the above argument, we also havẽ u û, ⇒ũ =û. This is an alternative, more direct proof of the uniqueness of positive and negative solutions of problem (44).
As in the proof of Proposition 7, we introduce the Carathéodory function We setF + (z, x) = x 0f + (z, s)ds and consider the C 1 -functionalφ + : Using the direct method of the calculus of variations, we obtain u 0 ∈ H 1 ( ) such that As in the proof of Proposition 7, we show that Similarly, using the Carathéodory function we produce a negative solution Moreover, reasoning as in the proof of Proposition 9, we produce extremal constant sign solution for problem (41) u + ∈ D + andv − ∈ −D + .
As in the proof of Proposition 10, using these two extremal constant sign solutions, we introduce the functionalφ and using it we produce a nodal solution. Note that in (33) we replace ϑ > 0 by μ 0 > 0 and we set p = 2. Claims 1 and 2 in the proof of Proposition 10 remain valid (as before, since m 2, we haveφ + (ũ + ) < 0 =φ + (0) and soũ + = 0). Finally, we apply the mountain pass theorem (see Theorem 1) and obtain Therefore we have  (40) is no longer true. We need to compute the critical groups ofφ at u = 0.
Let λ ∈ (λ m ,λ m+1 ) and consider the C 2 -functional ψ : We consider the homotopy Suppose that we can find {t n } n 1 ⊆ [0, 1] and {u n } n 1 ⊆ H 1 ( ) such that t n → t, u n → 0 in H 1 ( ) and h u (t n , u n ) = 0 for all n ∈ N.
But we have assumed that Kφ is finite (otherwise, on account of the definition ofφ and (33) with μ 0 > 0 replacing ϑ > 0 and p = 2 and using Claim 1 in the proof of Proposition 10, we see that we have an infinity of nodal solutions of (41) and so we are done). Therefore we have a contradiction and this means that (49) cannot happen. Then using the homotopy invariance property of critical groups (see Gasinski and Papageorgiou [10,Theorem 5.125, p. 836]), we have Note that ψ ∈ C 2 (H 1 ( )). Since λ ∈ (λ m ,λ m+1 ), u = 0 is a nondegenerate critical point of ψ. So, from Gasinski and Papageorgiou [10, Theorem 5.106, p. 832], we have (52)).
Moreover, as we did for y 0 , using hypothesis H ( f ) 2 (iii), we show that The proof is now complete.

Infinitely Many Nodal Solutions
In this section we return to problem (1) and by introducing a symmetry condition on f (z, ·), we produce a whole sequence of distinct nodal solutions for problem (1). The hypotheses on the reaction term are the following: 3 : f : × R → R is a Carathéodory function such that for almost all z ∈ , f (z, 0) = 0 and (i) there exist η > 0 and a η ∈ L ∞ ( ) + such that for almost all z ∈ , f (z, ·)| [−η,η] is odd and | f (z, x)| a η (z) for almost all z ∈ , and all |x| η; (ii) with τ ∈ (1, p) as in hypothesis H (a)(iv) we have lim x→0 f (z, x) |x| τ −2 x = +∞ uniformly for almost all z ∈ .

Remark 5
The symmetric condition on f (z, ·), permits the relaxation of the condition We have also dropped hypothesis Fix λ(·) an even continuous function such that ) is a Carathéodory function with the following properties: • for all z ∈ ,f (z, ·) is odd; •f (z, x) = f (z, x) for all z ∈ , |x| c; •f (z, x) = ξ(z)|x| p−2 x for all z ∈ , |x| η.
We consider problem (1) with f replaced byf . Note that given anyη > 0 and r > p, we can find c 14 = c 14 (η, r ) > 0 such that f (z, x)x η|x| τ − c 14 |x| r for almost all z ∈ , and all x ∈ R.
Then we introduce the following Carathéodory function Using this μ(·, ·), we consider the auxiliary Robin problem (11). As in Proposition Recalling that we consider problem (1) with f (z, x) replaced byf (z, x), as before we introduce the following sets: S + = the set of positive solutions of (1) in [0, η], S − = the set of negative solutions of (1) in [−η, 0].
If S + = ∅ and S − = ∅, then we havẽ u u for all u ∈ S + and v ṽ for all v ∈ S − (see Proposition 8).
This leads to the existence of extremal constant sign solutions u + ∈ D + andv − ∈ −D + .
Using these extremal constant sign solutions, we consider the Carathéodory function (z, x) as in (33) with f (z, x) replaced byf (z, x) (see the proof of Proposition 10) and then introduce the C 1 -functionalφ : W 1, p ( ) → R defined bỹ for all u ∈ W 1, p ( ).
This completes the proof.
Now we are ready for the multiplicity result producing a whole sequence of distinct nodal solutions.

Remark 6
Recently, Papageorgiou and Rȃdulescu [21] have proved an analogous result for problems with no potential term (that is, ξ ≡ 0) and with a reaction term with zeros. Theorem 14 generalizes the result of Papageorgiou and Rȃdulescu [21]. It also extends Theorem 2.10 of Wang [32] where the equation is driven by the p-Laplacian with no potential term (that is, ξ ≡ 0). Wang produced a sequence of nontrivial solutions {u n } n 1 , not necessarily nodal, such that ||u n || ∞ → 0.