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 ∞ (Ω) 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 ∈ Ω, 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 ∂Ω. 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 (∂Ω), with 0 < α < 1 and β(z) 0 for all z ∈ ∂Ω.
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

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".
Then c m ρ and c is a critical value of ϕ.
Hence G(·) is the primitive of a(·) and so by a well-known property of convex functions, we have The following lemma is an easy consequence of hypotheses H(a) and summarizes the main properties of a(·) (see Papageorgiou and Rȃdulescu [20]).
Then this lemma and (3) lead to the following growth restrictions for the primitive G(·).
Next, we present some characteristic examples of differential operators which fit in the framework provided by hypotheses H(a) (see Papageorgiou and Rȃdulescu [20]).
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 (Ω) given by The Banach space C 1 (Ω) is an ordered Banach space with positive (order) cone given by This cone has a nonempty interior containing the set On ∂Ω 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 (∂Ω) (for 1 q ∞). From the theory of Sobolev spaces 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 γ 0 (u) = u| ∂Ω for all u ∈ W 1,p (Ω) ∩ C(Ω).
Hence the trace map assigns boundary values to any Sobolev function. The trace map is compact into L q (∂Ω) for all q ∈ 1, p(N − 1) N − p if p < N and for all q ∈ [1, +∞) if p N . Also, we have In what follows, for the sake of notational simplicity, we drop the use of the map γ 0 . All restrictions of Sobolev functions on ∂Ω are understood in the sense of traces.
f 0 (z, s)ds and consider the C 1 -functional ψ 0 : W 1,p (Ω) → R defined by The following result is due to Papageorgiou and Rȃdulescu [22] and is an outgrowth of the nonlinear regularity theory of Lieberman [15].
Proposition 2.5. Assume that hypotheses H(a)(i), (ii), (iii), H(ξ), H(β) hold and u 0 ∈ W 1,p (Ω) is a local C 1 (Ω)-minimizer of ψ 0 , that is, there exists ρ 0 > 0 such that Then u 0 ∈ C 1,η (Ω) for some η ∈ (0, 1) and u 0 is also a local W 1,p (Ω)-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.
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 satisfŷ λ 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 have , 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 (Y 1 , Y 2 ) be a topological pair such that For ϕ ∈ C 1 (X, R) and c ∈ R we introduce the following sets: 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, Motreanu and Papageorgiou [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 Q(t) = k∈N0 β k t k is a formal series in t ∈ R with nonnegative integer coefficients.
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).
and δ 0 > 0 such that Remark 3.1. 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 Evidently, we can find ϑ 0 > 0 such that (9) f (z, x)x η 0 |x| q − ϑ 0 |x| p for almost all z ∈ Ω, and all |x| η.
Then we define Note that µ(z, x) is a Carathéodory function and for all z ∈ Ω, µ(z, ·) is odd. We consider the following auxiliary Robin problem Proof. We first show the existence of a positive solution. So, let ϑ > 0 be such that We introduce the following Carathéodory function We setM + (z, x) = x 0μ + (z, s)ds and consider the C 1 -functional ψ + : W 1,p (Ω) → R defined by Corollary 2.3, hypothesis H(β) and (12) imply that ψ + is coercive.
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 (Ω) such that (13) ψ Hypothesis H(a)(iv) implies that we can find c 7 > 0 and δ ∈ (0, δ 0 ] such that (14) G(y) c 7 |y| τ for all y ∈ R N with |y| δ .
Suppose thatũ,û ∈ W 1,p (Ω) are two positive solutions of the auxiliary problem (11). From the first part of the proof we have Therefore for every h ∈ C 1 (Ω) and for |t| small, we havẽ u + th ∈ dom j andû + th ∈ dom j.
Because of the convexity of j(·), we see that j(·) is Gâteaux differentiable atũ τ and atû τ in the direction h. Using the chain rule and the nonlinear Green's identity (see Gasinski and Papageorgiou [9,p. 210]), we get Recall that j(·) is convex, hence j ′ (·) is monotone. Hence we have (10) and (18)).
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 ∈ Ω, and all 0 x η.
Therefore we have proved that ∅ = S + ⊆ D + . For negative solutions we consider the Carathéodory function − (z, s)ds and consider the C 1 -functionalφ − : for all u ∈ W 1,p (Ω).
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. Proof. Let u ∈ S + and consider the following Carathéodory function We set Γ + (z, x) = Evidently,k + is coercive (see Lemma 2.2, (26) and recall thatθ > 0). Also, it is sequentially weakly lower semicontinuous. So, we can findû ∈ W 1,p (Ω) such that (27)k + (û) = inf{k + (u) : u ∈ W 1,p (Ω)}.

Similarly we show that
v ṽ for all v ∈ S − . 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, Rȃdulescu and Repovš [24] 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 3.1 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 3.3).
We have for all h ∈ W 1,p (Ω), n ∈ N.
As before, via the nonlinear maximum principle, we havē Similarly we producev The proof is now complete.

Existence of 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 − andū + means that this solution y 0 is necessarily nodal. Proof. Letv − ∈ −D + andū + ∈ D + be the two extremal constant sign solutions produced in Proposition 3.4 and let ϑ > ||ξ|| ∞ . We introduce the following Carathéodory function We also consider the positive and negative truncations of ℓ(z, ·), that is, the Carathéodory functions We set Consider the C 1 -functionalsφ,φ ± : W 1,p (Ω) → R defined bỹ Let u ∈ Kφ. We have for all h ∈ W 1,p (Ω).
So, we can formulate the following multiplicity theorem for problem (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 (41) −∆u(z) + ξ(z)u(z) = f (z, u(z)) in Ω, ∂u ∂n + β(z)u = 0 on ∂Ω.
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 4.2 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: • H(ξ) ′ : ξ ∈ L s (Ω) with s > N, ξ + ∈ L ∞ (Ω); • H(β) ′ : β ∈ W 1,∞ (Ω), β(z) 0 for all z ∈ ∂Ω.
Remark 5.1. Again we can have β ≡ 0, which corresponds to the Neumann problem.
We have the following multiplicity theorem for problem (41).
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 (see Papageorgiou and Rȃdulescu [19]) Note that hypotheses H(f ) 2 (i), (ii) imply that |f (z, x)| c 12 |x| for almost all z ∈ Ω, all x ∈ [−η, η], some c 12 > 0.
We introduce the following Carathéodory function and then consider the following auxiliary Robin problem (44) −∆u(z) + ξ(z)u(z) = µ(z, u(z)) in Ω, ∂u ∂n + β(z)u = 0 on ∂Ω (see (10) and (11)  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 have Let t * > 0 be the biggest real number such that (46) t * û ũ.
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 3.2, we show that Similarly, using the Carathéodory function we produce a negative solution v 0 ∈ [−η, 0] ∩ (−D + ).
In addition, as in the proof of Proposition 3.3, we show that u u for all u ∈ S + and v ṽ for all v ∈ S − .
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).
Remark 6.1. The symmetric condition on f (z, ·), permits the relaxation of the condition near zero (compare H(f ) 3 (ii) with H(f ) 1 (ii)). We have also dropped hypothesis H(f ) 3 (iii).
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 ∈ Ω, and all x ∈ R.
Then we introduce the following Carathéodory function Using this µ(·, ·), we consider the auxiliary Robin problem (11). As in Proposition 3.1 we show that the auxiliary problem has a unique positive solutionũ ∈ [0, η] ∩ D + and due to the oddness of the equation,ṽ = −ũ ∈ [−η, 0] ∩ (−D + ) is the unique negative solution of the auxiliary problem.
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 3.3).
This leads to the existence of extremal constant sign solutions u + ∈ D + andv − ∈ −D + .