Nodal solutions for the Robin $p$-Laplacian plus an indefinite potential and a general reaction term

We consider a nonlinear Robin problem driven by the $p$-Laplacian plus an indefinite potential. The reaction term is of arbitrary growth and only conditions near zero are imposed. Using critical point theory together with suitable truncation and perturbation techniques and comparison principles, we show that the problem admits a sequence of distinct smooth nodal solutions converging to zero in $C^1(\overline{\Omega})$.

The potential function ξ ∈ L ∞ (Ω) is indefinite (that is, sign changing) and the reaction term f (z, x) is a Carathéodory function (that is, for all x ∈ R, the mapping z → f (z, x) is measurable and for almost all z ∈ Ω, x → f (z, x) is continuous). We do not impose any global polynomial growth condition on f (z, ·). All the conditions on f (z, ·) concern its behaviour near zero. In the boundary condition, ∂u ∂np denotes the generalized normal derivative defined by extension of the map with n(·) being the outward unit normal on ∂Ω. The boundary coefficient β ∈ C 0,α (∂Ω) (with 0 < α < 1) satisfies β(z) ≥ 0 for all z ∈ ∂Ω. When β = 0, we have the usual Neumann problem. Using variational methods, together with suitable truncation and perturbation techniques and comparison principles, and an abstract result of Kajikiya [5], we show that the problem admits an infinity of smooth nodal (that is, sign changing) solutions converging to zero in C 1 (Ω). Our starting point is the recent work of Papageorgiou and Rȃdulescu [9], where the authors produced an infinity of nodal solutions for a nonlinear Robin problem with zero potential (that is, ξ ≡ 0) and a reaction term of arbitrary growth. They assumed that the reaction term f (z, x) is a Carathéodory function and there exists η > 0 such that for almost all z ∈ Ω, f (z, ·)| [−η,η] is odd and f (z, η) ≤ 0 ≤ f (z, −η) (the second inequality follows from the first inequality and the oddness of f (z, ·)). Moreover, they assumed that for almost all z ∈ Ω, f (z, ·) exhibits a concave (that is, a strictly (p − 1)-superlinear) term near zero. So, f (z, ·) has zeros of constant sign and it presents a kind of oscillatory behaviour near zero. In the present work we introduce in the equation an indefinite potential term ξ(z)|x| p−2 x and we remove the requirement that f (z, η) ≤ 0 for almost all z ∈ Ω. We point out that this was a very convenient hypothesis, since the constant functionũ ≡ η > 0 provided an upper solution for the problem andṽ = −η < 0 a lower solution. With them, the analysis of problem (1) was significantly simplified. The absence of this condition in the present work, changes the geometry of the problem and so we need a different approach. We should mention that in Papageorgiou and Rȃdulescu [9], the differential operator is more general and is nonhomogeneous. It is an interesting open problem whether our present work can be extended to equations driven by nonhomogeneous differential operators, as in [9].
Wang [13] was the first to produce an infinity of solutions for problems with a reaction of arbitrary growth. He used cut-off techniques to study semilinear Dirichlet problems with zero potential driven by the Laplacian. More recently, Li and Wang [6] produced infinitely many nodal solutions for semilinear Schrödinger equations. We also refer to our recent papers [11,12], which deal with the qualitative analysis of nonlinear Robin problems.
We denote by · the norm of the Sobolev space W 1,p (Ω) defined by u = u p p + Du p p 1 /p for all u ∈ W 1,p (Ω).
The Banach space C 1 (Ω) is an ordered Banach space with positive (order) cone This cone contains the open set D + = {u ∈ C + : u(z) > 0 for all z ∈ Ω}.
Also, letD + ⊆ C + be defined bŷ Evidently,D + ⊆ C 1 (Ω) is open and D + ⊆D + . 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 s (∂Ω), 1 ≤ s ≤ ∞. 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 an the "trace operator", such that So, the trace operator assigns "boundary values" to every Sobolev function. The trace operator is compact into L s (∂Ω) for all s ∈ 1, In the sequel, for the sake of notational simplicity, we will drop the use of operator γ 0 . All restrictions of Sobolev functions on ∂Ω, are understood in the sense of traces.
Given h 1 , h 2 ∈ L ∞ (Ω), we write that h 1 ≺ h 2 if and only if for every compact set K ⊆ Ω, we can find = (K) > 0 such that We see that, if h 1 , h 2 ∈ C(Ω) and h 1 (z) < h 2 (z) for all z ∈ Ω, then h 1 ≺ h 2 . The next strong comparison theorem can be found in Fragnelli, Mugnai and Papageorgiou [3].
As we have already mentioned in the introduction, the sequence of nodal solutions will be generated by using an abstract result of Kajikiya [5], which is essentially an extension of the symmetric mountain pass theorem (see also Wang [13]). Recall that, if X is a Banach space and ϕ ∈ C 1 (X, R), we say that ϕ satisfies the "Palais-Smale condition" ("PS-condition", for short), if the following holds: admits a strongly convergent subsequence".
Theorem 2.1. Let X be a Banach space and suppose that ϕ ∈ C 1 (X, R) satisfies the P S-condition, is even and bounded below, ϕ(0) = 0, and for every n ∈ N there exist an n-dimensional subspace V n of X and ρ n > 0 such that Then there exists a sequence {u n } n≥1 ⊆ X such that In what follows, we denote by A : It is well-known (see, for example, Gasinski and Papageorgiou [4]), that A(·) is monotone continuous and of type (S) For x ∈ R, we set x ± = max{±x, 0}. Then, given u ∈ W 1,p (Ω), we can define is the critical set of ϕ.
3. Infinitely many nodal solutions. In this section we prove our main result, namely the existence of a whole sequence of distinct nodal solutions {u n } n≥1 which converge to zero in C 1 (Ω).
The uniqueness of this positive solution of problem (4) follows from Theorem 1 of Diaz and Saa [1].
Since problem (4) is odd (note that k(z, ·) is odd, see (3)), it follows that is the unique negative solution of (4).
Using the two constant sign solutions of problem (4) produced by Proposition 2, we introduce the following truncation-perturbation of the reaction term f (z, ·) (recall thatξ 0 > ||ξ|| ∞ ) This is a Carathéodory function. We also consider the positive and negative truncations off (z, ·), that is, the Carathéodory functionŝ for all u ∈ W 1,p (Ω).
In a similar fashion we show that v ≤ṽ for all v ∈ Kφ − . Proposition 6. If hypotheses H(ξ), H(β), H(f ) hold and V ⊆ W 1,p (Ω) is a nontrivial finite dimensional subspace, then we can find ρ V > 0 such that