Ground-state nodal solutions for superlinear perturbations of the Robin eigenvalue problem

We consider a perturbed version of the Robin eigenvalue problem for the p-Laplacian. The perturbation is (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. Using the Nehari manifold method, we show that for all parameters λ<λ^1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\lambda < {\hat{\lambda }}_1$$\end{document} (= the principal eigenvalue of the differential operator), there exists a ground-state nodal solution of the problem.

On the set [Du = 0] of critical points, this operator is degenerate for p > 2 and is singular if 1 < p < 2. The analysis developed in this paper includes the borderline case p = N . In this situation, the Dirichlet energy Ω |Du| N dx is conformally invariant. The borderline case is important in the theory of quasi-conformal mappings.
Problem (P λ ) contains the perturbation u → ξ(z)|u| p−2 u with the potential function ξ ∈ L ∞ (Ω), ξ(z) 0 for a.a. z ∈ Ω. In the reaction (right-hand side of problem (P λ )), we have the combined effects of a parametric term u → λ|u| p−2 u and of a Carathéodory perturbation f (z, x). (That is, for all x ∈ R the mapping z → f (z, x) is measurable and for a.a. z ∈ Ω the function x → f (z, x) is continuous. ) We assume that f (z, x) exhibits (p − 1)-superlinear growth as x → ±∞.
The nonlinear Robin boundary condition in problem (P λ ) is motivated by certain nonlinear patterns in which the flux across the boundary is not linearly proportional to the density function. A typical example is Boltzmann's fourth power law in heat transfer problems, where where h 0 is the surrounding temperature; seeÖzisik [10]. Another example is based on the Michaelis-Menten hypothesis in some biochemical reaction problems where the substrate concentration satisfies the boundary condition see Ross [15]. We are looking for ground-state (that is, least energy) nodal (sign-changing) solutions of problem (P λ ). Using the Nehari manifold method, we show that if λ <λ 1 (hereλ 1 is the principal eigenvalue of the differential operator u → −Δ p u + ξ(z)|u| p−2 u with Robin boundary condition), then problem (P λ ) has a ground-state nodal solution. We prove this result by relaxing the usual Nehari monotonicity hypothesis which is the following: This condition was used by Szulkin and Weth [16] to have uniqueness of the projection on the Nehari manifold. Instead, in the present paper, we assume that the quotient is simply increasing.
In the past, the problem of the existence of ground-state solutions for such parametric problems was investigated only in the context of semilinear Dirichlet problems driven by the Laplace differential operator. We mention the work of Szulkin and Weth [16], who produce a ground-state solution using the stronger monotonicity condition (N), but they do not show that their ground-state solution is nodal. Later, Tang [17] obtained a ground-state solution using the relaxed monotonicity condition, but the ground-state solution need not be nodal. Ground-state nodal solutions under the relaxed monotonicity hypothesis were obtained recently by Lin and Tang [7]. All the aforementioned works deal with semilinear equations (that is, p = 2), and the boundary condition is Dirichlet. Ground-state nodal solutions under the strong Nehari monotonicity condition (see hypothesis (N) above) were obtained by Liu and Dai [8] (Dirichlet problems) and Gasiński and Papageorgiou [4] (problems with a nonlinear boundary condition). In both these works, the reaction is nonparametric and has a different structure. We also mention the work of Papageorgiou, Rȃdulescu and Repovš [12], who studied problem (P λ ) when p = 2 (semilinear equation) looking for positive solutions and proved a bifurcation-type result with critical parameter beinĝ λ 1 . Finally, we point out that eigenvalue problems with nonlinear Robin boundary condition naturally arise in the study of reaction-diffusion equation where a distributed absorption competes with a boundary source; see Lacey et al. [5] for details.
Our main result in this paper is the following theorem. Hypotheses H 0 and H 1 on the data of the problem can be found in Sect. 2. (b) If, in addition, e(z, x) > 0 for a.a. z ∈ Ω, all x = 0, then u * has two nodal domains; here, e(z,

Mathematical preliminaries and hypotheses
The main space in the analysis of problem (P λ ) is the Sobolev space W 1,p (Ω). By · , we denote the norm of W 1,p (Ω) defined by Also, we will use the boundary Lebesgue spaces L p (∂Ω). 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), 1 q +∞. From the theory of Sobolev spaces, we know that there exists a unique continuous linear operatorγ 0 : W 1,p (Ω) → L p (∂Ω), known as the "trace operator," such that So, the trace operator extends the notion of boundary values to all Sobolev functions. We know that this operator is compact into L r (∂Ω) for r < The trace operator is not surjective, and we have We introduce our hypotheses on the potential function ξ(·) and on the boundary coefficient β(·).

Remark 1.
We see that these hypotheses incorporate also the Neumann problem (β ≡ 0).
In what follows, by γ p : W 1,p (Ω) → R we denote the C 1 -functional defined by Using Lemma 4.11 of Mugnai and Papageorgiou [9] and Proposition 2.4 of Gasiński and Papageorgiou [3], we have Another way to see this is via a simple contradiction argument. So, suppose that we could find By homogeneity, we may assume that u n = 1 for all n ∈ N. So, we may assume that u n w − → u in W 1,p (Ω) and u n → u in L p (Ω) and in L p (∂Ω).
Then, in the limit as n → ∞ and since the norm in a Banach space is weakly lowers semicontinuous, we obtain But then we have a contradiction to the fact that u n = 1 for all n ∈ N.
We consider the nonlinear eigenvalue problem We say thatλ ∈ R is an eigenvalue of the operator u → −Δ p u+ξ(z)|u| p−2 u with Robin boundary condition, if problem (2) admits a nontrivial solutionû ∈ W 1,p (Ω), known as an eigenfunction corresponding to the eigenvalueλ.
By using the Lagrange multiplier rule, we see that problem (2) has a smallest eigenvalueλ 1 , which is characterized variationally byλ On account of (1), we see thatλ 1 > 0. Also, this eigenvalue is isolated in the spectrum and simple. The infimum in (3) is realized on the corresponding one dimensional eigenspace and so the eigenfunctions corresponding toλ 1 > 0 have fixed sign. Byû 1 , we denote the positive, L p -normalized (that is, û 1 p = 1) eigenfunction corresponding toλ 1 . The nonlinear regularity theory (see Lieberman [6]) and the nonlinear maximum principle (see Pucci and Serrin [14]) imply thatû 1 ∈ int C + = u ∈ C 1 (Ω) : u(z) > 0 for all z ∈ Ω (the interior of positive (order) cone C + = u ∈ C 1 (Ω) : u(z) 0 for all z ∈ Ω of C 1 (Ω)). The Ljusternik-Schnirelmann minimax scheme implies the existence of a whole strictly increasing sequence λ k k∈N of eigenvalues of problem (2) such thatλ k → +∞, known as "variational eigenvalues." We do not know if they exhaust the spectrum of the operator. We know that ifλ =λ 1 is a nonprincipal eigenvalue, then the corresponding eigenfunctionsû ∈ C 1 (Ω) (regularity theory) are nodal functions. Details can be found in Fragnelli et al. [2].
Let A p : W 1,p (Ω) → W 1,p (Ω) * be the nonlinear map defined by It is well known that this map is bounded (maps bounded sets to bounded sets), continuous, monotone (thus maximal monotone too) and of type (S) + , that is We have for all u, h ∈ W 1,p (Ω). Now, we introduce our hypotheses on the perturbation f (z, x). Recall that if N p (the critical Sobolev exponent for p).
We will prove our existence theorem first using the strong Nehari monotonicity condition (see (N )), and then via approximations of the perturbation, we will establish the result for the general case. For this reason, we introduce the following set of hypotheses: H 1 : f : Ω × R → R is a Carathéodory function such that f (z, 0) = 0 for a.a. z ∈ Ω, hypotheses H 1 (i), (ii), (iii) are the same as the corresponding hypotheses in H 1 and (iv) hypothesis (N ) holds.
If u ∈ W 1,p (Ω), then we define u ± = max {±u, 0} and we have We denote by | · | N the Lebesgue measure on R N . Let ϕ λ : W 1,p (Ω) → R be the energy (Euler) functional defined by Evidently, ϕ λ ∈ C 1 (W 1,p (Ω)). We introduce the following two sets: We see that N 0 ⊆ N . The set N is known as the "Nehari manifold" for the functional ϕ λ (·). Note that every nontrivial solution of problem (P λ ) belongs to the Nehari manifold. Since we look for nodal solutions, we introduce the Nehari submanifold N 0 . Hypotheses H 0 , H 1 imply that ∅ = N 0 ⊆ N (see also Proposition 5 and Papageorgiou et al. [11]).

Ground-state nodal solutions
We look for an element of N 0 which realizes the infimumm 0 λ and which is a critical point of ϕ λ . Such a function will be a ground-state nodal solution of problem (P λ ).

Proposition 2.
If hypotheses H 0 , H 1 hold, then for all τ, t 0 and all u ∈ W 1,p (Ω) we have Using the fact that {u Similarly, we have Finally, we have Let x = 0 and μ 0. Then, Returning to (7) and using (8), we obtain Finally, we use (5), (6) and (9) in (4) and obtain This proof is now complete.
From this proposition, we infer at once the following two useful corollaries.
Using the chain rule, we see that for all t > 0 the following equivalence holds: On account of hypothesis H 1 (iv) = (N ), the integral in the right-hand side of (10) is strictly increasing in t > 0.
Recall that η > 0 is arbitrary. So, choosing η > 0 large we have We conclude that there exists unique τ u > 0 (see (10)) such that In a similar fashion, working this time with the fibering function we produce a unique t u > 0 such that We conclude that This ends the proof of the proposition.

Proposition 7.
If hypotheses H 0 , H 1 hold and λ <λ 1 , then there existsû ∈ N 0 such that ϕ λ (û) =m 0 λ > 0. Proof. Let {u n } n∈N ⊆ N 0 be a minimizing sequence. We show that this sequence is bounded in W 1,p (Ω). Arguing by contradiction, suppose that up to a subsequence, we have u n → +∞. Let v n = un un , n ∈ N. We have that v n = 1 for all n ∈ N and so we may assume that v n w − → v in W 1,p (Ω) and v n → v in L r (Ω) and L p (∂Ω). (15) Suppose that v = 0. Using (11), we see that for every ρ > 0 we have (15) and recall that v = 0).