Elliptic $p$-Laplacian systems with nonlinear boundary condition

In this paper we study quasilinear elliptic systems given by \begin{equation*} \begin{aligned} -\Delta_{p_1}u_1&=-|u_1|^{p_1-2}u_1 \quad&&\text{in } \Omega,\newline -\Delta_{p_2}u_2&=-|u_2|^{p_2-2}u_2 \quad&&\text{in } \Omega,\newline |\nabla u_1|^{p_1-2}\nabla u_1 \cdot \nu&=g_1(x,u_1,u_2)&&\text{on } \partial\Omega,\newline |\nabla u_2|^{p_2-2}\nabla u_2 \cdot \nu&=g_2(x,u_1,u_2)&&\text{on } \partial\Omega, \end{aligned} \end{equation*} where $\nu(x)$ is the outer unit normal of $\Omega$ at $x \in \partial\Omega$, $\Delta_{p_i}$ denotes the $p_i$-Laplacian and $g_i\colon \partial\Omega \times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ are Carath\'{e}odory functions that satisfy general growth and structure conditions for $i=1,2$. In the first part we prove the existence of a positive minimal and a negative maximal solution based on an appropriate construction of sub- and supersolution along with a certain behavior of $g_i$ near zero related to the first eigenvalue of the $p_i$-Laplacian with Steklov boundary condition. The second part is related to the existence of a third nontrivial solution by imposing a variational structure, that is, $(g_1,g_2)=\nabla g$ with a smooth function $(s_1,s_2)\mapsto g(x,s_1,s_2)$. By using the variational characterization of the second eigenvalue of the Steklov eigenvalue problem for the $p_i$-Laplacian together with the properties of the related truncated energy functionals, which are in general nonsmooth, we show the existence of a nontrivial solution whose components lie between the components of the positive minimal and the negative maximal solution.


Introduction
Let Ω ⊆ R N be a bounded domain with a C 2 -boundary ∂Ω.For i = 1, 2 and 1 < p i < ∞ we consider the following p i -Laplacian system with nonlinear boundary conditions where ν(x) is the outer unit normal of Ω at x ∈ ∂Ω, ∆ pi denotes the p i -Laplacian given by and g i : ∂Ω×R×R → R are Carathéodory functions that satisfy appropriate growth and structure conditions, see Sections 3 and 4 for the detailed assumptions.
We are interested in the multiplicity of solutions of the system (1.1).In the first part, under general local conditions on the vector field (g 1 , g 2 ), we prove the existence of a positive minimal and a negative maximal solution (see Definition 2.4) by constructing suitable pairs of sub-and supersolution to the system (1.1) using a specific behavior of g i near zero corresponding to the first eigenvalue of the p i -Laplacian with Steklov boundary condition (see (2.6)).In the second part of this paper we suppose a variational structure of the system (1.1) which means that (g 1 , g 2 ) = ∇g with a smooth function (s 1 , s 2 ) → g(x, s 1 , s 2 ).Then, by means of the extremal positive and negative solutions obtained in the first part, we are going to show the existence of a third nontrivial solutions whose components lie between the components of the positive minimal and the negative maximal solution of (1.1).The proof uses a variational characterization of the second eigenvalue of the Steklov eigenvalue problem for the p i -Laplacian together with the properties of the corresponding truncated energy functionals.The main difficulty is the fact that the truncated energy functionals turn out to be nonsmooth independently of the smoothness of ∇g.This situation is different to the scalar case and needs further investigations in terms of Clarke's generalized gradient of locally Lipschitz functionals.
Our work is motivated by the papers of Carl-Motreanu [6] and Winkert [35].In [6] the authors study a Dirichlet system of the form in Ω, in Ω, where f i : Ω × R × R → R are Carathéodory functions having a certain local behavior near zero.It is shown that the system (1.2) has at least three nontrivial solutions whereby the first and the second eigenvalue of the p i -Laplacian with Dirichlet boundary condition have been used.On the other hand, in [35], a scalar equation with nonlinear boundary condition of the form has been considered.Here, the nonlinearities f : Ω × R → R and g : ∂Ω × R → R are Carathéodory functions which are bounded on bounded sets and which satisfy appropriate conditions near zero and at infinity.If λ is larger than the second eigenvalue of the eigenvalue problem of the p-Laplacian with Steklov boundary condition, then the existence of three nontrivial solutions has been shown whereby two of them have constant sign and the third one turns out to be sign-changing.In our paper we combine the ideas of both papers to show multiplicity of solutions for the coupled system given in (1.1).We also refer to El Manouni-Papageorgiou-Winkert [12] which extends problem (1.3) to more general, nonhomogeneous operators of type (p, q).As far as we know there are only few works for elliptic systems with nonlinear boundary condition and with a variational structure.In 2016, de Godoi-Miyagaki-Rodrigues [10] studied the following Laplacian system in Ω, where is a positive definite matrix for a.a.x ∈ Ω and the nonlinearities f : Ω × R 2 → R 2 , g : ∂Ω × R 2 → R 2 satisfy suitable growth and structure conditions.The authors prove existence results for (1.4) when resonance or nonresonance conditions occur by using variational tools.For systems with nonlinear boundary conditions but without a variational structure we refer to the works by Guarnotta-Livrea-Winkert [17] who developed a sub-supersolution method for variable exponent double phase systems and Frisch-Winkert [14] for boundedness, existence and uniqueness results for coupled gradient dependent elliptic systems, see also the paper of Guarnotta-Marano-Moussaoui [20] for singular convective systems based on perturbation techniques along with fixed point arguments.Systems with homogeneous Neumann boundary conditions have been studied in the papers by Chabrowski [7] by constrained minimization based on the concentration compactness principle, by Guarnotta-Marano [18,19] getting infinitely many solutions for convection problems by appropriate pairs of sub-supersolution and by Motreanu-Perera [28] who studied p-Laplace systems via Morse theory.Finally, in case of systems with Dirichlet boundary conditions, we refer to the works by Carl-Motreanu [5] for convective p-Laplace systems based on a sub-supersolution approach, de Morais Filho-Souto [11] using the concentration compactness principle, Gambera-Marano-Motreanu [16] for (p, q)-problems via Brouwer's fixed point theorem, Hai-Shivaji [21] for parametric p-Laplacian systems, Liu-Nguyen-Winkert-Zeng [24] for coupled double phase obstacle systems involving nonlocal functions and convection terms, Marino-Winkert [25] for existence and uniqueness results of convection systems, Motreanu-Moussaoui-Pereira [27] for p-Laplacian systems via sub-supersolution method and the Leray-Schauder topological degree, Motreanu-Vetro-Vetro [30,31] for systems involving (p, q)-Laplacians, see also the references therein.
The paper is organized as follows.In Section 2 we present the main tools which are needed in the sequel including the properties of the eigenvalue problem for the r-Laplacian (1 < r < ∞) with Steklov boundary condition.Section 3 deals with the existence of extremal positive and negative solutions where positive (resp.negative) means that both components are positive (resp.negative).Finally, in Section 4 we are going to assume a variational structure of (1.1) and prove the existence of a third nontrivial solution whose components lie between the related components of the positive and the negative solution.

Preliminaries
In this section we recall the main tools that will be needed in the sequel.For 1 ≤ r < ∞ we denote by L r (Ω) and L r (Ω; R N ) the usual Lebesgue spaces with norm • r and by W 1,r (Ω) the corresponding Sobolev space with norm • 1,r = ∇ • r + • r .We equip the spaces V i := W 1,pi (Ω) with the equivalent norms where 1 < p 1 , p 2 < ∞.Moreover, we denote by L r (∂Ω) the boundary Lebesgue space with norm • r,∂Ω for any r ∈ [1, ∞].For s ∈ R, we set s ± = max{±s, 0} and for u ∈ W 1,r (Ω) we define u ± (•) = u(•) ± .We have The space L pi (Ω) is endowed with the natural partial ordering given by the positive cone which implies a related partial ordering in its subspace W 1,pi (Ω).The positive cone induces the componentwise partial ordering on the product space This implies the componentwise partial ordering in the subspace and hold true for all (ϕ 1 , ϕ 2 ) ∈ W and all the integrals in (2.1) and (2.2) are finite.
Next, we introduce the notion of weak sub-and supersolution to (1.1).
) is a pair of weak sub-and supersolution, then the order interval i be the operator given by for u i , ϕ i ∈ V i , where • , • Vi denotes the duality pairing between V i and its dual space V * i .The following proposition summarizes the main properties of A pi , see, for example, Carl-Le-Motreanu [4,Lemma 2.111].
Proposition 2.3.Let p i ∈ (1, ∞) and let A pi : V i → V * i be given by (2.5).Then A pi is well-defined, bounded, continuous, monotone and of type (S + ), that is, Next, we want to explain the notion of minimal and maximal constant sign solutions.
Definition 2.4.An element m ∈ W is said to be a minimal positive solution of (1.1) if m is a positive solution of (1.1) and if for any positive solution u with u ≤ m it follows that m = u.Similarly, we define a maximal negative solution.
Let C 1 (Ω) be equipped with norm • C 1 (Ω) and let C 1 (Ω) + be its positive cone defined by This cone has a nonempty interior given by int Let us recall some basic facts about the Steklov eigenvalue problem for the r-Laplacian with r ∈ (1, ∞) which is given by (2.6) From Lê [22] we know that the set of eigenvalues of (2.6), denoted by σ(r), has a smallest element λ 1,r which is positive, isolated, simple and can be characterized by We further point out that every eigenfunction corresponding to the first eigenvalue λ 1,r does not change sign in Ω.In fact it turns out that every eigenfunction associated to an eigenvalue λ = λ 1,r changes sign on ∂Ω.
In what follows we denote by u 1,r the normalized (i.e., u 1,r r,∂Ω = 1) positive eigenfunction corresponding to λ 1,r .As shown in Lê [22], thanks to the nonlinear regularity theory and the nonlinear maximum principle, we can suppose that u 1,r ∈ int C 1 (Ω) + .Additionally, due to the fact that λ 1,r is isolated, the second eigenvalue λ 2,r is well-defined by Then, due to Martínez-Rossi [26], we have a variational characterization of λ 2,r given by Next, we recall some basic notions in nonsmooth analysis that are required in the sequel.We refer to the monograph of Carl-Le-Motreanu [4].For a real Banach space (X, • X ), we denote by X * its dual space and by •, • the duality pairing between X and X * .A function f : X → R is said to be locally Lipschitz if for every x ∈ X there exist a neighborhood U x of x and a constant L x ≥ 0 such that For a locally Lipschitz function f : X → R on a Banach space X, the generalized directional derivative of f at the point x ∈ X along the direction y ∈ X is defined by , y for all y ∈ X, then the usual directional derivative f ′ (x; y) given by t exists and coincides with the generalized directional derivative f • (x; y).
If f 1 , f 2 : X → R are locally Lipschitz functions, then we have y) for all x, y ∈ X.The generalized gradient of a locally Lipschitz function f : X → R at x ∈ X is the set ∂f (x) := {x * ∈ X * : x * , y ≤ f • (x; y) for all y ∈ X} .Based on the Hahn-Banach theorem we easily verify that ∂f (x) is nonempty.An element x ∈ X is said to be a critical point of a locally Lipschitz function f : X → R if there holds f • (x; y) ≥ 0 for all y ∈ X or, equivalently, 0 ∈ ∂f (x), see Chang [8].
The nonsmooth mountain-pass theorem due to Chang is stated as follows [8, Theorem 3.4].
Theorem 2.5.Let X be a reflexive real Banach space and let J : X → R be a locally Lipschitz functional satisfying the nonsmooth Palais-Smale condition.If there exist x 0 , x 1 ∈ X and a constant r > 0 such that x 1 − x 0 > r and max{J(x 0 ), J(x 1 )} < inf x∈∂Br(x0) J(x), then J has a critical point u 0 ∈ X such that

Constant-sign solutions
In this section we prove the existence of maximal and minimal constant sign solutions for problem (1.1).We suppose the following hypotheses: (H 0 ) For i = 1, 2, the functions g i : ∂Ω × R × R → R are Carathéodory functions such that g i (x, 0, 0) = 0 for a.a.x ∈ ∂Ω and uniformly for a.a.x ∈ ∂Ω and for all s 2 ∈ (0, k 2 ], lim inf uniformly for a.a.x ∈ ∂Ω and for all uniformly for a.a.x ∈ ∂Ω and for all s 1 ∈ (0, uniformly for a.a.x ∈ ∂Ω and for all ) is needed for the usage of the regularity results of Lieberman [23].
Let us verify this just for u 1 , the case for u 2 works in the same way.First, from the boundedness of u 1 and (3.1) along with Theorem 2 in Lieberman [23], we know that u ∈ C 1,α (Ω) for some α ∈ (0, 1).From the first line in (1.1) for a.a.x ∈ Ω. Taking β(s) = s p1−1 for all s > 0, we get from Vázquez's strong maximum principle (see [33]) Applying again the maximum principle we obtain ∇u 1 (x 0 ) • ν(x 0 ) < 0. In view of hypothesis (H 2 ) first line, for ε > 0 small enough such that c 1 − ε > 0, there exists δ > 0 such that for all s 1 ∈ (0, δ) we get which yields by the continuity of g 1 as The continuity of g 1 then shows that g 1 (x 0 , 0, that is, we have g 1 (x 0 , 0, u 2 (x 0 )) ≥ 0, and thus from the third line of (1.1) it follows Theorem 3.2.Let hypotheses (H 0 ), (H 1 ) and (H 2 ) be satisfied.Then there exist a positive solution (u 1 , u 2 ) ∈ W and a negative solution (v 1 , v 2 ) ∈ W of the system (1.1).
From the Steklov eigenvalue problem for the p i -Laplacian multiplied with ε pi−1 > 0 we know that holds for all ϕ i ∈ V i with ϕ i ≥ 0 and i = 1, 2. We choose ε > 0 small enough such that εu 1,pi (x) < δ for all x ∈ Ω and i = 1, 2.
On the other hand, we get from (3.2) and (2.4) that in Ω and for all (w 1 , w 2 ) ∈ W such that u i ≤ w i ≤ u i for i = 1, 2. Therefore, (u 1 , u 2 ) ∈ W and (u 1 , u 2 ) ∈ W form a pair of sub-and supersolution related to Definition 2.2.From Guarnotta-Livrea-Winkert [17] (for µ ≡ 0) we know that a solution (u 1 , u 2 ) ∈ W of the system (1.1) exists such that u i ≤ k i .Moreover, the nonlinear regularity theory implies that Similarly, one can show that (d 1 , d 2 ) and (−εu 1,p1 , −εu 1,p2 ) form a pair of suband supersolution in the sense of Definition 2.2 for the system (1.1) provided the parameter ε > 0 is sufficiently small.Therefore, we obtain a negative solution Next, we are going to prove the existence of a minimal positive and of a maximal negative solution of the system (1.1) in the trapping region constructed in the proof of Theorem 3.2.Theorem 3.3.Let hypotheses (H 0 ), (H 1 ) and (H 2 ) be satisfied.Then, for a given solution ] for some ε > 0, there exists a maximal solution Proof.We are going to prove just the first assertion of the theorem, the second one can be shown using similar arguments.
We choose ε > 0 sufficiently small (like in the proof of Theorem 3.2).Then, Theorem 3.2 guarantees that a solution (u Apparently, S ε is not empty.We are going to prove that S ε has a minimal element by applying Zorn's Lemma.For this purpose, let C be a chain in S ε .Then we can find a sequence 2) with ϕ 2 = u k 2 and using (H 0 ) together with the trace theorem, we get that Therefore, up to a subsequence if necessary, not relabeled, we may assume that for a.a.x ∈ ∂Ω. (3.7) and ûi ≤ u i for i = 1, 2. Furthermore, testing the corresponding weak formulations with u k i − ûi and using (3.7) along with (H 0 ) we get that lim sup Combining this with (3.7) and the fact that A pi fulfills the (S + )-property on V i , see Proposition 2.3, we conclude that ).In order to get maximal and minimal solutions of (1.1), we have to suppose further conditions on the vector field (g 1 , g 2 ) near zero as follows.
which is minimal among the positive solutions of (1.1).Moreover, problem (1.1) admits a negative solution which is maximal among the negative solutions of (1.1).
Proof.As before, we only show the existence of a minimal positive solution of (1.1), the proof for the maximal negative solution works in a similar way.The application of Theorems 3.2 and 3.3 gives us a sequence {(u n 1 , u n 2 )} n≥n0 ⊆ W for n 0 sufficiently large such that for every integer n ≥ n 0 we have that (u n 1 , u n 2 ) is a solution of (1.1) that is minimal in the trapping region [ From this and (H 0 ) we may suppose, for a subsequence if necessary, not relabeled, that, for i = 1, 2, in L pi (Ω) and pointwisely a.e. in Ω, in L pi (∂Ω) and pointwisely a.e. in ∂Ω.
for some (u 1,+ , u 2,+ ) ∈ W. As in the proof of Theorem 3.3 by applying the (S + )property of A pi on V i , see Proposition 2.3, we conclude that (u 1,+ , u 2,+ ) is a solution of (1.1).Claim : u i,+ = 0 for i = 1, 2. Suppose this is not the case and assume that u 1,+ = 0.For each n ≥ n 0 we set and Clearly the sequence {w n } n≥n0 ⊆ V 1 is bounded and due to hypotheses (H 2 ) and (H 3 ) we may assume that in L p1 (Ω) and pointwisely a.e. in Ω, in L p1 (∂Ω) and pointwisely a.e. in ∂Ω, for some w ∈ V 1 and ξ ∈ L (3.10) From (3.10) and (3.9) we obtain that Thus, again by the (S + )-property of A p1 on V 1 it follows that w n → w in V 1 which implies that w = 0 since w n 1,p1 = 1.Moreover, from the strong convergence in V 1 and the fact that (u n 1 , u n 2 ) ∈ W is a solution of (1.1) as well as the representation Taking (H 2 ) and (H 3 ) into account, for any given ε > 0 there exists an integer n(x) for a.a.x ∈ ∂Ω such that for every n ≥ n(x) it holds Since ε > 0 is arbitrary, letting n → ∞, we get via Mazur's theorem Hence w is an eigenfunction associated to the eigenvalue 1 of the weighted eigenvalue problem with weight µ(x) > 0 We consider now the V (x)-weighted eigenvalue problem with V (x) > 0, λ(V ) the eigenvalue for the weight V (x) and w V the corresponding eigenfunctions.In the following, we call λ 1 (V ) the first eigenvalue of (3.12).Since w is nonnegative, due to Fernández Bonder-Rossi [13, Theorem 1.2 and Proposition 3.1], we know that λ 1 (µ) = 1 because of (3.11).We consider now problem (3.12) with weights c 1 and λ 1,p1 and related first eigenvalues λ 1 (c 1 ) and λ 1 (λ 1,p1 ), respectively.Since λ 1,p1 < c 1 ≤ µ(x) for a.a.x ∈ ∂Ω, we have with [13, Theorem Since λ 1,p1 is the smallest eigenvalue of (2.6) with eigenfunction u 1,p1 > 0 we see that λ 1 (λ 1,p1 ) = 1.This is a contradiction to (3.13).Hence, u i,+ = 0 for i = 1, 2. Since u i,+ ∈ [0, k i ] for i = 1, 2, by the nonlinear regularity theory, see Remark 3.1, we conclude that (u It remains to show that (u 1,+ , u 2,+ ) is a minimal positive solution of problem (1.1).To this end, let (v 1 , v 2 ) ∈ W be any positive solution of (1.1) such that v 1 ≤ u 1,+ and v 2 ≤ u 2,+ .Again, by the nonlinear regularity theory and the strong maximum principle, we know that (v whenever n is sufficiently large.However, since ( , we get from (3.14) that u n i ≤ v i for i = 1, 2. But then, again because of (3.14), it follows that u 1,+ = v 1 and u 2,+ = v 2 .This completes the proof of the theorem.

Another nontrivial solution
In this section we are interested in a third nontrivial solution of the system (1.1) under the assumption that (1.1) has a variational structure.To be more precise we consider the system where with g : ∂Ω×R×R → R being a Carathéodory function which is twice differentiable with respect to the second and third variable (s 1 , s 2 ) ∈ R 2 .Moreover, we suppose that the partial derivatives g s1 , g s2 , g s1s1 , g s1s2 , g s2s2 are Carathéodory functions on ∂Ω × R 2 and g s1 , g s2 are supposed to be bounded on bounded sets.Without any loss of generality, we assume that g(x, 0, 0) = 0 for a.a.∂Ω.
To avoid having to write down all the conditions again, let us now assume that (H 0 )-(H 3 ) hold true replacing (g 1 , g 2 ) by (g s1 , g s2 ).Taking Theorem 3.4 into account, we can find a minimal positive solution (u 1,+ , u 2,+ ) of problem (4.1) with u i,+ ≤ k i for i = 1, 2. Based on this, we introduce the truncation function τ In the same way, by applying the maximal negative solution (u 1,− , u 2,− ) of (4.1) with u i,− ≥ d i for i = 1, 2 obtained in Theorem 3.4, we define the truncation function τ − : ∂Ω × R 2 → R 2 as the projection τ − (x, s 1 , s 2 ) of (s 1 , s 2 ) on the closed convex subset [u 1,− (x), 0] × [u 2,− (x), 0] of R 2 .Lastly, we introduce the truncation function τ 0 : ∂Ω × R 2 → R 2 as the projection τ 0 (x, s 1 , s 2 ) of (s 1 , s 2 ) on the closed convex subset With the help of the truncation functions τ + , τ − , τ 0 : ∂Ω × R 2 → R 2 we can introduce truncated functions related to g : ∂Ω × R × R → R in the following way: These truncated mappings g − , g + , g 0 : ∂Ω×R 2 → R are Carathéodory functions being locally Lipschitz continuous with respect to the variables (s 1 , s 2 ) ∈ R 2 .Therefore, their generalized gradients in the sense of Clarke exist.Applying Clarke's calculus according to [9, Theorem 2.5.1],we have the following representations: for a.a.x ∈ ∂Ω and for all (s 1 , s for a.a.x ∈ ∂Ω and for all (s 1 , s Taking the modified truncated functions g − , g + , g 0 : ∂Ω × R 2 → R into account, we introduce the related truncated, nonsmooth functionals E + , E − , E 0 : W → R defined by These functionals are locally Lipschitz and so their generalized gradients exist.Before we consider the location of the critical points of these functionals, we need to suppose an additional condition: for a.a.x ∈ ∂Ω and for all for a.a.x ∈ ∂Ω and for all s 2 ∈ [d 2 , k 2 ].Next, we are interested in the location of critical points of the functionals E + , E − , E 0 : W → R. Proposition 4.1.Let hypotheses (H 0 )-(H 3 ) be satisfied, where (g 1 , g 2 ) is replaced by ∇g and suppose (H 4 ).Then, the following assertions hold: for a.a.x ∈ ∂Ω.
Proof.We only prove the assertion in (i), the cases (ii) and (iii) can be shown using similar arguments.To this end, let (v 1 , v 2 ) be a critical point of E + , that is, (0, 0) where which follows from Clarke [9, Theorem 2.7.5].Note that the precise expressions of the functions h 1 and h 2 can be found in Carl [2,3].Using the fact that (u 1,+ , u 2,+ ) solves problem (4.1), we obtain, due to (4.5) and (4.6), by choosing the test function Therefore, v 1 ≤ u 1,+ .In the same way, using (H 4 ) (ii), we show that v 2 ≤ u 2,+ .
In the next proposition we are going to compare the minimal and maximal constant-sign solutions obtained in Theorem 3.4 with the minimizers of the constructed nonsmooth functionals.Proof.Due to the truncated function g + : ∂Ω × R 2 → R, it is clear that the functional E + : W → R is coercive and sequentially weakly lower semicontinuous.This guarantees the existence of a global minimizer (w 1 , w 2 ) ∈ W of E + which is a critical point of E + in the sense of nonsmooth analysis, see Section 2. From Proposition 4.1 (i) and u i,+ ≤ k i for i = 1, 2, it follows that 0 ≤ w i (x) ≤ u i,+ (x) ≤ k i for a.a.x ∈ ∂Ω and for i = 1, 2.
For the next result, we need associated scalar problems of (4.1) defined by and We have the following result.
Proposition 4.3.Let hypotheses (H 0 )-(H 3 ) be satisfied, where (g 1 , g 2 ) is replaced by ∇g and suppose (H 4 ).Then there exists (u such that u + is a solution of (4.8) and v + is a solution of (4.9) satisfying Furthermore, there exists (u ) and v − is a solution of (4.9) satisfying Proof.Note that E + (•, 0) : W 1,p1 (Ω) → R is coercive and sequentially weakly lower semicontinuous.Hence, we can find u + ∈ W 1,p1 (Ω) such that . By means of (H 4 ) we have with nonnegative test function Using this and the fact that u + solves we get, with the test function (u This implies 0 ≤ u + (x) ≤ u 1,+ (x) for a.a.x ∈ ∂Ω.Therefore, u + is a solution of (4.8).Applying again the regularity results, as before, we get that u + ∈ int C 1 (Ω) + .The proofs for v + , u − and v − can be done in a very similar way.
For our main result, we need the following sub-homogeneous conditions on the right-hand sides of (4.8) and (4.9).
Example 4.5.For the sake of simplicity, we have omitted the x-dependence on g and consider the problem Then, the potential is given by Then, for any constants k 1 , k 2 > 0 and d 1 , d 2 < 0, Hypotheses (H 0 )-(H 6 ) are satisfies provided α > 0 is sufficiently large.Let us prove this for g 1 , the same arguments can be used for g 2 .
We now set This shows (H 5 ).Hypothesis (H 6 ) can be shown in a similar way.

Proposition 4 . 2 .
Let hypotheses (H 0 )-(H 3 ) be satisfied, where (g 1 , g 2 ) is replaced by ∇g and suppose (H 4 ).Then the minimal positive solution (u 1,+ , u 2,+ ) of problem (4.1) is the unique global minimizer of E + and a local minimizer of E 0 while the maximal negative solution (u 1,− , u 2,− ) of problem (4.1) is the unique global minimizer of E − and a local minimizer of E 0 .
+ in the interval (m, c] associated to critical points with one positive component and the other one equal to zero.This proves the Claim.Because of the Claim, we can now apply the nonsmooth version of the second deformation lemma to the functional E + , see Gasiński-Papageorgiou [15, Theorem 2.1.1].This gives us a continuous map η 199) Note that we supposed hypothesis (H 2 ) with λ 1,pi replaced by λ 2,pi .Then it follows that we can find δ > 0 such that