Sign-Changing Solutions for Nonlinear Elliptic Problems Depending on Parameters

Example 2.1. For the sake of simplicity we drop the x dependence for the function f in the right-hand side of (1.4). The function f : ℝ × (0, +∞) → ℝ given by

with c > 0 and r > p − 1, satisfies hypotheses H(f). Next we give an example of function f : ℝ × (0, +∞) → ℝ verifying assumptions H(f) which is generally not odd with respect to s: with λ > 0, a₁ ≥ 1, a₂ ≥ 1, c₁ > 0, c₂ > 0, r₁ > p − 1, r₂ > p − 1.

Lemma 2.2. Let the data c, r, and a(·, λ) be as in H(f)(ii). Then for every constant θ > 0 there is $$ with the property that if λ ∈ (0, λ₀), one can choose ξ₀ = ξ₀(λ)∈(0, θ) such that

Proof. On the contrary there would exist a constant θ > 0 and a sequence λ_n ↓ 0 as n → ∞ such that

Letting n → ∞ we get $$ for all ξ ∈ (0, θ) because we have ∥a(·,λ)∥_∞ → 0 as λ ↓ 0. Since r > p − 1, a contradiction is achieved as ξ ↓ 0. Therefore (2.9) holds true.

Lemma 2.3. Assume H(f)(i) and (ii) and the following weaker form of hypothesis H(f)(iii): for all $$ there exist μ₀ = μ₀(λ) > λ₁ and Ω_λ ⊂ Ω with Ω∖Ω_λ of Lebesgue measure zero such that

uniformly with respect to x ∈ Ω_λ.

Fix a constant θ > 0 and consider the corresponding number $$ obtained in Lemma 2.2. Then for any λ ∈ (0, λ₀) the function $$, with ξ₀ ∈ (0, θ) given by Lemma 2.2, is a supersolution for problem (1.4), and the function $$ is a subsolution of problem (1.4) provided that the number ε > 0 is sufficiently small.

Proof. For a fixed λ ∈ (0, λ₀), from (2.9) and H(f)(ii) we derive

which says that $$ is a supersolution for problem (1.4).

On the other hand, by hypothesis we can find μ = μ(λ) > λ₁ and δ = δ(λ) > 0 such that

Choose $$. Then by (2.14) we have which ensures that $$ is a subsolution of problem (1.4).

Theorem 2.4. Assume H(f)(i) and (ii) and the following weaker form of H(f)(iii): for all $$ there exist constants μ₀ = μ₀(λ) > λ₁, ν₀ = ν₀(λ) > μ₀ and a set Ω_λ ⊂ Ω with Ω∖Ω_λ of Lebesgue measure zero such that

uniformly with respect to x ∈ Ω_λ. Then for all b > 0 there exists a number $$ with the property that if λ ∈ (0, λ₀), then there is a constant ξ₀ = ξ₀(λ) ∈ (0, b/∥e∥_∞) such that problem (1.4) has a least positive solution $$ in the order interval [0, ξ₀e] and a greatest negative solution $$ in the order interval [−ξ₀e, 0].

Proof. Since the proof of the existence of the greatest negative solution follows the same lines, we only provide the arguments for the existence of the least positive solution.

Applying Lemma 2.3 for θ = b/∥e∥_∞  we find $$ as therein. Fix λ ∈ (0, λ₀). Lemma 2.3 ensures that $$ is a supersolution for problem (1.4), with ξ₀ ∈ (0, b/∥e∥_∞) given by Lemma 2.2, and $$ is a subsolution for problem (1.4) if ε > 0 is small enough. Passing eventually to a smaller ε > 0, we may assume that εφ₁ ≤ ξ₀e. Then by the method of sub-supersolution we know that in the order interval [εφ₁, ξ₀e] there is a least (i.e., smallest) solution $$ of problem (1.4) (see [29]).

We thus obtain that for every positive integer n sufficiently large there is a least solution $$ of problem (1.4) in the order interval [(1/n)φ₁, ξ₀e]. Clearly, we have

with some function u₊ : Ω → ℝ satisfying 0 ≤ u₊ ≤ ξ₀e. First we claim that Taking into account that u_n solves (1.4), and the fact that u_n belongs to the order interval [0, ξ₀e], from H(f)(ii) we see that which implies the boundedness of the sequence (u_n) in V₀. Then due to (2.17) we have that u₊ ∈ V₀ as well as Since u_n solves problem (1.4), one has Setting φ = u_n − u₊ in (2.21) gives As already noticed that the sequence (f(·, u_n(·), λ) is uniformly bounded on Ω, so (2.20) and (2.22) yield The S₊-property of −Δ_p on V₀ implies The strong convergence in (2.24) and Lebesgue′s dominated convergence theorem permit to pass to the limit in (2.21) that results in (2.18).

By (2.18) and the nonlinear regularity theory (cf., e.g., Theorem 1.5.6 in [27]) it turns out $$. The choice of ξ₀ guarantees that

Thus, from (2.18), assumptions H(f)(ii) and (iii), and the boundedness of u₊, we get with a constant $$. Applying the nonlinear strong maximum principle (cf. [28]) we conclude that either u₊ = 0 or $$.

We claim that

Assume on the contrary that u₊ = 0. Then (2.17) becomes Since u_n ≥ (1/n)φ₁, we may consider Along a relabelled subsequence we may suppose for some $$. Moreover, one can find a function w ∈ L^p(Ω) ₊ such that $$ for almost all x ∈ Ω. Relation (2.21) reads Setting $$ leads to By H(f)(iii) we know that there exist constants c₀ = c₀(λ) > λ₁ and α = α(λ) > 0 such that while H(f)(ii) entails for a.a. x ∈ Ω and for all |s| ≥ α. Combining the two estimates gives with a constant c₁ = c₁(λ) > 0. Since u_n ∈ [(1/n)φ₁, ξ₀e], r > p − 1 and (2.35) holds, there exists a constant C > 0 such that We see from (2.36) that Then, because the right-hand side of the above inequality is in L¹(Ω), by means of (2.30) and (2.36) we can apply Lebesgue′s dominated convergence theorem to get Consequently, from (2.32) we obtain The S₊-property of −Δ_p on V₀ implies On the basis of (2.31) and (2.40) it follows Notice from (2.36) that for a.a. x ∈ Ω and for all φ ∈ V₀. We are thus allowed to apply Fatou′s lemma which in conjunction with (2.28), (2.30), and (2.16) ensures for all φ ∈ V_0,+ : = V₀∩L^p(Ω) ₊. Thanks to (2.41) we obtain Owing to (2.42) we may once again use Fatou′s lemma; so according to (2.28), (2.30), and the last part of (2.16), we find for all φ ∈ V_0,+. Then (2.41) ensures Combining (2.44) and (2.46) results in which guarantees to have $$ (see [27, Theorem 1.5.5]). Since by (2.47) we know that $$, we are in a position to address Theorem 1.5.6 in [27], which provides $$ with some β ∈ (0,1). This regularity up to the boundary and the fact that $$ a.e. in Ω and (2.47) enable us to refer to the strong maximum principle (see Theorem 5 of Vázquez [28]). Recalling that $$ does not vanish identically on Ω (because $$) we deduce that $$ for all x ∈ Ω and $$ for all x ∈ ∂Ω which amounts to saying $$. Consequently, there exist constants k₀ > 0 and k₁ > 0 such that Following [30] let us denote whenever (u, v) ∈ D_I, where Relation (2.48) justifies that $$. Then Proposition 1 of Anane [30] implies $$. On the other hand a direct computation based on (2.48) and (2.47) shows This contradiction proves that the claim in (2.27) holds true.

In view of (2.18) it remains to establish that u₊ is the smallest positive solution of problem (1.4) in the interval $$. Let u ∈ V₀ be a positive solution to (1.4) in $$. Since u ∈ L^∞(Ω), then (1.4) and H(f)(ii) allow to deduce that −Δ_pu ∈ L^∞(Ω). Using Theorem 1.5.6 of [27] leads to $$. Then, as u is a solution to (1.4) and $$, with $$, by means of hypotheses H(f)(ii) and (iii), we are able to apply the strong maximum principle. So we get $$, hence $$ for n sufficiently large. The fact that u_n is the least solution of (1.4) in $$ ensures u_n ≤ u. Taking into account (2.17), we obtain u₊ ≤ u. This completes the proof.

Theorem 2.5. Under hypotheses H(f), for all b > 0, there exists a number $$ with the property that if λ ∈ (0, λ₀), then problem (1.4) has a (positive) solution $$, a (negative) solution $$ and a nontrivial sign-changing solution $$ satisfying ∥u₊∥_∞ < b, ∥u₋∥_∞ < b, ∥u₀∥_∞ < b.

Proof. Let b > 0. Consider the positive number λ₀ given by Theorem 2.4 and fix λ ∈ (0, λ₀). Let $$ and $$ be the two extremal solutions determined in Theorem 2.4. We introduce on Ω × ℝ the truncation functions

and then define the following associated functionals: It is clear that E₊, E₋, E₀ ∈ C¹(V₀).

We observe that if v is a critical point of E₊, then

which implies v ≤ u₊. Similarly, it follows that v ≥ 0. This leads to

Since the function E₊ is coercive and weakly lower semicontinuous, there exists a global minimizer z₊ ∈ V₀ of it. Using (2.14), it is seen that

and so z₊ ≠ 0. Relation (2.55) shows that z₊ is a nontrivial solution of problem (1.4) belonging to the order interval [0, u₊]. Via assumptions H(f)(ii) and (iii) and the boundedness of z₊, we may apply the strong maximum principle which ensures z₊ > 0 on Ω. In view of the minimality property of u₊ as stated in Theorem 2.4, it follows that z₊ = u₊. In fact, u₊ is the unique global minimizer of E₊.

Since $$, there exists a neighborhood 𝒰 of u₊ in the space $$ such that $$. Therefore E₀ = E₊ on 𝒰, which guarantees that u₊ is a local minimizer of E₀ on $$. It results that u₊ is also a local minimizer of E₀ on the space V₀ (see [27], pages 655-656 ). Employing the functional E₋ and proceeding as in the case of u₊, we establish that u₋ is a local minimizer of E₀ on V₀.

As in the case of (2.55), we verify that every critical point of E₀ belongs to the set {u ∈ V₀ : u₋(x) ≤ u(x) ≤ u₊(x)  a.e.  x ∈ Ω}, which implies that every critical point of E₀ is a solution to problem (1.4). The functional E₀ is coercive, weakly lower semicontinuous, with $$. Thus E₀ has a global minimizer y₀ ∈ V₀ with y₀ ≠ 0. The above properties ensure that y₀ is a nontrivial solution of problem (1.4) belonging to the order interval [u₋, u₊]. Assume y₀ ≠ u₊ and y₀ ≠ u₋. We claim that y₀ changes sign. Indeed, if not, y₀ would have constant sign, for instance y₀ ≥ 0 a.e. on Ω. Using assumptions H(f)(ii) and (iii) and the boundedness of y₀, we may apply the strong maximum principle which leads to y₀ > 0 on Ω. This is impossible because it contradicts the minimality property of the solution u₊ as given by Theorem 2.4. According to the claim, we obtain the conclusion of the theorem setting u₀ = y₀.

Thus, the proof reduces to consider the cases y₀ = u₊ or y₀ = u₋. To make a choice, suppose y₀ = u₊. We may also admit that u₋ is a strict local minimizer of E₀. This is true since on the contrary we would find (infinitely many) critical points x₀ of E₀ belonging to the order interval [u₋, u₊] which are different from 0, u₋, u₊, and if x₀ does not change sign, taking into account the strong maximum principle, the extremality properties of the solutions u₋, u₊ given in Theorem 2.4 will be contradicted. A straightforward argument allows then to find ρ ∈ (0, ∥u₊ − u₋∥) such that

where ∂B_ρ(u₋) = {u ∈ V₀   :   ∥u − u₋∥ = ρ}. Relation (2.57) in conjunction with the Palais-Smale condition (which holds for E₀ due to its coercivity) enables us to apply the mountain pass theorem to the functional E₀ (see, e.g., [31]). In this way we get u₀ ∈ V₀ satisfying E′₀(u₀) = 0 and where We infer from (2.57) and (2.58) that u₀ ≠ u₋  and u₀ ≠ u₊.

The next step in the proof is to show that

By the equality in (2.58), it suffices to produce a path $$ such that Let $$, where $$, and $$ be endowed with the topologies induced by V₀ and $$, respectively. We set Making use of the first inequality in assumption H(f) (iii), we fix numbers μ > λ₂ and δ > 0 such that (2.14) holds, and then let ρ₀ ∈ (0, μ − λ₂). We recall the following variational expression for λ₂ given by Cuesta et al. [32]: where By (2.63) there exists γ ∈ Γ₀ such that Choose some number r with 0 < r ≤ (λ₂ + ρ₀) ^1/p − (λ₂ + ρ₀/2) ^1/p. The density of S_C in S implies that Γ_0,C is dense in Γ₀; so there is γ₀ ∈ Γ_0,C satisfying Then the choice of r establishes The boundedness of the set $$ in ℝ  ensures the existence of some ε₁ > 0 such that Since $$ (see Theorem 2.4), for every u ∈ γ₀([−1,1]) and any bounded neighborhood V_u of u in $$ there exist positive numbers h_u and j_u such that whenever h ≥ h_u, j ≥ j_u, and v ∈ V_u. This fact and the compactness of γ₀([−1,1]) in $$ allow to determine a number ε₀ > 0 for which one has We now focus on the continuous path εγ₀ in $$ joining −εφ₁  and εφ₁ with a fixed constant ε satisfying 0 < ε < min{ε₀, ε₁}. By (2.70), (2.67), (2.68), (2.14) with μ > λ₂, and taking into account the choice of ρ₀ as well as $$ we obtain

At this point we apply the second deformation lemma (see, e.g., [27, page 366]) to the C¹ functional E₊ : V₀ → ℝ. Towards this let us denote

It was already shown that u₊ is the unique global minimizer of E₊, and so we have m₊ < c₊. Taking into account (2.55), E₊ has no critical values in the interval (m₊, c₊] (for, otherwise, the minimality of the positive solution u₊ of (1.4) would be contradicted). Using also that the functional E₊ satisfies the Palais-Smale condition (because it is coercive), the second deformation lemma can be applied to E₊  yielding a continuous mapping $$ such that η(0, u) = u and η(1, u) = u₊ for all $$ as well as E₊(η(t, u)) ≤ E₊(u) whenever t ∈ [0,1] and $$. Introducing γ₊ : [0,1] → V₀ by for all t ∈ [0,1], it is seen that γ₊ is a continuous path in V₀ joining εφ₁ and u₊. (Note the mapping w ↦ w⁺ is continuous from V₀ into itself.) The properties of the deformation η imply for all t ∈ [0,1]. Similarly, applying the second deformation lemma to the functional E₋, we construct a continuous path γ₋ : [0,1] → V₀ joining u₋ and −εφ₁ such that The union of the curves γ₋, εγ₀, and γ₊ gives rise to a path $$. We see from (2.75), (2.71), and (2.74) that (2.61) is satisfied. Hence (2.60) holds, and so u₀ ≠ 0. Recalling that the critical points of E₀ are in the order interval {u ∈ V₀   :   u₋(x) ≤ u(x) ≤ u₊(x)  a.e.  x ∈ Ω} we derive that u₀ is a nontrivial solution of (1.4) distinct from u₋ and u₊, with u₋ ≤ u₀ ≤ u₊. By the nonlinear regularity theory we have that $$. The extremality properties of the constant sign solutions u₋ and u₊ as described in Theorem 2.4 force u₀ to be sign-changing. This completes the proof.

Theorem 2.6. Assume that hypotheses H'(f) are fulfilled. Then there exists a number $$ with the property that if λ ∈ (0, λ₀), then problem (1.4) has a minimal (positive) solution $$, a maximal (negative) solution $$ and two nontrivial sign-changing solutions $$ satisfying ∥u₊∥_∞ < b, ∥u₋∥_∞ < b, u₋ ≤ u₀ ≤ u₊ a.e. in Ω (so ∥u₀∥_∞ < b) and ∥w₀∥_∞ ≥ b, where b : = min {b₊, |b₋|}.

Proof. Since hypotheses H'(f) are stronger than H(f), we can apply Theorem 2.5 with b = min {b₊, |b₋|}, which ensures the existence of a number $$ such that for every λ ∈ (0, λ₀), problem (1.4) possesses a positive solution $$, a negative solution $$ and a sign-changing solution $$ with −b < u₋ ≤ u₀ ≤ u₊ < b. The proof of Theorem 2.5 shows that u₊ and u₋ can be chosen to be the minimal positive solution and the maximal negative solution, respectively.

On the other hand, hypotheses H'(f) enable us to apply Theorem 1.1 of Bartsch et al. [33]. It follows that there exists a sign-changing solution $$ (by the nonlinear regularity theory) with $$ and $$. Therefore, we have ∥w₀∥_∞ ≥ b, which shows that the sign-changing solutions u₀ and w₀ are different. This completes the proof.

Remark 2.7. In fact, under hypotheses H'(f), for λ ∈ (0, λ₀), problem (1.4) admits at least six nontrivial solutions: two positive solutions, two negative solutions, and two sign-changing solutions, as seen in Theorem 5 in [34].

Lemma 3.1. Suppose u_n → u in V₀, w_n⇀w in L^q′(Ω), and w_n ∈ ∂𝒢(u_n) for all n ∈ ℕ. Then w ∈ ∂𝒢(u).

Definition 3.2. A function u ∈ V₀ is called a solution of (1.7) if there is an η ∈ L^q′(Ω) such that

Remark 3.3. Due to assumption (g3) we have g₁(x, 0) ≤ 0 ≤ g₂(x, 0) for almost all x ∈ Ω. Hence, in view of (3.8), problem (1.7) always possesses the trivial solution.

Definition 3.4. A function $$ is called a subsolution of (1.7) if $$, and if there is an $$ such that

Definition 3.5. A function $$ is called a supersolution of (1.7) if $$, and if there is an $$ such that

Lemma 3.6. Let e be the uniquely defined solution of (2.6). If a > λ₁, then there exists a constant α_a > 0 such that for any b ∈ ℝ the function α_ae is a positive supersolution of problem (1.7).

Proof. Let a > λ₁. By (g4) there is s_a > 0 such that

and by (g2) we get which implies and thus in view of the definition of g₂ we obtain Let $$, where α_a is a positive constant to be specified. Then we get which shows that for $$ the function α_ae is in fact a supersolution of (1.7) with $$.

Lemma 3.7. Let e be the uniquely defined solution of (2.6). If b > λ₁, then there exists a constant β_b > 0 such that for any a ∈ ℝ the function −β_be is a negative subsolution of problem (1.7).

Lemma 3.8. Let φ₁ be the normalized positive eigenfunction corresponding to the first eigenvalue λ₁ of (−Δ_p, V₀). If a > λ₁, then for ε > 0 sufficiently small and any b ∈ ℝ the function εφ₁ is a positive subsolution of problem (1.7). If b > λ₁, then for ε > 0 sufficiently small and any a ∈ ℝ the function −εφ₁ is a negative supersolution of problem (1.7).

Proof. By (g3) there is a constant δ_a > 0 such that

which implies Define $$ with ε > 0. Applying (3.19) and the definition of g₁ we get provided 0 ≤ εφ₁ ≤ δ_a. The latter can be satisfied by choosing ε sufficiently small such that $$, where $$ stands for the supremum-norm of φ₁. This proves that εφ₁ is a subsolution if $$. In a similar way one can show that for ε sufficiently small the function −εφ₁ is a negative supersolution.

Theorem 3.9. Let hypotheses (g1)-(g2) be satisfied and assume the existence of a subsolution $$ and supersolution $$ of (1.7) such that $$. Then there exist extremal solutions of (1.7) within $$.

Theorem 3.10. Let hypotheses (g1)–(g4) be fulfilled. For every a > λ₁ and b ∈ ℝ there exists a smallest positive solution $$ of (1.7) within the order interval [0, α_ae] with the constant α_a > 0 as in Lemma 3.6. For every b > λ₁ and a ∈ ℝ there exists a greatest negative solution $$ of (1.7) within the order interval [−β_b  e, 0] with the constant β_b > 0 as in Lemma 3.7.

Proof. Let a > λ₁. Lemmas 3.6 and 3.8 ensure that $$ is a supersolution of problem (1.7) and $$ is a subsolution of problem (1.7) provided that ε > 0 is sufficiently small. We may choose ε > 0 such that, in addition, εφ₁ ≤ α_ae. Thus by Theorem 3.9 there exists a smallest and a greatest solution of (1.7) within the ordered interval [εφ₁, α_ae]. Let us denote the smallest solution by u_ε. Moreover, the nonlinear regularity theory for the p-Laplacian (cf., e.g., [27, Theorem 1.5.6]) and Vázquez′s strong maximum principle [28] ensure that $$. Thus for every positive integer n sufficiently large there is a smallest solution $$ of problem (1.7) within [(1/n)φ₁, α_ae]. In this way we inductively construct a sequence (u_n) of smallest solutions which is monotone decreasing; that is, we have

with some function u₊ : Ω → ℝ satisfying 0 ≤ u₊ ≤ α_ae. As $$ and u_n are solutions of (1.7) we have where η_n ∈ L^q′(Ω) and η_n(x) ∈ ∂G(x, u_n(x)) for almost all x ∈ Ω. Since u_n ∈ [(1/n)φ₁, α_ae], the last equation together with (g2) implies that the sequence (u_n) is bounded in V₀. Taking into account (3.21) we obtain that u₊ ∈ V₀ and The solution u_n of (1.7) satisfies which yields with φ = u_n − u₊ in (3.24) the equation Using the convergence properties (3.23) of (u_n) and (g2) as well as the uniform boundedness of the sequence (u_n), we get by applying Lebesgue′s dominated convergence theorem which by the S₊-property of −Δ_p on V₀ implies Since u_n are uniformly bounded, from (g2) we see that there exists a constant c > 0 such that and thus we get (for some subsequence if necessary) η_n⇀η₊ in L^q′(Ω). By the strong convergence (3.27), Lemma 3.1 can be applied to show that η₊(x) ∈ ∂G(x, u₊(x)) for almost every x ∈ Ω. Passing to the limit in (3.24) (for some subsequence if necessary) proves Claim 1.

As u₊ belongs, in particular, to L^∞(Ω), Claim 1 and Assumption (g2) implies Δ_pu₊ ∈ L^∞(Ω). The nonlinear regularity theory (cf., e.g., Theorem 1.5.6 in [27]) ensures that $$ for some γ ∈ (0,1), so $$. In view of (g2) (g3) a constant $$ can be found such that

which yields in conjunction with Claim 1 that We now apply Vázquez′s strong maximum principle [28] where in its statement the function β is chosen as $$ for all s > 0, which is possible because $$. This result guarantees that if u₊ ≠ 0, then u₊ > 0 in Ω and ∂u₊/∂n < 0 on ∂Ω which means that $$. Suppose that Claim 2 does not hold. Then by Vázquez′s strong maximum principle we must have u₊ = 0, and thus the sequence (u_n) satisfies Setting we may suppose that along a relabelled subsequence one has with some $$, and there is a function w ∈ L^p(Ω) ₊ such that Since u_n are positive solutions of (1.7), we get for $$ the following variational equation: With the special test function $$ in (3.35) we obtain From (3.29) and (3.34) we get the estimate As the right-hand side of the last inequality is in L¹(Ω), we may apply Lebesgue′s dominated convergence theorem, which in conjunction with (3.33) yields From (3.33) and (3.36) we conclude which in view of the S₊-property of −Δ_p on V₀ results in and therefore, in particular, $$. Taking into account (g3), (3.31), and (3.40), we may pass to the limit in (3.35) which results in As $$, relation (3.41) expresses the fact that $$ is an eigenfunction of (−Δ_p, V₀) corresponding to the eigenvalue a. As a > λ₁, this is impossible according to Anane [30], because $$ must change sign. This contradiction proves that Claim 2 holds true. Note that unlike in the proof of Theorem 2.4, here the contradiction is achieved by the sign-changing property of eigenfunctions belonging to eigenvalues bigger than λ₁. We already know that u₊ ∈ [0, α_ae]. Assume that u ∈ V₀ is any positive solution of (1.7) belonging to [0, α_ae]. Since u ∈ L^∞(Ω), then by (1.7) and (g3) we deduce Δ_pu ∈ L^∞(Ω). Using [27, Theorem 1.56] we derive $$, and applying Vázquez′s strong maximum principle [28] we infer $$, which yields u ∈ [(1/n)φ₁, α_ae] for n sufficiently large. This in conjunction with the fact that u_n is the least solution of (1.7) in [(1/n)φ₁, α_ae] ensures u_n ≤ u if n is large enough. In view of (3.21), we obtain u₊ ≤ u, which proves Claim 3.

The proof of the existence of the greatest negative solution $$ of (1.7) within the ordered interval [−β_be, 0] can be done in a similar way. This completes the proof of the theorem.

Lemma 3.11. Let u₊ and u₋ be the extremal constant-sign solutions of (1.7). Then the following holds.

• (i)

  A critical point v ∈ V₀ of E₊ is a (nonnegative) solution of (1.7) satisfying 0 ≤ v ≤ u₊.

• (ii)

  A critical point w ∈ V₀ of E₋ is a (nonpositive) solution of (1.7) satisfying u₋ ≤ w ≤ 0.

• (iii)

  A critical point z ∈ V₀ of E₀ is a solution of (1.7) satisfying u₋ ≤ z ≤ u₊.

Proof. To prove (i) let v be a critical point of E₊, that is, 0 ∈ ∂E₊(v), which results in

for some w ∈ L^q′(Ω) and such that w(x) ∈ ∂G₊(x, v(x)) almost everywhere in Ω, with Let us show that v ≤ u₊ holds. As u₊ is a positive solution of (1.7), it satisfies the first equation in (3.43), and by subtracting that equation from (3.47) and applying the special test function φ = (v − u₊) ⁺ we get By the definition of the truncations introduced above we have $$ and w(x) − η₊(x) = 0 for a.a. x ∈ {v > u₊}, and thus the right-hand side of (3.49) is zero which leads to and hence it follows ∇(v − u₊) ⁺ = 0. Because (v − u₊) ⁺ ∈ V₀, this implies (v − u₊) ⁺ = 0 which proves v ≤ u₊. To prove 0 ≤ v we test (3.47) with φ = v⁻ = max {−v, 0} ∈ V₀ and get which results in $$, and thus v⁻ = 0, that is, 0 ≤ v. Thus the critical point v of E₊ which is a solution of the Dirichlet problem (3.47) satisfies 0 ≤ v ≤ u₊, and therefore τ₊(x, v) = v. Because ∂G₊(x, v(x)) ⊂ ∂G(x, v(x)), it follows w ∈ ∂G(x, v(x)), and therefore v must be a solution of (1.7). This proves (i). In just the same way one can prove also (ii) and (iii) which is omitted.

Lemma 3.12. Let a > λ₁ and b > λ₁. Then the extremal positive solution u₊ of (1.7) is the unique global minimizer of the functional E₊, and the extremal negative solution u₋ of (1.7) is the unique global minimizer of the functional E₋. Both u₊ and u₋ are local minimizers of E₀.

Proof. The functional E₊ : V₀ → ℝ is bounded below, coercive, and weakly lower semicontinuous. Thus there exists a global minimizer v₊ ∈ V₀ of E₊, that is,

As v₊ is a critical point of E₊, so by Lemma 3.11 it is a nonnegative solution of (1.7) satisfying 0 ≤ v₊ ≤ u₊. Since a − λ₁ > 0, there is a ν_a > 0 such that a − λ₁ − ν_a > 0. By (g3) we infer the existence of a $$ such that and thus for ε > 0 sufficiently small such that we obtain (note ∥φ₁∥_p = 1) Hence it follows that E₊(v₊) < 0, and thus v₊ ≠ 0. Applying nonlinear regularity theory for the p-Laplacian (cf., e.g., [27, Theorem 1.5.6]) and Vázquez′s strong maximum principle, we see that $$. As u₊ is the smallest positive solution of (1.7) in [0, α_ae] and 0 ≤ v₊ ≤ u₊, it follows v₊ = u₊, which shows that the global minimizer v₊ must be unique and equal to u₊. By similar arguments one can show that the global minimizer v₋ of E₋ must be unique and v₋ = u₋. It remains to prove that u₊ and u₋ are local minimizers of E₀. Let us show this last assertion for u₊ only. By definition we have Since u₊ is a global minimizer of E₊ and $$, it follows that u₊ is a local minimizer of E₀ with respect to the C¹ topology. Due to a result by Motreanu and Papageorgiou in [39, Proposition 4], we conclude that u₊ is also a local minimizer of E₀ with respect to the V₀ topology. This completes the proof of the lemma.

Lemma 3.13. The functional E₀ : V₀ → ℝ has a global minimizer v₀ which is a nontrivial solution of (1.7) satisfying u₋ ≤ v₀ ≤ u₊.

Proof. One easily verifies that E₀ : V₀ → ℝ is coercive and weakly lower semicontinuous, and thus a global minimizer v₀ exists which is a critical point of E₀. Apply Lemma 3.11(iii) and note that, for example, E₀(u₊) = E₊(u₊) < 0, which shows that v₀ ≠ 0.

Theorem 3.14. Let hypotheses (g1)–(g4) and (H) be satisfied. Then problem (1.7) has a smallest positive solution $$ in [0, α_a  e], a greatest negative solution $$ in [−β_be, 0], and a nontrivial sign-changing solution $$ with u₋ ≤ u₀ ≤ u₊.

Proof. Clearly the existence of the extremal positive and negative solution u₊ and u₋ follows from Theorem 3.10, because (H), in particular, implies that a > λ₁ and b > λ₁. As for the existence of a sign-changing solution we first note that by Lemma 3.13 it follows that the global minimizer v₀ of E₀ is a nontrivial solution of (1.7) satisfying u₋ ≤ v₀ ≤ u₊. Therefore, if v₀ ≠ u₊ and v₀ ≠ u₋, then v₀ = u₀ must be a sign-changing solution as asserted, because u₋ is the greatest negative and u₊ is the smallest positive solution of (1.7). Thus, we still need to prove the existence of sign-changing solutions in case that either v₀ = u₋ or v₀ = u₊.

Let us consider the case v₀ = u₊ only, since the case v₀ = u₋ can be treated quite similarly. By Lemma 3.12, u₋ is a local minimizer of E₀. Without loss of generality we may even assume that u₋ is a strict local minimizer of E₀, because on the contrary we would find (infinitely many) critical points z of E₀ that are sign-changing solutions thanks to u₋ ≤ z ≤ u₊ and the extremality of the solutions u₋, u₊ obtained in Theorem 3.10 which proves the assertion.

Therefore, it remains to prove the existence of sign-changing solutions under the assumptions that the global minimizer v₀ of E₀ is equal to u₊, and u₋ is a strict local minimizer of E₀. This implies the existence of ρ ∈ (0, ∥u₋ − u₊∥) such that

where ∂B_ρ(u₋) = {u ∈ V₀   :   ∥u − u₋∥ = ρ}. The functional E₀ satisfies the Palais-Smale condition, because it is bounded below, locally Lipschitz continuous, and coercive; see, for example, [40, Corollary 2.4]. Thus in view of (3.57) we may apply the nonsmooth version of Ambrosetti-Rabinowitz′s Mountain-Pass Theorem (see, e.g., [41, Theorem 3.4]) which ensures the existence of a critical point u₀ ∈ V₀ satisfying 0 ∈ ∂E₀(u₀) and where It is clear from (3.57) and (3.58) that u₀ ≠ u₋  and u₀ ≠ u₊, and thus u₀ is a sign-changing solution provided u₀ ≠ 0. To prove the latter we claim for which it suffices to construct a path $$ such that The construction of such a path $$ can be done by adopting an approach due to the authors in [3] and applying the Second Deformation Lemma for locally Lipschitz functionals as it can be found in [42, Theorem 2.10]. This completes the proof.

Remark 3.15. The multiplicity results and the existence of sign-changing solutions obtained in this section generalize very recent results due to the authors obtained in [3, 4, 7]. Moreover, if the function t ↦ g(x, t) is continuous on ℝ and a = b = λ, then ∂G(x, ξ) = {g(x, ξ)}, and problem (1.7) reduces to

However, even in this setting the results obtained here are more general than obtained in [6, Theorem 3.9], because we do not assume that g(x, t)t ≥ 0 for all t ∈ ℝ.

Remark 3.16. Theorem 3.14 improves also Corollary 3.2 of [8]. In fact, let p = 2, let u ∈ V₀ be a solution of (1.7) in case a = b = λ and g(x, t) ≡ g(t), (x, t) ∈ Ω × ℝ with η ∈ L^q′(Ω) satisfying η(x) ∈ ∂G(u(x)). By definition of Clarke′s gradient we have, for any φ ∈ V₀,

As u is a solution, the following holds: u ∈ V₀ and (p = 2), which yields That is, u turns out to be a solution of the hemivariational inequality studied in [8]. Since the hypotheses of [8, Corollary 3.2] imply (g1)–(g4), the assertion follows.

Remark 3.17. Multiplicity results for a nonsmooth version of problem (1.4) in form of (1.2) can be established under structure conditions for Clarke′s gradient ∂j similar to H(f).

Multiplicity and sign-changing solutions results have been obtained recently in [43] for the following nonlinear Neumann-type boundary value problem: find u ∈ V∖{0} and parameters a, b ∈ ℝ such that

For problem (3.66) conditions on the parameters have been given in terms of the “Steklov-Fučik” spectrum to ensure multiplicity results.