Multiplicity of Solutions for an Elliptic Kirchhoff Equation

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Z. Liu proved, in [9], assuming that f (0) > 0 and under the hypotheses in [6] that all those solutions are ordered, is that to say, w 1,λ < u 1,λ < w 2,λ < u 2,λ < w 3,λ < · · · < u m−1,λ < w m,λ . E. N. Dancer and K. Schmitt have shown in [5] that this relation between the area of the positive hump and the previous negative one is necessary for the existence of a positive solution with norm in the interval corresponding to the positive hump. Indeed, they showed that if 0 < α < β are two consecutive zeroes of f such that f (s) > 0 for every s ∈ (α, β) then a necessary condition for the existence of solution (1.1) In [7] J. García-Melián and L. Iturriaga have studied problem (P λ,0 ) for a nonnegative function f with r positive zeros, α 1 , . . . , α r . They showed that, for large enough λ, there exist at least two solutions w k,λ , u k,λ with w k,λ L ∞ (Ω) < α k < u k,λ L ∞ (Ω) if f verifies a suitable non-integrability condition near each of its zeros. Specifically this condition is: there exists δ > 0 such that for k = 1, ..., r. In addition, they obtained that They used the sub-and supersolutions method and topological degree arguments combined with the use of a suitable Liouville theorem. The above condition (1.2) was improved in [3] by B. Barrios, J. García-Melián and L. Iturriaga under the assumption that f has an isolated positive zero α such that They proved, still for nonnegative f , that for large enough λ there exist at least two ordered positive solutions w α,λ < u α,λ verifying the properties w α,λ L ∞ (Ω) < α < u α,λ L ∞ (Ω) and w α,λ (x), u α,λ (x) → α uniformly on compact subsets of Ω as λ → +∞. The existence of these solutions holds independently of the behavior of f near zero or infinity.
In this paper we are interested in the multiplicity results for the Kirchhoff problem (P λ,γ ) (in the case γ > 0) providing a description of the behavior of solutions for λ → +∞. Thus, in addition to f (0) ≥ 0 we only assume (H) There exists 0 < α < β such that f (α) = f (β) = 0, f (s) > 0 for every s ∈ (α, β) and We observe that f may change sign and that hypothesis (H) implies that (1.1) is satisfied for every r ∈ (r α , β).
In order to state the main result we use the following notation: Theorem 1.1. Assume that f is a locally Lipschitz continuous real function with f (0) ≥ 0 and satisfies (H). Assume also that G ∈ C 1 (R + ) with G, G ≥ 0 and γ ≥ 0. Then there exists λ > 0 such that, for every λ > λ, problem (P λ,γ ) admits at least . Moreover, one of the following alternatives holds: 1. There exists an unbounded continuum Σ ⊂ S such that Σ λ has at least two elements. 2. S λ is infinite.
We observe that in contrast with the semilinear case, γ = 0, due to the lack of a convenient comparison result for the Kirchhoff problem (P λ,γ ) we have no information on whether the solutions are ordered or not.
With respect to the plan of the paper, in the second section we show the main nonexistence results for (P λ,γ ), we establish the existence of λ and main properties of the solution w λ as a minimum of a truncated functional. The third section is devoted to the existence of the second solution u λ using some Leray-Schauder degree computations which leads to the continuum alternative. Finally, in the last section we collect all the results leading to the proof of the main result.

Qualitative Properties of Solutions
In this section we collect the main results concerning with the existence, nonexistence and main properties of solutions to the problem (P λ,γ ).
Let us remark that we are mainly interested in positive bounded solutions and the interaction between its L ∞ (Ω) norm and the positive zeroes of f . Thus, for 682 D. Arcoya et al. Vol. 90 (2022) every k > 0, we may take and consider the truncated problem This truncated problem provides us with bounded solutions to (P λ,γ ) as it is stated in the following lemma.
The next result is related to the nonexistence of bounded solutions with a specific L ∞ (Ω) norm.

Lemma 2.2.
There is no bounded solution to (P λ,γ ) with L ∞ (Ω) norm equal to r > 0 in the following cases: Thus, the first item is a direct consequence of the strong maximum principle (f (r) ≤ 0, see [2]) and the second is proved in [5] ( r s 0f (s)ds ≤ 0). Remark 2.3. Combining the results in the previous two lemmas, in the case f (k) = 0, any solution u to (P k ) it is also a solution to (P λ,γ ) satisfying u L ∞ (Ω) < k. In addition, we may consider the functional I k : H 1 0 (Ω) → R given by and observe that I k ∈ C 1 (H 1 0 (Ω)) and critical points of I k are solutions to (P k ). The existence of solution to (P k ) when f (k) = 0 is then standard as it is stated in the following lemma: Proof. Observe that, since f k is bounded (which also implies that any solution of (P k ) belongs to L ∞ (Ω)), the functional I k is coercive. In addition, I k is weak lower semicontinuous which implies that I k has a global minimum w λ which, taking into account Remark 2.3, is a solution of (P λ,γ ).
The solution given by Lemma 2.4 may be the trivial one and in the next result we give sufficient conditions to prove that it is not the trivial one. The proof follows closely that of [6] and we include it here for the reader convenience. Lemma 2.5. Assume that f (k) = 0 for some k > 0 and that for some Proof. Assume that w λ L ∞ (Ω) < k − ε and observe that, in this case Therefore, using (2.2) we have: Now we fix δ > 0 such that η ≡ |Ω| s 1 s 0 f (t)dt − C|Ω δ | > 0 and we have that On the other hand for λ > λ ε ≡ 1 This implies that w λ L ∞ (Ω) > k − ε for every λ > λ ε and we finish the proof.
The following result is concerned with the existence of solution having L ∞ (Ω)norm between two consecutive zeros 0 < α < β when (H) is satisfied. Proof. Using Lemma 2.4 we have the existence of a global minimum w λ of I β which is a solution to problem (P λ,γ ). This solution may be the trivial one and we will prove now that, for large enough λ, w λ L ∞ (Ω) ∈ (r α , β). This is a consequence of Lemma 2.5 with k = β, ε = β − α and s 1 > r α . (Moreover, as in the proof of Lemma 2.5 we also have that lim λ→+∞ I β (w λ ) = −∞). As a consequence [λ, +∞) ⊂ Λ for any λ > λ . In particular Λ = ∅ and we can take λ = inf Λ. Observe that, since (P 0,γ ) admits only the trivial solution, 0 ∈ Λ. Now we prove that Λ is closed and as a consequence λ > 0. Let λ n be a convergent sequence in Λ and denote λ = lim n→∞ λ n . Let take u n ∈ H 1 0 (Ω) ∩ L ∞ (Ω) with r α < u n L ∞ (Ω) < β satisfying . In addition, applying [8, Theorem 6.1] we deduce that the sequence u n is bounded in C 0,γ (Ω). Consequently, Ascolí-Arzelá Theorem assures that u n possesses a subsequence converging strongly in Moreover, taking u n as test function and using that f is continuous we get that u n is bounded in H 1 0 (Ω). This implies, in particular, that u ∈ H 1 0 (Ω) and, up to a subsequence, u n → u weakly in H 1 0 (Ω). In addition, taking u n − u as test function, we can also assure that u n strongly converges to u in H 1 0 (Ω) and we can pass to the limit and it follows that Since u solves (P λ,γ ) the strong maximum principle allows us to assure that r α = u L ∞ (Ω) = β (see Lemma 2.2). Therefore we have proved that λ ∈ Λ, i.e. Λ is closed and we conclude the proof.

Remark 2.8.
Observe that when f (k) = 0, Lemma 2.4 also gives a bounded solution to (P k ) which is not necessarily a solution to (P λ,γ ) unless it is less than k. Moreover, Lemma 2.5 is also true for solutions to (P k ). Thus, if α < k < β, with α, β as in Remark 2.7 we have that the norm of this bounded solution to (P k ) has to be greater than k.

Leray-Schauder Degree Computations
In this section we perform the Leray Schauder degree computations to obtain the multiplicity result, again we follow closely [6]. Vol. 90 (2022)

Multiplicity of Solutions for an Elliptic Kirchhoff Equation 685
Observe that u solves (P λ,γ ) and u L ∞ (Ω) ≤ k if and only if Moreover, Φ λ,k : H 1 0 (Ω) → R is compact and we can use Leray-Schauder degree theory.
Proof. Let us consider H(t, u) = u − tΦ λ,k (u), for every t ∈ [0, 1] and u ∈ H 1 0 (Ω). Since solutions to H(t, u) = 0 are bounded in L ∞ (Ω) we have that they are also bounded in H 1 0 (Ω) and, for big enough R > 0, H is a valid homotopy and the result follows.
Observe that R depends on λ and k but it can be chosen constant in compact sets. Now we denote by S k to the set of solutions to (P λ,γ ) with L ∞ (Ω)-norm less that k. Recall that, when f (k) = 0, S k is the set of solutions to (P k ). Let us denote Let us consider H(t, u) = u − tΦ λ,α (u) + (1 − t)Φ λ,β (u), for every t ∈ [0, 1] and u ∈ H 1 0 (Ω). We claim that H is a valid homotopy for small ε which implies the final result. Indeed, assume on the contrary that for some ε n → 0, t n → t ∈ [0, 1] and u n ∈ ∂U ε n ,α we have that Whenever u n L ∞ (Ω) ≤ α, since f β (u n ) = f α (u n ) we have that u n ∈ S α which is a contradiction with the fact that u n ∈ ∂U ε n ,α , in particular u n L ∞ (Ω) > α. Moreover, since f β (s) = f α (s) = 0 for every s ≥ β, using Lemma 2.2 we have that u n L ∞ (Ω) < β.
Moreover, taking u n as test function we get that u n is bounded in H 1 0 (Ω). This implies, in particular, that u ∈ H 1 0 (Ω) and, up to a subsequence, u n → u weakly 686 D. Arcoya et al. Vol. 90 (2022) in H 1 0 (Ω). In addition, taking u n − u as test function, we can also assure that u n strongly converges to u in H 1 0 (Ω) and we can pass to the limit and it follows that − 1 + γG ∇u 2 Even more, since ε n → 0 and u n ∈ U ε n ,α we have that u ∈ S α which also implies that u L ∞ (Ω) < α and this is a contradiction.

There exists un unbounded continuum
Proof. Observe that whenever w λ 0 is isolated, taking into account that it is a minimum for I β , it is proved in [10, Theorem 1.1] that Hence we may use [1,Theorem 4.4.1] to deduce the existence of the unbounded continuum Σ with (λ 0 , w λ 0 ) ∈ Σ. For the existence of the second solution v λ 0 with (λ 0 , v λ 0 ) ∈ Σ we use the ideas of [1,Theorem 4.4.2] in the interval [a, b] with a < λ and b = λ 0 , U 1 = B ε (w λ 0 ) and taking into account that U may depend continuously on λ. Therefore we take By construction we have that (P λ,γ ) has no solution on ∂V ε,α (λ) for every λ ∈ [a, b] and (P a,γ ) has no solutions in V ε,α (a) and the result follows as in [1,Theorem 4.4.2].
As a consequence of Lemma 2.5 (see also Remark 2.7) we have that Thus, only remains to prove that, if we assume that f is nonnegative and it satisfies (1.3) then lim u μ n L ∞ (Ω) = α for some divergent sequence μ n . Indeed, in this case r α = α and this is a consequence of the results in [3] where it is proved the existence of v λ solution to −Δu = λf (u) in Ω, with lim λ→+∞ v λ L ∞ (Ω) = α and v λ L ∞ (Ω) > α. Thus, given λ n → +∞ we can and u μ n = v λ n . Since G ≥ 0 we have that μ n → +∞ and the proof is completed.
Remark 4.1. Let us observe that the multiplicity result cannot be deduced directly from the know results for the semilinear case. More precisely, given u a solution to the semilinear problem (4.1) then u is also a solution to with μ = 1 + γG ∇u 2 λ. This means that, for fixed λ, multiple solutions to (4.1) may lead to solutions to (4.2) for different values of the parameter μ. This is not enough to describe the global behavior in the case γ = 0 since the reverse depend on G and u H 1 0 (Ω) : , and we can assure that u μ is a solution to (4.1) for this value of λ but we cannot assure that it is precisely v λ .

Remark 4.2.
As in the semilinear case, depending on the behavior of f below α and above β, the results proved in this paper may lead to the existence of solutions with L ∞ (Ω) norm less that α or greater than β. Even more, the existence of multiple zeroes of f may lead to a more general multiplicity result.
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