Existence, uniqueness, localization and minimization property of positive solutions for non-local problems involving discontinuous Kirchhoff functions

Let $\Omega\subset {\bf R}^n$ be a smooth bounded domain. In this paper, we prove a result of which the following is a by-product: Let $q\in ]0,1[$, $\alpha\in L^{\infty}(\Omega)$, with $\alpha>0$, and $k\in {\bf N}$. Then, the problem $$\cases {-\tan\left(\int_{\Omega}|\nabla u(x)|^2dx\right)\Delta u= \alpha(x)u^q&in $\Omega$\cr&\cr u>0&in $\Omega$\cr&\cr u=0&on $\partial \Omega$ \cr&\cr (k-1)\pi<\int_{\Omega}|\nabla u(x)|^2dx<(k-1)\pi+{{\pi}\over {2}} \cr}$$ has a unique weak solution $\tilde u$ which is the unique global minimum in $H^1_0(\Omega)$ of the functional $$u\to {{1}\over {2}}\tan\left (\int_{\Omega}|\nabla\tilde u(x)|^2dx\right)\int_{\Omega}|\nabla u(x)|^2dx-{{1}\over {q+1}}\int_{\Omega}\alpha(x)|u^+(x)|^{q+1}dx\ ,$$ where $u^+=\max\{0,u\}$.

https://doi.org/10.1515/anona-2023-0104 received March 20, 2023; accepted December 6, 2023 Abstract: Let ⊂ R Ω n be a smooth bounded domain.In this article, we prove a result of which the following is a by-product: Let ∈ q 0, 1 ] [, ∈ ∞ α L Ω ( ), with > α 0, and ∈ k N.Then, the problem has a unique weak solution u ˜, which is the unique global minimum in H Ω

Introduction
First, we stress that we have chosen the above long title just to summarize the main features and novelties of our results.
We define the weak solution to this problem as any This is a Kirchhoff-type problem, with K being the Kirchhoff function.Unquestionably, it is among the most studied nonlinear problems of the last two decades.For a lucid introduction to the subject jointly with the relevant bibliography, we refer to the recent article by Pucci and Rădulescu [5].
Here is a most remarkable corollary of our main result.
has a unique weak solution u ˜, which is the unique global minimum in H Ω 0 1 ( ) of the functional Moreover, u ˜satisfies the inequality where The main novelties of Theorem 1. Actually, as far as we know, the continuity of K in +∞ 0, [ [ is an assumption present in each article devoted to this subject, except the one by Ricceri [9].More precisely, in the article by Ricceri [9], we assumed that, for some > r 0, K is continuous and increasing in r 0, cannot be covered by any of the results known up to now.Thanks to Theorem 1.1, each of these problems has a unique weak solution u ˜, which minimizes the functional (1.1).But even when = +∞ I 0, ] [, Theorem 1.1 turns out to be new.Actually, the known results for the problem , with > a b , 0 ([10], Theorem 1.2; [11], Corollary 1.1) or with > a 0 and ≥ b 0 ([4], Theorem 5.4).

Results
If X is a topological space, a function is compact.We will obtain our main result via the following abstract theorem.
Proof.Clearly, the function − Ψ 1 is increasing and continuous in +∞ , we think of − Ψ 1 as an increasing and continuous function in +∞ 0, ( ) is concave, while, for each > λ 0, the function ⋅ φ λ , ( ) is lower semicontinuous, inf-compact and admits a unique global minimum.Consequently, in view of Theorem 1.1 of [8], we have But, by continuity, we have and so from which it follows that Of course, By assumption, there is some ∈ , and so we have ) is sup-compact.Then, from equality (2.1), it follows that there exists and Note that > λ ˜0.Indeed, otherwise, by equality (2.3), x ˜would be a global maximum of J , against an assump- tion.On the other hand, by equality (2.2), it follows that , and equality (2.3) provides the conclusion for the existence part.Now, let us prove the uniqueness of x ˜.So, let ∈ y X ˜be such that ∈ y I Φ ( ) and Arguing by contradiction, assume that ≠ y x ˜˜.We claim that , and has a unique weak solution u ˜, which is the unique global minimum in Proof.First of all, extend f to R putting = f ξ 0 ( ) for all < ξ 0. We are going to apply Theorem 2.1, taking = X H Ω 0 1 ( ) endowed with the weak topology, = K Ψ , and defining Φ and J by for all ∈ u H Ω 0 1 ( ), where The functionals Φ and J are C 1 with derivatives given by ).Moreover, due to the sub-critical growth of f , J is sequentially weakly continuous.Fix , where λ 1 is the constant defined in Theorem 1.1.Since , there is and so for all ∈ u H Ω 0 1 ( ).Hence, due to the choice of ε, we have Non-local problems involving discontinuous Kirchhoff functions  5 This fact, jointly with the reflexivity of H Ω 0 1 ( ) and the Eberlein-Smulyan theorem, implies that the sequentially weakly lower semicontinuous functional − λ J Φ is weakly inf-compact.We now show that it has a unique global minimum in H Ω 0 1 ( ).Indeed, its critical points are exactly the weak solutions of the problem In turn, since the right-hand side of the equation is non-negative, the non-zero weak solutions of the problem are positive in Ω.Moreover, since, for each ∈ x Ω, the function This shows that 0 is not a global minimum for the functional − λ J Φ .Consequently, the global minimum of this functional agrees with its only non-zero critical point.Finally, let us show that J has no global maxima.Arguing by contradiction, suppose that ∈ u H ˆΩ 0 1 ( ) is a global maximum of J .Clearly, > J u ˆ0 ( ) .Consequently, the set 1 are the lack of continuity of K in +∞ 0and the property that u ˜minimizes the functional (1.1) (which depends on u ˜itself).

Remark 2 . 1 .
)) } has a positive measure.Fix a closed set ⊂ C A of positive measure.Let ∈ v absurd.Therefore, each assumption of Theorem 2.1 is satisfied.As a consequence, there exists a unique ∈ u what seen above, the function u ˜satisfies the conclusion.□ In Theorem 1.1, the inequality satisfied by the solution follows directly from Theorem 1.2 of[7].
[2]orem 1 of[2]ensures that the problem has at most one positive weak solution, and so it has at most one non-zero weak solution.As a consequence, we infer that the functional