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This paper deals with the existence of entire nontrivial solutions for critical quasilinear systems (𝓢) in the Heisenberg group ℍn, driven by general (p, q) elliptic operators of Marcellini types. The study of (𝓢) requires relevant topics of nonlinear functional analysis because of the lack of compactness. The key step in the existence proof is the concentration–compactness principle of Lions, here proved for the first time in the vectorial Heisenberg context. Finally, the constructed solution has both components nontrivial and the results extend to the Heisenberg group previous theorems on quasilinear (p, q) systems.

In this paper, we prove the existence of a nontrivial (weak) solution of the Brézis–Nirenberg equation for the $ (m, p) $ Laplacian in the whole space $ \mathbb R^N $. Critical problems have been intensively studied in the last decades, starting with the pioneering paper by Brézis and Nirenberg for the Dirichlet Laplacian problems in bounded domains of $ \mathbb R^N $. Even if we obtain existence of solutions for the $ (m, p) $ Laplacian Brézis–Nirenberg equation in the whole space via standard variational techniques, the last step of the proof is pretty involved and requires new ideas and a delicate use of the Talenti functions. [ABSTRACT FROM AUTHOR]

In this paper we establish some variants of the celebrated concentration-compactness principle of Lions - CC principle briefly - in the classical and fractional Folland-Stein spaces. In the first part of the paper, following the main ideas of the pioneering papers of Lions, we prove the CC principle and its variant, that is the CC principle at infinity of Chabrowski, in the classical Folland- Stein space, involving the Hardy-Sobolev embedding in the Heisenberg setting. In the second part, we extend the method to the fractional Folland-Stein space. The results proved here will be exploited in a forthcoming paper to obtain existence of solutions for local and nonlocal subelliptic equations in the Heisenberg group, involving critical nonlinearities and Hardy terms. Indeed, in this type of problems a triple loss of compactness occurs and the issue of finding solutions is deeply connected to the concentration phenomena taking place when considering sequences of approximated solutions. [ABSTRACT FROM AUTHOR]

We complete the study started in the paper [P. Pucci, L. Temperini, On the concentration-compactness principle for Folland-Stein spaces and for fractional horizontal Sobolev spaces, Math. Eng. 5 (2023), Paper no. 007], giving some applications of its abstract results to get existence of solutions of certain critical equations in the entire Heinseberg group. In particular, different conditions for existence are given for critical horizontal p-Laplacian equations. [ABSTRACT FROM AUTHOR]

In this article, we investigate multiplicity results for Choquard-Kirchhoff type equations, with Hardy-Littlewood-Sobolev critical exponents, − (a + b ∫RN |∇u|2dx) Δu = αk(x)|u|q−2u + β (∫RN |u(y)|2*μ/|x−y|μ dy) |u|2*μ−2u, x ∈ RN, where a > 0, b ≥ 0, 0 < μ < N, N ≥ 3, α and β are positive real parameters, 2*μ = (2 N − μ) / (N − 2) is the critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, k ∈ Lr(ℝN), with r = 2∗/(2∗ − q) if 1 < q < 2* and r = ∞ if q ≥ 2∗. According to the different range of q, we discuss the multiplicity of solutions to the above equation, using variational methods under suitable conditions. In order to overcome the lack of compactness, we appeal to the concentration compactness principle in the Choquard-type setting. [ABSTRACT FROM AUTHOR]

In this paper we complete the study started in [P. Pucci, L. Temperini, Existence for (p,q) critical systems in the Heisenberg group, Adv. Nonlinear Anal. 9 (2020), 895-922] on some variants of the concentration-compactness principle in bounded PS domains of the Heisenberg group Hn. The concentration-compactness principle is a basic tool for treating nonlinear problems with lack of compactness. The results proved here can be exploited when dealing with some kind of elliptic systems involving critical nonlinearities and Hardy terms. [ABSTRACT FROM AUTHOR]

The paper is concerned with existence of nonnegative solutions of a Schrödinger–Choquard–Kirchhoff-type fractional p-equation. As a consequence, the results can be applied to the special case (a + b ⁢ ∥ u ∥ s p ⁢ (θ - 1) ) ⁢ [ (- Δ) p s ⁢ u + V ⁢ (x) ⁢ | u | p - 2 ⁢ u ] = λ ⁢ f ⁢ (x , u) + (∫ ℝ N | u | p μ , s * | x - y | μ ⁢ 𝑑 y ) ⁢ | u | p μ , s * - 2 ⁢ u in ⁢ ℝ N , (a+b\|u\|_{s}^{p(\theta-1)})[(-\Delta)^{s}_{p}u+V(x)|u|^{p-2}u]=\lambda f(x,u)% +\Bigg{(}\int_{\mathbb{R}^{N}}\frac{|u|^{p_{\mu,s}^{*}}}{|x-y|^{\mu}}\,dy% \Biggr{)}|u|^{p_{\mu,s}^{*}-2}u\quad\text{in }\mathbb{R}^{N}, where ∥ u ∥ s = ( ∬ ℝ 2 ⁢ N | u ⁢ (x) - u ⁢ (y) | p | x - y | N + p ⁢ s ⁢ 𝑑 x ⁢ 𝑑 y + ∫ ℝ N V ⁢ (x) ⁢ | u | p ⁢ 𝑑 x ) 1 p , \|u\|_{s}=\Bigg{(}\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^{p}}{|x-y|^{N+ps}}% \,dx\,dy+\int_{\mathbb{R}^{N}}V(x)|u|^{p}\,dx\Biggr{)}^{\frac{1}{p}}, a , b ∈ ℝ 0 + {a,b\in\mathbb{R}^{+}_{0}} , with a + b > 0 {a+b>0} , λ > 0 {\lambda>0} is a parameter, s ∈ (0 , 1) {s\in(0,1)} , N > p ⁢ s {N>ps} , θ ∈ [ 1 , N / (N - p ⁢ s)) {\theta\in[1,N/(N-ps))} , (- Δ) p s {(-\Delta)^{s}_{p}} is the fractional p-Laplacian, V : ℝ N → ℝ + {V:\mathbb{R}^{N}\rightarrow\mathbb{R}^{+}} is a potential function, 0 < μ < N {0<\mu

In this paper we complete the study started in [Existence of entire solutions for quasilinear equations in the Heisenberg group, Minimax Theory Appl. 4 (2019)] on entire solutions for a quasilinear equation (ελ) in ℍn, depending on a real parameter λ, which involves a general elliptic operator Α in divergence form and two main nonlinearities. Here, in the so called sublinear case, we prove existence for all λ > 0 and, for special elliptic operators Α, existence of infinitely many solutions (uk)k. [ABSTRACT FROM AUTHOR]

The paper deals with Kirchhoff type equations on the whole space R N , driven by the p -fractional Laplace operator, involving critical Hardy–Sobolev nonlinearities and nonnegative potentials. We present different variational approaches to overcome the lack of compactness at critical levels, due to the presence of critical terms as well as the possibly degenerate nature of the Kirchhoff problem. [ABSTRACT FROM AUTHOR]

Abstract: Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that admits a positive weak solution in of class , whenever , and . The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if is a weak solution in of , then when either , or and u is also of class . [Copyright &y& Elsevier]

This paper deals with the existence of entire nontrivial solutions for fractional (p , q) systems with critical Hardy terms in R N. Existence of solutions for system (S) is derived via the mountain pass lemma, whose application is pretty involved and requires some variants of the concentration–compactness principle of Lions and of the concentration–compactness principle at infinity of Chabrowski for vectorial fractional spaces, here proved. Indeed, the "triple loss of compactness" in (S), caused by the simultaneous presence of Hardy and critical terms in the whole R N , forces to study the exact behaviour of the P S sequences, in the spirit of Lions. We also present a simplified version (S ˜) of the main system (S), which is anyway interesting, since the existence proof for (S ˜) is simpler and more direct. Finally, both existence theorems improve or complement previous results for fractional (p , q) scalar and vectorial problems. [ABSTRACT FROM AUTHOR]