Existence of positive radial solution for Neumann problem on the Heisenberg group

The existence of at least one positive radial solution of the Neumann problem −ΔHnu+R(ξ)u=a(|ξ|Hn)|u|p−2u−b(|ξ|Hn)|u|q−2u,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ -\Delta _{\mathbb{H}^{n}} u+R(\xi ) u=a \bigl( \vert \xi \vert _{\mathbb{H}^{n}} \bigr) \vert u \vert ^{p-2} u - b\bigl( \vert \xi \vert _{\mathbb{H}^{n}}\bigr) \vert u \vert ^{q-2}u, $$\end{document} is proved on the Heisenberg group Hn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathbb{H}^{n}$\end{document}, via the variational principle, where a(|ξ|Hn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$a(|\xi |_{\mathbb{H}^{n}})$\end{document}, b(|ξ|Hn)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$b(|\xi |_{\mathbb{H}^{n}})$\end{document} are nonnegative radial functions and R(ξ)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$R(\xi )$\end{document} satisfies suitable conditions.


Introduction
Semilinear elliptic equations were the first nonlinear generalization of linear elliptic partial differential equations. They are of fundamental importance for the study of engineering, geometry, life sciences, economics, physics and mechanics; see [1,[23][24][25][26][27]. Some examples, like the theory of Bose-Einstein condensates, obey The variational approach to these equations has experienced spectacular success in recent years, reaching a high level of complexity and refinement, with a multitude of applications (see [10,[14][15][16]22]).
In [3] and [12] some problems which depend on continuous component of time like coherent states in quantum optics are probed. These problems are studied in a space which have a component of time and are known as Heisenberg group. Important topics where the Heisenberg group reveals itself as an essential factor are quantum mechanics, ergodic theory, representation theory of nilpotent Lie group, foundation of abelian harmonic analysis, and the theory of partial differential equations. We are now interested in the last one.
In this paper, the existence of at least one radial solution of a generalized well-known supercritical Neumann problem is proved where Ω is the Korányi ball in the Heisenberg group. Recently, the existence of a nontrivial weak solution of following singular boundary value problem on the Heisenberg group was proved (see [29]): In [18] problem (1.1) is studied where R(ξ ) = 1 on the unit ball in R n , n ≥ 2, see [2,17,21,22] for some applications.
Here we recall some definitions and results on the Heisenberg group (see [4,7,30]). Let us recall some features of the Heisenberg group. The Heisenberg group H n = (R 2n+1 , •) is the space R 2n+1 with the noncommutative law of product where x 1 , x 2 , y 1 , y 2 ∈ R n , t 1 , t 2 ∈ R and , denotes the standard inner product in R n . This operation endows H n with the structure of a Lie group. The Lie algebra of H n is generated by the left-invariant vector fields These generators satisfy the noncommutative formula Let z = (x, y) ∈ R 2n and ξ = (z, t) ∈ H n . The parabolic dilation satisfies δ λ (ξ 0 • ξ ) = δ λ ξ • δ λ ξ 0 , and |ξ | H n = |z| 4 + t 2 1 4 = x 2 + y 2 2 + t 2 1 4 , is a norm with respect to the parabolic dilation which is known as Korányi gauge norm N(z, t). The Heisenberg distance between two points (z, t) and (z , t ) is given by Clearly, the vector fields X i , Y i , i = 1, 2, . . . , n are homogeneous of degree 1 under the norm | · | H n and T is homogeneous of degree 2. The Korányi ball of center ξ 0 and radius r is defined by and it satisfies The Heisenberg gradient and Kohn-Laplacian (Heisenberg-Laplacian) operator on H n are given by respectively. Let Ω ⊂ H n , n ≥ 1, be a smooth bounded open set. Its associated Sobolev space is defined as follows: and closure of C ∞ 0 (Ω) in H 1 (Ω, H n ) under the norm The following norm is a norm on H 1 0 (Ω, H n ): which is equivalent to the standard one. The dual space of H 1 0 (Ω, H) will be denoted by H -1 (Ω, H). Here we recall Hardy's inequality and some results on the Heisenberg group.
Q-2 is critical exponent of Q = 2n + 2, which is a homogeneous dimension of H n . We denote the Sobolev embedding constant of the above compact embedding by C s > 0; i.e.
Now we recall results we need later. Let V be a real Banach space and V * be its topological dual and pairing between V and V * denoted by , for each u ∈ V and any sequence {u n } converging to u in the weak topology σ (V , V * ). Let ψ : V → R ∪ {+∞} be a proper convex function. The subdifferential of ψ at u is the set The multivalued mapping ∂ψ :

Lemma 1.3 ([1]) Let X be a reflexive Banach space and I
Suppose the functional I : V → (-∞, +∞] is defined by Notice that a global minimum point of I is a critical point of I.

Definition 1.2 ([11]) We say that I satisfies the Palais-Smale compactness condition (PS)
if every sequence {u n } is such that where ε n → 0, then {u n } possesses a convergent subsequence.
The following Mountain Pass Geometry (MPG) theorem is proved in [28].

Theorem 1.3 Suppose that I : V → (-∞, +∞] is of the form (1.3) and satisfies the PS condition and the following conditions:
Subsequently, we need the following theorem.
where c ≥ 0 is a small enough constant.

Positive radial solution
Here, we recall the following variational principle established in [5].
over the convex and weakly closed set K = u ∈ V : u ≥ 0, u is increasing with respect to the radius = |ω| H n . (2.1) To adapt Theorem 2.1 to our problem, we define ψ, ϕ : V → R by Finally, let us introduce the functional I : V → (-∞, +∞] as follows: where ψ K is defined as (1.2). We should be aware that I is indeed the Euler-Lagrange functional corresponding to our problem restricted to K . Theorem 2.1 implies the following corollary.
(Ω), and K be the convex closed subset of V given in (2.1). Assume that the functional has a critical point u ∈ V and there exists v ∈ K satisfying the linear equation in the weak sense. Then u ∈ K is a solution of the equation Notice that satisfying Eq. (2.3) shows that the triple (ψ K , ϕ, K) satisfies the point-wise invariance condition at u, where G = 0. We need some lemmas and theorems before proving our main result.

Lemma 2.1 Assume R(ξ ) is given in Theorem
for all t ∈ R. Then the problem

4)
admits at least one solution.
Proof First notice that by integration, there exist α 1 , β 1 > 0 such that ds. Now consider the following energy functional corresponding to problem (2.4) on D 1,2 0 (Ω) ∩ D 2,2 (Ω): So J is coercive and weakly lower semi-continuous on D 1,2 0 (Ω). So according to the Weierstrass theorem, J has a global minimum point which means problem (2.4) admits at least one solution.
where ϕ and ψ K are given in Corollary 2.1. Then I has a nontrivial critical point.
Proof We utilize Theorem 1.3 to prove this lemma. First recall that Dϕ(u) = a |ξ | H n |u| p-2 u, and therefore ϕ is a C 1 -function on the space V . Also notice that ψ is a proper, convex and lower semi-continuous and K is closed in V . We are going to prove this lemma in two steps: Step 1. We verify that I satisfies in MPG conditions. It is clear that I(0) = 0. Take e ∈ K . It follows that Now since p > q ≥ 2, for t sufficiently large I(te) is negative. We now prove condition (iii) of the MPG theorem. Take u ∈ Dom(ψ) with u V = ρ > 0. Notice that from Lemma 2.2 for u ∈ K we have provided ρ > 0 is small enough as 2 < p and C 1 , C 2 , C 3 are positive constants. If u / ∈ Dom(ψ). Then clearly I(u) > 0. Therefore, MPG holds for the functional I.
Step 2. We verify the PS compactness condition. Suppose that {u n } is a sequence in K such that I(u n ) → c ∈ R, n → 0 and (2.7) We show that {u n } has a convergent subsequence in V . First notice that u n ∈ Dom(ψ), then Thus, for large values of n we have Now consider the function g(r) = r qp(r -1)-1 on the interval (1, +∞) and set r * = ( p q ) 1 q-1 . It is easy to see that for every r ∈ (1, r * ) we have g(r) < 0. We choose a number r for which we have r > 1 and r q -1 < p(r -1). In (2.7) set v = ru n . Then (1r) Dϕ(u n ), u n + r q -1 ψ K (u n ) ≥n (r -1) u n V .
in the weak sense.
Proof Let u ∈ K so 0 ≤ u ∈ K ⊂ D 2,2 . According to Theorem 2.4, it is enough to show that h(ξ ) = a(|ξ | H n )u(ξ ) p-1 belongs to L 2 (Ω) and f (v) = -b(|ξ | H n )v(ξ ) q-1 satisfies the inequality f (t) ≤ α + β|t| Q * -1 and f (t)t ≤ 0 for all t ∈ R. Clearly since q < Q * both hold. On the other hand, as u ∈ L Q * (Ω) and Ω ⊂ H n one can see following estimate: where ω 2n is the measure of unit ball in R 2n+1 = H n , and C, C , C , C * are positive constants, and the fundamental theorem of calculus and Hardy's inequality (1.1) are applied. Recent inequalities are satisfied since u is a radial function so |∇ H n u| = r |φ | where = |ξ | H n = |(z, t)| H n , r = |z| (see [6]). Because of the last inequality and since p < Q * , it is easy to see that h ∈ L 2 (Ω). This means the desired result has been obtained. on V , it is converted to a reflexive Banach space. Note that K = u ∈ V : ∂u ∂n = 0 on ∂Ω and u is a radially increasing function is a convex closed subset and we define the functional I = ψ Kϕ which is the Euler-Lagrange functional restricted to K . It follows from the MPG theorem and the PS compactness condition in Lemma 2.3 that the functional I has a critical point u, also it is guaranteed that I(u) > 0. Lemma 2.4 implies that there exists v ∈ Dom(ψ) satisfying the linear equation Dψ K (v) = Dϕ(u). Setting C(ξ ) := 1a(|ξ | H n )u(ξ ) p-1 we have -H n u + C(ξ ) u = 0 in Ω. We show u > 0 in Ω. Otherwise, by applying the strong maximum principle, we deduce u is identically zero in Ω. On the other hand u is a radially increasing function. This a contradiction and so u must be a nontrivial nonnegative solution of (1.1). So the conditions of Corollary 2.1 hold and the proof is completed.
Remark 2.1 Corollary 2.1 implies problem (1.1) has at least one nontrivial radially increasing solution.