Entire solutions to sublinear elliptic problems on harmonic NA groups and Euclidean spaces

<jats:p>We give necessary and sufficient conditions for the existence of entire solutions bounded or large of the equation <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {L}}u - p\psi (u) =0$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:mi>L</mml:mi>
                    <mml:mi>u</mml:mi>
                    <mml:mo>-</mml:mo>
                    <mml:mi>p</mml:mi>
                    <mml:mi>ψ</mml:mi>
                    <mml:mo>(</mml:mo>
                    <mml:mi>u</mml:mi>
                    <mml:mo>)</mml:mo>
                    <mml:mo>=</mml:mo>
                    <mml:mn>0</mml:mn>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula>, where <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathcal {L}}$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mi>L</mml:mi>
                </mml:math></jats:alternatives></jats:inline-formula> is either the Laplace operator on <jats:inline-formula><jats:alternatives><jats:tex-math>$${\mathbb {R}} ^d$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:msup>
                    <mml:mrow>
                      <mml:mi>R</mml:mi>
                    </mml:mrow>
                    <mml:mi>d</mml:mi>
                  </mml:msup>
                </mml:math></jats:alternatives></jats:inline-formula>, <jats:inline-formula><jats:alternatives><jats:tex-math>$$d\ge 3$$</jats:tex-math><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML">
                  <mml:mrow>
                    <mml:mi>d</mml:mi>
                    <mml:mo>≥</mml:mo>
                    <mml:mn>3</mml:mn>
                  </mml:mrow>
                </mml:math></jats:alternatives></jats:inline-formula> or the Laplace–Beltrami operator on the harmonic <jats:italic>NA</jats:italic> group and <jats:italic>p</jats:italic> is a function whose oscillation tends to zero at infinity at a specified rate. The results apply to noncompact rank one symmetric spaces.</jats:p>


Introduction
Let L be the Laplace operator on ℝ d ( (d ≥ 3) ) or the Laplace-Beltrami operator on a harmonic NA group. As a Riemannian manifold such a group is ℝ d with an appropriate leftinvariant metric. 1  Our aim is to give sufficient necessary and sufficient conditions for the existence of entire solutions bounded or "large". A solution u to (1.1) is called large if u(x) → ∞ when d(x, 0) → ∞ where d is the Riemannian distance on NA or the Euclidean distance on ℝ d .
Large solutions, i.e. the boundary blow-up problems are of significant interest due to its various scientific applications in different fields. Such problems arise in the study of Riemannian geometry [3], non-Newtonian fluids [2], the subsonic motion of a gas [13] and the electric potential in some bodies [12].
Throughout this paper, satisfies the following hypotheses: (H 1 ) For every t 0 ∈ [0, ∞) , x ↦ (x, t 0 ) ∈ K loc d (S) , i.e. it is locally in the Kato class in S.
Let Ω be a domain in S . We recall that a Borel measurable function on Ω is locally in the Kato class in Ω if for every open bounded set D, D ⊂ Ω . (H 1 ) makes locally integrable against the Green function 2 for L which plays an important role in our approach.
There is a number of results that indicate that bounded and large solutions cannot occur at the same time. This holds, in particular, in a more general setting of elliptic operators with smooth coefficients, if for every x, (x, ⋅) is concave as a function of the second variable or (x, t) = p(x) (t) and is sublinear [5].
In our setting, we are able to say more. Both spaces ℝ d and harmonic NA have a common feature-global geodesic coordinates and a phenomenon of radiality with respect to the appropriate Riemannian distance d. Suppose that, for every t, (⋅, t) is radial as a function of x, i.e. (x, t) = (d(0, x), t) , see [6]. Then an entire solution always exists, but if it is bounded or large, it depends on the growth of (⋅, c) measured by I (⋅,c) , where for a radial function g and r = d(x, 0) , I g is defined by on Euclidean S and on harmonic S. To obtain I (⋅,c) we fix c and we put g(r) = (r, c) in (1.2) In both spaces, in the radial case, I (⋅,c) is finite if and only if the Green potential of (⋅, c) is well defined. The difference in I (⋅,c) for ℝ d (1.2) and for NA (1.3) is due to the properties of the fundamental solution G to L on each space. More precisely, in both spaces G is radial. On ℝ d , G(x) = a d |x| −d+2 where a d is a constant depending only on the dimension. For NA, we have a precise estimates for G proved in Theorems 21 and 22 in [6].
For radial with respect to the first variable, Theorem 1 goes considerably beyond the previous results. The aim of this paper is to prove an analogous result for (x, t) = p(x) (t) and (H 4 ) or (H 5 ) satisfied without assuming radiality of p. The strategy is to "radialize" p and to make use of Theorem 1. For that we have to control the oscillation of p at infinity, i.e.
where and We define a radial function [see (3.5) and (3.6)] that may grow quite fast at infinity and we assume that which means that p osc is "close" at infinity to a radial function. Our result is For existence of nontrivial bounded solutions, weaker conditions are sufficient, see Theorem 3. For p radial the above result reduces to Theorem 1.
For the Laplace operator in ℝ d and (x, t) = p(x)t where 0 < < 1 and p satisfying some additional hypotheses, the problem of entire solutions was considered in [7,8,11]. Theorems 2 and 3 not only generalize the previous results to a considerably larger class of nonlinearities but give an analogous characterization for harmonic NA groups.
In particular, the result applies to rank one symmetric spaces and It has a bounded solution if I p * < ∞ and a large solution if I p * = ∞ provided I ( p osc ) < ∞ and, if p is additionally bounded then is polynomially growing or exponentially growing depending whether < 1 or = 1.
These types of problems have been studied in a more general framework [5,9,10] where L is a second order elliptic operator with C ∞ coefficients defined in a Greenian The present paper is based on those results. Some of them are in "Appendix", but for most of them we refer to [5,9,10].
At last, the authors want to express their gratitude to Krzysztof Bogdan, Konrad Kolesko, Mohamed Selmi and Mohamed Sifi for their helpful and kindly suggestions.

Bounded solutions
Existence of entire nontrivial bounded solutions to the equation (1.1) is guaranteed by essentially weaker assumptions than those of Theorem 2. In particular does not need to be of the product form. We consider the radial function Proof Suppose that there exists a nonnegative nontrivial bounded solution to (1.1). We have By Proposition 16 in [10], there exists a nonnegative nontrivial bounded solution to the equation Consequently, by Theorem 5 in [6], there exists c 1 > 0 such that I * (⋅,c 1 ) < ∞. Now let us focus on sufficiency of the condition. We are going to apply Theorem 14, see "Appendix" which is a basic tool in obtaining bounded solutions. It is enough to prove that where G S is the Green function for S.
Let G ∶ S → ℝ + be the fundamental solution for L given by G(s) = G S (s, 0) , s ∈ S . G is radial and for S = NA it satisfies the estimates and see [6].
where dm L is the left Haar measure on S. See [6] for more properties of G on NA. From now we continue the proof for NA groups. For ℝ d , we use estimates of the corresponding fundamental solution and follow the same argument. Suppose that there exists c 1 > 0 such that I * (⋅,c 1 ) < ∞. Then by (2.1) I * (⋅,c 1 ) < ∞ 3 and so by (2.5) Writing dm L in the radial coordinates see Section 3 in [6] we have Hence By (H 1 ) , the integral on B(0, 1) is finite and so we conclude that 3 * does not necessarily satisfy (H 1 ) otherwise we could use Proposition 16 in [10] as before.

Large solutions
In this section, we prove Theorem 2. To some extent we follow the ideas of [8]. However, now the nonlinearity is more general and we require an additional regularity of to define and to formulate the result precisely. This is done in Proposition 8 and Corollary 10. Before, we need to develop some preparatory material.
In addition to (1.6) we consider the following equations Proof We have Function is the second ingredient to construct . In fact, we need to dominate by a more regular function 1 . In [8], for (t) = t , such regularization was not needed.

Lemma 7 Suppose that is a positive function satisfying (H 2 ) and
∫ 1 0 (y) dy > 0 . Let 0 < < 1 and let be defined by Then 1 = * ∈ C 1 (ℝ) satisfies: Now we are ready to prove that all radial supersolutions of (3.1) having the same value at 0 can be dominated by means of the function V.

Remark 9
Notice that in view of Lemma 7, and are well defined. 4 For satisfying (H 2 ) -(H 4 ) , we note If in addition, is C 1 integrable at 0 then we can take (r) = ∫ r 0 1 (y) dy.
In particular, for (t) = t for some 0 < < 1 we obtain as in [8].
More generally, notice that and V are strictly increasing and so is F. The growth of 1 is the same as the growth of and it is sublinear. So the growth of may be like log r which makes • to grow exponentially. If grows like t then • grows polynomially. Introducing F as above we cover a much larger range of nonlinearities than in [8].
is continuous. By Lemma 14 of [6] we get Letting tend to zero we get ◻ In fact, the condition (H 4 ) may be replaced by sublinearity of . Given that satisfies (H 2 ) − (H 3 ) and there exists a constant C > 0 such that (t) ≤ C(1 + t) we can find ̃ satisfying (H 2 ) − (H 4 ) and such that ≤̃ , see [5]. C(1 + t) . Let ̃ be a function satisfying (H 2 ) − (H 4 ) and such that ≤̃ . Let F the function constructed in Proposition 8 but for ̃ . Let u be a radial supersolution of (3.1) such that Lu ∈ L 1 loc (S) then Proof Let ̃ and g be as in the proof of Proposition 8 but defined for ̃ . Then

Remark 13
Notice that V defined as in Remark 6 is dominated by (d − 2) −1 ∫ r 0 tp * (t) dt so we may take as it was done in [8].
Equation (3.7) means that p osc must decay quite fast at ∞ and the rate of decay depends both on and p * . Suppose that p * is bounded and not integrable (at infinity). Then by (2.5) and the formula for m L , I p * = ∞ . Then V(r) basically grows linearly. If is linear at ∞ , then • or •̃ grows exponentially; if (t) = t , then • grows polynomially. The same applies to which determines the behavior of p osc at ∞ . If p * is unbounded V(r) may grow faster than linearly and faster than exponentially.
Proof Since F ≥ 1 ( resp. F ≥ 1 ), necessity of condition follows from Theorem 3. Indeed, suppose that there is a large solution and I p * < ∞ . Then by Theorem 3 there is a bounded solution to (1.1). But this is in contradiction with Theorem 2 (resp. Theorem 1) in [5]. So let us focus on sufficiency. Suppose that I p * = ∞. By Theorem 1, there exists a large radial solution v to (3.1) satisfying v(0) = 1 5 . Moreover, so v is a radial supersolution to Eq. Consequently, by Lemma 15, (u n ) is a positive decreasing sequence bounded above by v. Let u = lim n→+∞ u n . By Lemma 8 in [10], u is a radial solution of (3.2) satisfying u ≤ v . Furthermore and , i.e. u is a subsolution to (1.1) and v is a supersolution to (1.1). We argue that there is a solution w to (1.1) such that Indeed, Let w n = U p D n v . Due to Lemma 15 , (w n ) is a positive decreasing sequence satisfying u ≤ w n ≤ v . Hence again by Lemma 8 in [10], w = lim n→+∞ w n is a solution to (1.1). 5 If I p * = ∞ , for any ∈ ℝ * + , we can get a large solution v to Lu − p * (u) = 0 such that v(0) = , see Theorem 8 in [6]. Now, we need to prove that w is a large solution. Actually, it is enough to prove that However, I p * = ∞ ⇒ I p * = ∞, then by Theorem 1, u is either trivial or unbounded. Moreover, u is L-subharmonic radial in S, hence, by the maximum principle for elliptic operators, it follows that hence u is either trivial or (3.8) is satisfied. It remains to prove that u is nontrivial. By Theorem 14, we have in D n Though v(0) = 1 then in D n Additionally, u n ≤ v in D n so In view of Proposition 8 (resp. Proposition 10) applied to v, we have 6 Moreover, writing the left Haar measure in radial coordinates, we have Since G(r) sinh p+q ( r 2 ) cosh q ( r 2 ), is bounded on ℝ + then by (3.7) We conclude that Letting n → ∞ in (3.10), by the dominated convergence theorem we have And so u is not trivial. ◻ = v(0) + G D n (p * (v))(0). u n (0) + G D n (p osc (u n ))(0) = 1 + G D n (p * ( (v) − (u n )))(0).  Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.
We need also a Lemma that gives comparison between sub-solutions and super-solutions to (4.3) in D. It holds in a considerable generality: we require only that the function ∶ D × ℝ ↦ ℝ is increasing with respect to the second variable, see [10]:.
Lemma 15 (Comparison with values on the boundary) Let u, v ∈ C(D) , Lu, Lv ∈ L 1 loc (D) and let ∶ D × ℝ → ℝ be an increasing function with respect to the second variable . If Then: