Positive radial solutions of a nonlinear boundary value problem

In this work we study the following quasilinear elliptic equation: \begin{document}$\left\{ {\begin{array}{*{20}{l}}{ - {\rm{div}}(\frac{{|x{|^\alpha }\nabla u}}{{{{(a(|x|) + g(u))}^\gamma }}}) = |x{|^\beta }{u^p}}&{{\rm{in}} \ \Omega }\\{u = 0}&{{\rm{on}}\;\;\;\;\partial \Omega }\end{array}} \right.$ \end{document} where \begin{document}$ a $\end{document} is a positive continuous function, \begin{document}$ g $\end{document} is a nonnegative and nondecreasing continuous function, \begin{document}$ Ω = B_R $\end{document} , is the ball of radius \begin{document}$ R>0 $\end{document} centered at the origin in \begin{document}$ \mathbb{R} ^N $\end{document} , \begin{document}$N≥3 $\end{document} and, the constants \begin{document}$ α,β∈\mathbb{R} $\end{document} , \begin{document}$ γ∈(0,1) $\end{document} and \begin{document}$ p>1 $\end{document} . We derive a new Liouville type result for a kind of "broken equation". This result together with blow-up techniques, a priori estimates and a fixed-point result of Krasnosel'skii, allow us to ensure the existence of a positive radial solution. In this paper we also obtain a non-existence result, proven through a variation of the Pohozaev identity.

 −div |x| α ∇u (a(|x|) + g(u)) γ = |x| β u p in Ω u = 0 on ∂Ω where a is a positive continuous function, g is a nonnegative and nondecreasing continuous function, Ω = B R , is the ball of radius R > 0 centered at the origin in R N , N ≥ 3 and, the constants α, β ∈ R, γ ∈ (0, 1) and p > 1. We derive a new Liouville type result for a kind of "broken equation". This result together with blow-up techniques, a priori estimates and a fixed-point result of Krasnosel'skii, allow us to ensure the existence of a positive radial solution. In this paper we also obtain a non-existence result, proven through a variation of the Pohozaev identity.
1. Introduction. Consider the following quasilinear problem: where B R denotes the open ball of radius R > 0 centered at the origin in R N , with N ≥ 3. In this paper we study the existence and non-existence of positive radial solutions, for the case when A(x, u) = 1766 P. CERDA, L. ITURRIAGA, S. LORCA AND P. UBILLA γ ∈ (0, 1), α, β ≥ 0 with g and a continuous functions. In other words, here we study    −div |x| α ∇u (a(|x|) + g(u)) γ = |x| β u p in B R u = 0 on ∂B R (1.2) We will say that u is a solution of (1.2) if u ∈ C 1 (B R ) ∩ C 0 (B R ) and solve the equation in Problem (1.2) in the weak sense.
In 1996, Clement, de Figueiredo and Mitidieri [7], under the assumption that γ = 0 and N + α − 2 > 0 and β − α + 2 > 0 , (1.3) using variational techniques, proved the existence of positive radial solutions for this class of quasilinear elliptic equations. In addition, they defined the critical exponent associated to (1.2) by (1. 4) In 2003, Alvino, Boccardo, Ferone, Orsina and Trombetti, (see [1]), studied the existence of positive solutions of Problem (1.1) where b is independent on the u variable and the main part having degenerated coercivity. They proved that the existence and regularity of solutions, depending on the summability of the datum b. Subsequently, Boccardo (see [3]) considered a nonlinearity b belonging to L m (Ω), with 1 ≤ m ≤ N 2 (independent on u), and using approximation techniques, proved the existence of a entropy solution. In [4], Boccardo and Brezis studied the case that the datum b belongs to L m (Ω), with m > N 2 . We have noted that in the literature, the problems of the form (1.1), for example, when γ = 0 are studied using variational methods. However, when γ = 0 it is not possible to use those methods. We emphasize that, in our case, the operators have degenerated coercivity and nonlinearity depends on x and u. More precisely, by considering coefficients such as those in the Problems studied by Boccardo (see examples [1,2,3,4,5,6], among others).
Note that our approach, which is nonvariational, allows us to extend the class of operators studied initially by Clement, de Figueiredo and Mitidieri [7].
On the other hand, regarding the non-existence of positive solutions for problems like (1.1), in general, when the nonlinearity has a powerlike behavior to infinity, there is an extensive literature dealing with this kind of problems. Among them, we mention the work of Gidas and Spruck [17], where they consider the non-existence of positive solutions of problem where 1 < p < N +2 N −2 , either in R N or in the half space, see also [16]. In the case of an exterior domain Bidaut-Veron [12] proved that if 1 < p < N N −2 then (1.5) has no nontrivial solution. Now, in the case of degenerate elliptic equations in the form of with 1 < m < N , Mitidieri and Pohozaev proved in [20] that the problem (1.6) has no positive supersolution in . For an exterior domain, Bidaudt-Veron and Pohozaev [13] proved that (1.6) has no positive supersolution if p ∈ (1, m * − 1].

POSITIVE RADIAL SOLUTIONS OF A NONLINEAR BOUNDARY VALUE PROBLEM 1767
Other sharp results are established in Serrin-Zou ( [22]). They have proved that the problem (1.6) in R N or in an exterior domain has a positive supersolution if, and only if, p > m * − 1. They have also considered m−superlinear nonlinearities which are m−subcritical.
where c 1 and c 2 are positive constants. Now let us state our main results: 2) has at least one positive radial solution.
A delicate matter is to establish an Derrick-Pohozaev type inequality (see for example [9]) which allows us to get a non-existence results.
To prove our results, associated to our equation which is not variational, we will consider a parametrized truncated problem and we will obtain a priori bounds. Then, using a fixed-point argument, we prove the existence of a positive solution for the truncated problem. Finally, for some range of the parameter, Liouville type theorems show that the solution of the parametrized truncated problem is in fact a solution of the original problem. This study it's a delicate one because it requires studying the limiting problem, including a Liouville type theorems for a broken equation.
Note that all of the results above are valid for strictly positive nonlinearities in (0, ∞), which behave like powers to infinity or at zero, see e.g. [11,15] and references therein. In our case, we will need Liouville type results for the following equations: and For more details, see Theorem 2.1 and 2.3. Note that the equations above are limit problems to the truncated equations ((A) ξ k ) given in Section 3 and Section 5. This manuscript is organized as follows. In Section 2 we prove some Liouville type results. In Section 3, we truncate the problem (3.1) to obtain a priori estimates for a family of parameterized truncated problems (see problem ((A) ξ k )). In Section 4, we prove the existence of positive solution for the truncated Problem (A) 0 k . In the Section 5, we show that, for a large enough k, the solutions of the truncation problem are radial solutions of Problem (1.1), which proves our main result (Theorem 1.1). Finally in Section 5, we show some non-existence results of positive radial solutions of Problem (1.1).
On the other hand, if we suppose that exists y 0 > s 0 such that y 0 u (y 0 ) + ρ u(y 0 ) < m 0 , with m 0 < 0. We have that y u (y) + ρ u(y) < m 0 for all y ≥ y 0 .

POSITIVE RADIAL SOLUTIONS OF A NONLINEAR BOUNDARY VALUE PROBLEM 1769
Then u (y) < m 0 y −1 , and integrating from y 0 to y, we have that which is impossible for a nonnegative function. Thus, we have proved that function y u (y) + ρ u(y) is nonnegative and nonincreasing on (s 0 , +∞).
Now, we will show a result of a priori estimates for solutions of Problem 2.1, which will be fundamental to obtain Liouville type results.
Proof. Assume that u is a nonnegative solution of Problem (2.1), integrating equation (2.1) from y to t, with y ≥ s 0 , we obtain Thus, by Lemma 2.1, u (y) ≤ 0 and Since ρu(t) ≥ −tu (t) = t|u (t)|, we obtain Taking t = 2y and using the fact that u is a positive and decreasing function, we have and so, from the last inequality, we have Thus, , for all y > s 0 .
Moreover, multiplying the equation in problem (2.1) by yu (y) and integrating from s 0 to y, we obtain Thus, combining the last two equations, we obtain On the other hand, we note that Notice that, multiplying equation in (2.1) by u and integrating from s 0 to y, we obtain y s0

POSITIVE RADIAL SOLUTIONS OF A NONLINEAR BOUNDARY VALUE PROBLEM 1771
Then, by the last two equalities, we have Hence, we obtain does not have a positive solution.
Proof. Notice that by hypothesis (H 1 ), we have that N +2β−α+2−p(N +α−2) > 0, and by a priori estimate of Lemma 2.2, we obtain By Lemma 2.2 with s 0 = 0, we have that Where, by Lemma 2.1, we know that The last inequality, we obtain which is absurd, since u is a positive solution.
Now, we announce and prove a second result of non-existence, which will be fundamental in this paper. Theorem 2.4. Assuming the hypotheses (H 0 ), (H 1 ) and (H 2 ), the following problem Proof. Suppose u is a nonnegative solution of (P ). Considering the change is easy to see that v, is a nonnegative solution of the following problem for all y > s 0 , Since v is a decreasing function and by Lemma 2.1, we obtain v (y) y N +α−1 2 [yv (y) + (N + α − 2)v(y)] ≤ 0 for all y ∈ (s 0 , ∞). (2.13) Notice that, by inequality in (2.11) is easy to see that Also note that by hypothesis (H 1 ), we have Now, from equation (2.12) to facilitate the notation, we define Where it's easy to verify that by returning to the variable u, we obtain Our objective is to study the sign of L, for this, we observe that multiplying the equation (P ) by u and integrating from 0 to s 0 , we obtain (2.17) Then, using equation (2.17) in (2.16), we obtain Since u is a positive nonincreasing function, solution of (P), we have (2.19) Now, using equation (2.19) in (2.18), we have Then, since u is a nonnegative function, d > 1 and hypotheses (H 0 ) and (H 1 ), from the last inequality we have L > 0.

3.
A priori estimates. In this section we use the Liouville type results obtained in Section 2, to get a priori estimates for the radial solutions of Problem (1.1).
Notice that the radial problem associated with Problem (1.1) is as follows where the coefficient 1 (a(r)+g(u)) γ is not bounded from below. In order to overcome these difficulties, for each k ∈ N, we introduce the functions T k (s) := max{−k, min{k, s}} and g k (s) := (g • T k )(s) and consider the family of truncated problems parametrized by ξ ≥ 0, and q > p.
The following is a result of a priori bound for the auxiliary problem: Theorem 3.1. Assume hypotheses (H 0 ), (H 1 ) and (H 2 ). Then there is a positive constant C k which depends only on k, such that for every solution v of Problem (A) ξ k . Proof. Let k ∈ N be fixed and assume by contradiction that there is a sequence of positive solutions {v n } n of Problem (A) ξ k , so that ||v n || ∞ → +∞ when n → +∞. Note that by using the following changes of variables y = z n t n r , and defining w n (y) = v n (r) t n (3.3)

7)
From the properties of the function w n , it is easy to see that t n w n → ∞. Then, for k sufficiently large, the solution satisfies the following integral equation Since the function a bounded and Lebesgue's dominated theorem, w satisfies the following integral equation Then, w is a positive solution in [0, M ] to the following initial value problem where λ 0 is a positive constant. By using a diagonal iterative scheme, as in the last part of the proof of Proposition 4.1 in [8], w can be extended to all R + , as a positive solution of (2.6). Furthermore, using the argument of [8], it can be shown that w is indeed a positive solution of class C 2 (0, +∞) of (2.6).
Finally, for k > 0 fixed we conclude that there exists a constant C k > 0 (independent of ξ) such that for every solutions of Problem (A) ξ k , verifies ||v|| ∞ ≤ C k .

4.
Existence of positive solution of the truncated Problem (A) ξ k . The existence of positive solution of the truncated Problem (A) ξ k , is based on the following theorem due to Krasnosel'skii (See [10], [19]).  In order to use this result, we consider the Banach space X = C([0, 1], R) endowed with the L ∞ −norm, and the cone of nonnegative continuous functions given by Define also the operator F : X → X by Note that a simple calculation shows that the fixed point of operator F are the positive solutions of Problem (A) 0 k . Lemma 4.2. The operator F : X → X defined by (4.1) is compact, and the cone C 1 is invariant under F . Proof Outline. The compactness of F follows from the well known Ascoli Arzèla's theorem. The invariance of the cone C 1 is a consequence of the fact that the nonlinearities are nonnegative.
We will give an existence result of the truncated Problem (A) 0 k .
Note that H(t, v) is a compact homotopy and that H(0, v)(r) = F (v)(r), which verifies (b).
Hence, using Krasnosel'skii Theorem, we have that the operator (4.1) has a fixed-point v such that δ < ||v|| ∞ < η, which is solution of equation (A) 0 k .

5.
Proof of Theorem 1.1. In this section we will prove the existence of positive radial solution of Problem (1.1). Notice that if there is k 0 ∈ N, such that v k0 ∞ ≤ k 0 , where v k0 is a solution of Truncated Problem (A) 0 k0 , then v k0 is a radial solution of Problem (1.1).
Proof of Theorem 1.1. We will prove that there is k 0 ∈ N, such that v k0 ∞ ≤ k 0 , where v k0 is a solution of Truncated Problem (A) 0 k0 . For this, suppose by contradiction that v k ∞ > k for all k ∈ N, where v k is a solution of (A) 0 k . Using a similar change of variables given in (3.2), where t k := v k ∞ and z k = , we see that the function w k is a solution of the following problem From the last equation, it is not difficult to see that, w k (y) < 0 for all y ∈ (0, z k t k R) and Therefore, for any M > 0 there is a positive constant C 1 (M ) such that From the last inequality, we have the sequence {w k } k is equicontinuous. Since it is also uniformly bounded, by Ascoli Arzèla's theorem, we have that {w k } k contains a convergent subsequence, which we denote again by {w k } k , verifying Now, we will study the limiting problem associated with the Problem ((B) k ). Since t k > k for all k ∈ N, we have 0 < k t k < 1. Then, there are l ∈ [0, 1] and a subsequence, which we denote again by { k t k } k , such that k t k → l. Note that, since w k is a solution of Problem (B) k , for each k ∈ N there is 2) Now, we analyze the limit problem associated with Problem (B) k , depending on the value of l.
(1) If l = 0, that is k t k → 0, it is easy to see that s k → +∞. Thus, for any Then, since w k in a non-increasing function and by definition of g k , we have g k (t k w k (y)) = g(k) for all y ∈ [0, M ] and k ≥ k M . From equation in ((B) k ), is easy to see that, for k ≥ k M we have Using a diagonal iterative scheme, w can be extended to all R + , as a positive solution of (5.5), and using [8], it can be shown that w is indeed a positive solution of class C 2 (0, +∞) of (2.6). This is a contradiction with Theorem 2.3.
(2) If l = 1, that is k t k → 1. Here, is natural to expect that s k → 0. Indeed, integrating from 0 to s k ∈]0, M ] as in (5.3), we have Hence, we obtain s k → 0 as k → ∞.
Note that, the equation in ((B) k ), we have that w k satisfies the following integral equation As in the previous case, w can be extended to all R + , as a positive solution of (5.7). Furthermore, using [8], it can be shown that w is indeed a positive solution of class C 2 (0, +∞) of (5.7).
Let us consider the change of variables given in (2.9), we have that u ∈ C 2 (0, +∞) is a positive solution of This is a absurd by Theorem 2.3. (3) If 0 < l < 1. Since w k a solution of Problem ((B) k ) and w k ( z k R t k ) = 0, it is not difficult to verify that s k ∞. Then, there is a constant c 0 > 0, such that s k ≤ c 0 for all k ∈ N. Then by compactness, there is s 0 ∈ R + and a subsequence, which we denote again by {s k } k , such that s k → s 0 .
Observe that for y ∈ (0, s k ) we have that g k (t k w k (y)) = g(k). Then, we have that w k satisfies the integral equation (5.3) in (0, s 0 ). Proceeding as in case (1), we conclude that w is a positive solution of the following problem −(y N +α−1 w ) = y N +β−1 w p in (0, s 0 ), On the other hand, for each y ∈ (s k , ∞), it is easy to see that w(y) > 0, then we have t k w k (y) → ∞ for each y ∈ (s k , ∞). From the equation (5.6), by hypothesis (H 2 ), (5.1) and Lebesgue's dominated convergence theorem, we conclude that w satisfies de following integral equation Thus, w is a nonnegative nontrivial solution in (s 0 , ∞) to the initial value problem w(s 0 ) = l . (5.11) Then, we have that w is nontrivial solution of (P ). This is absurd by Theorem 2.4. Then there is k 0 ∈ N, such that v k0 ≤ k 0 , then v k0 is a positive solution of Problem 3.1.
6. non-existence via Pohozaev type identity. In this section, we prove Theorem 1.2 (see [21]), i.e. we stablish the non-existence of positive solutions of Problem (1.1). For this, using a variant of Pohozaev identity, we obtain the exponent p * defined in (1.4), which for the case γ = 0 correspond to the critical exponent associated to the Laplacian operator.