Positive radial solutions for Dirichlet problems via a Harnack‐type inequality

We deal with the existence and localization of positive radial solutions for Dirichlet problems involving ϕ$$ \phi $$ ‐Laplacian operators in a ball. In particular, p$$ p $$ ‐Laplacian and Minkowski‐curvature equations are considered. Our approach relies on fixed point index techniques, which work thanks to a Harnack‐type inequality in terms of a seminorm. As a consequence of the localization result, it is also derived the existence of several (even infinitely many) positive solutions.

Our approach here is based on fixed point index theory, namely, on compression-expansion-type homotopy arguments. The most known are those from Krasnosel'skiı's compression-expansion theorem in a conical annulus defined by using the max-norm of the space. Applications to one-dimensional -Laplace equations are given in the papers. 15,16 In the radial case considered in the present paper, the absence of a Harnack-type inequality in terms of the max-norm makes Krasnosel'skiı's theorem inoperative and forces us to use instead some other homotopy conditions and properties of the fixed point index.
The first paper in radial solutions that uses the compression-expansion technique, but in a variational form and only for p-Laplacian equations, is Precup et al. 17 As explained there, the difficulty in applying the compression-expansion method consists in the necessity that, for the considered differential operator, a Harnack-type inequality be available. In the present paper, such a key inequality is established for problem (1.2) with a general homeomorphism satisfying some additional conditions. With its help, a precise localization of positive solutions is possible, allowing in a natural way to obtain multiple solutions. The results apply in particular for the p-Laplacian and the Minkowski mean curvature operator.
Note that the homeomorphisms related to the p-Laplacian for p ≥ 2 and the Minkowski mean curvature operator both satisfy condition (H ). Contrarily, the bounded homeomorphisms with a = +∞, for example, the one involved by the Euclidian mean curvature operator, are not convex on [0, +∞) and thus they do not satisfy our assumption (H ).
A similar result has been established in Precup et al 17 for the particular case of the p-Laplacian with p > n, for which (s) = |s| p−2 s and a = b = +∞. More exactly, it has been proved that In this case, since by using Hölder's inequality one has a Harnack-type inequality in terms of the max-norm |u| ∞ = max r∈[0,1] |u (r)| can be immediately derived from (2.5), namely, It is an open problem to obtain an analog result for more general homeomorphisms satisfying (H ). At this moment we are only able to establish such an inequality in terms of a max-seminorm on C [0, 1] . For example, taking p = +∞ in (2.4), we have the following Harnack-type inequality related to a seminorm on C [0, 1] .

Corollary 2.2. Under the assumptions of Theorem 2.1, if for a fixed subinterval
This combined with (2.4) yields (2.6).
In the sequel, inequality (2.6) is a key ingredient for the localization and multiplicity of positive radial solutions. We will use the main ideas in Precup 17 in order to localize the solutions in terms of a norm and a seminorm.

POSITIVE RADIAL SOLUTIONS
Recall that by a (nonnegative) solution of (1.2), we mean a function u ∈ C 1 ([0, 1] , R + ) with u ′ (0) = u (1) = 0, |u ′ (r)| < a for all r ∈ [0, 1], such that r n−1 (u ′ ) ∈ C 1 [0, 1] and (1.2) is satisfied. We will say that a nonnegative solution is positive if it is distinct from the identically zero function. Let X be the Banach space of continuous functions X = C [0, 1] and K 0 its positive cone As proved in earlier studies, 6,8 the operator T is completely continuous.
Let us now consider a subcone of K 0 related to the Harnack inequality (2.6), namely,

Lemma 3.1. The operator T maps the cone K into itself.
Proof. Indeed, take u ∈ K and let us show that v ∶= Tu belongs to K. Since is nonnegative, v ≥ 0, and moreover, Therefore, v ∈ K, as claimed.
Now, for any number > 0, consider the set The operator T being completely continuous, the set T where the last equality is due to the normalization property of the fixed point index, since 0 ∈ U .
Next, for a number > 0, consider the set By the homotopy property of the fixed point index, one has Remark 3.1. If the operator T maps U into itself, theñ= and condition (3.4) reduces to By using the previous fixed point index computations, we deduce the following existence result. Proof. One has As a result, i In addition i or equivalently, T (w) = w, where w = |v| ∞ v and = |v| ∞ . Since |w| ∞ = and > 1, we arrived to a contradiction with (3.3). Therefore i (T , U ⧵ V , Ũ) = 1, which implies our conclusion. Now we give sufficient conditions in order to guarantee the assumptions of the previous lemmas hold. We will use the following notation. If b < +∞, denote Hence, |T(u)| ∞ < for all u ∈ K with |u| ∞ ≤ , which implies (3.3). In addition, on the basis of Remark 3.1, we can takẽ= .

Remark 3.2 (Asymptotic conditions). Existence of both positive numbers and satisfying inequalities (3.5) and (3.6)
is guaranteed if the following asymptotic conditions at zero and infinity hold: Obviously, if is a classical homeomorphism (a = b = +∞), conditions (3.5) and (3.6) can be rewritten as .
Hence, if we assume in addition that satisfies: then the existence of both positive numbers and is guaranteed under suitable asymptotic conditions about at zero and at infinity. Note that assumption (3.7) holds in the case of the p-Laplacian operator and so it is commonly employed in the literature, see for instance. 8,12 Theorem 3.6. Assume that n ≥ 3, conditions (H ) and (H ) are fulfilled, and is a classical homeomorphim. If (3.7) and hold, then problem (1.2) has at least one positive solution.
On the other hand, condition clearly implies the existence of a positive number satisfying (3.5) and such that < AB . Therefore, Theorem 3.5 ensures the existence of at least one positive solution for problem (1.2).

Corollary 3.7.
Assume that n ≥ 3, p ≥ 2, and (H ) holds. If has at least one positive radial solution.
Proof. It suffices to show that problem (1.2) has at least one positive solution with (x) = |x| p−2 x, p ≥ 2. Since is a classical homeomorphism which satisfies (H ) and (3.7), the conclusion follows from Theorem 3.6.
We show the applicability of our theory with an example involving radial solutions of p-Laplacian equations.

Example 3.8. Consider the function given by
with 0 ≤ q < p − 1 and p ≥ 2, which clearly satisfies condition (H ).
It is immediate to check that Therefore, problem (3.10) associated to this function has at least one positive radial solution, as a consequence of Corollary 3.7.
Finally, we highlight that due to the asymptotic behavior of at zero and at infinity, this problem falls outside the scope of the results in earlier studies. 12,17 On the other hand, it is worth to mention that in the case of a singular homeomorphism (i.e., with a < +∞, b = +∞), condition (3.5) is trivially satisfied for large enough. Hence, in that case, we only need to ensure the existence of the number in order to obtain positive solutions for problem (1.2).
Let us assume in the rest of this section that is singular. We present an existence result inspired by those in Bereanu et al. 8 Theorem 3.9. Assume that n ≥ 3, conditions (H ) and (H ) are fulfilled, and is a singular homeomorphim. If (3.7) and hold, then problem (1.2) has at least one positive solution.
Proof. Arguing as in the proof of Theorem 3.6, conditions (3.7) and (3.11) imply the existence of a positive number satisfying (3.6). Therefore, Theorem 3.5 ensures the existence of at least one positive solution for problem (1.2).
As a consequence, we derive a simple existence result concerning positive radial solutions for Dirichlet problems involving the Minkowski mean curvature operator.

13)
has at least one positive radial solution.
Proof. If suffices to show that problem has at least one positive solution. The conclusion follows from Theorem 3.9 with (x) = x∕ = for all > 0, and that for this homeomorphism , condition (3.12) implies (3.11).
Example 3.11. Consider problem (3.13) with a function of the form where 0 ≤ q < 1 and g is a nonincreasing positive and continuous function. Clearly, satisfies (H ) and the asymptotic condition (3.12), so Corollary 3.10 ensures the existence of a positive radial solution.

MULTIPLICITY RESULTS
Obviously, the localization of positive solutions provided by Theorem 3.5 makes possible to obtain multiplicity results for problem (1.2) if there exist several (perhaps infinitely many) well-ordered pairs of numbers ( , ) satisfying (3.5) and (3.6). Nevertheless, a suitable computation of the fixed point index related to the operator T allows us to establish a three-solution-type result under less stringent assumptions. First, we present the three-solution-type fixed point theorem concerning the operator T defined in (3.1). then T has at least three fixed points u 1 , u 2 , and u 3 such that Proof. By Lemma 3.4, T has a fixed point u 1 such that Moreover, assumption (4.1) ensures that i (T , U 0 , Ũ) = 1 and thus the operator T has a fixed point u 2 in U 0 , that is, Since 0 < , one has U 0 ⊂ V , and so the properties of the fixed point index together with its computation in Lemma 3.3 imply Therefore, the existence property of the fixed point index ensures that the operator T has a third fixed point u 3 located in V ⧵ U 0 .
As a consequence, we obtain a three-solution-type result for problem (1.2).
then problem (1.2) has at least three solutions u 1 , u 2 , and u 3 such that A multiplicity result can be also obtained under a suitable behavior of the nonlinearity at zero and infinity. Proof. By the asymptotic behavior of at zero and at infinity given by (4.2), there exist 0 < 0 < (sufficiently small) and 1 > ∕(AB) (sufficiently large) such that −1 ( (0, i )) < i , i = 0, 1.
Therefore, the conclusion follows from Theorem 4.2.
Next, we emphasize the multiplicity result in the remarkable particular cases of Dirichlet problems involving the p-Laplacian and Minkowski mean curvature operators. Corollary 4.4. Assume that n ≥ 3, p ≥ 2, and condition (H ) holds. In addition, suppose that there exists > 0 satisfying condition (3.6) and (3.10) has at least two positive radial solutions. Corollary 4.5. Assume that n ≥ 3 and condition (H ) holds. In addition, suppose that there exists > 0 satisfying condition (3.6) and Then problem (3.13) has at least two positive radial solutions.
Proof. The conclusion follows from Corollary 4.3 with (x) = x∕ is trivially satisfied.
We illustrate the applicability of the previous multiplicity results with the following example. where q > 1 and > 0. We shall study the existence of two positive solutions for problem (1.2) with as above and being the singular homeomorphism given by (x) = x∕ √ 1 − x 2 , −1 < x < 1. Clearly, satisfies (H ) and it is immediate to check that Finally, taking = 1∕4 and = 3∕4, it is a simple matter to see that = 1∕16 satisfies condition (3.6) for any large enough (e.g., with > 4 (2n+1)q+n−1∕2 ∕(n − 2) q √ 15). Therefore, Corollary 4.5 guarantees that problem −div has at least two positive radial solutions for any q > 1 provided that > 0 is sufficiently large.
Finally, the existence of infinitely many positive solutions for (1.2) is obtained if the nonlinearity has an oscillating behavior at zero or at infinity. then (1.2) has a sequence of positive solutions u k such that |u k | ∞ → +∞ as k → ∞.
Proof. Let us prove claim (a) (case (b) is analogous). By (4.3), there exist two decreasing sequences { i } i∈N and { i } i∈N tending to zero such that i+1 < i < AB i and Therefore, Theorem 3.5 can be applied to each pair ( i , i ) and so the conclusion is immediately obtained. To finish, we provide an example concerning the existence of infinitely many positive radial solutions for a Dirichlet problem involving the relativistic operator. Therefore, Corollary 4.7 ensures that the corresponding problem (3.13) associated to this nonlinearity has a sequence of positive radial solutions u k such that |u k | ∞ → 0 as k → ∞ provided that ≥ ( + 1), ( − ) n−1 ( + ) > 1 c (1 − ) and − < 1.