Generalized Picone inequalities and their applications to $(p,q)$-Laplace equations

We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the $(p,q)$-Laplace type operators. With its help, as well as with the help of several other known generalized Picone inequalities, we provide some nontrivial facts on the existence and nonexistence of positive solutions to the zero Dirichlet problem for the equation $-\Delta_p u -\Delta_q u = f_\mu(x,u,\nabla u)$ in a bounded domain $\Omega \subset \mathbb{R}^N$ under certain assumptions on the nonlinearity and with a special attention to the resonance case $f_\mu(x,u,\nabla u) = \lambda_1(p) |u|^{p-2} u + \mu |u|^{q-2} u$, where $\lambda_1(p)$ is the first eigenvalue of the $p$-Laplacian.


Picone inequalities
Throughout this section, we denote by Ω a nonempty connected open set in R N , N ≥ 1. The nowadays classical version of the Picone inequality (also commonly referred to as the Picone identity) for the p-Laplacian can be stated as follows. In the linear case p = 2, the inequality (1.1) is a direct consequence of the simple identity (1.2) whose one-dimensional version was used by M. Picone in [22,Section 2] to prove the Sturm comparison theorem. Subsequently, due to the nontrivial and convenient choice of the test function v p u p−1 , the identity (1.2) and the inequality (1.1) appeared to be effective in the study of many other properties of various ordinary and partial differential equations and systems of both linear and nonlinear nature. In particular, one can mention the uniqueness and nonexistence of positive solutions, Hardy type inequalities, bounds on eigenvalues, Morse index estimates, etc. Such a wide range of applications particularly motivated a search of reasonable generalizations of the Picone inequality, see, e.g., the works [3,4,8,10,11,17,18,23,24], although this list is far from being comprehensive.
On the other hand, during the last few decades, there has been growing interest in the investigation of various composite type operators such as the sum of the p-and q-Laplacians with p = q, the so-called (p, q)-Laplacian. The motivation for corresponding studies comes from both the intrinsic mathematical interest and applications in natural sciences, see, for instance, [1,5,6,7,8,13,16,21,25] and references therein, to mention a few. Clearly, most of the properties indicated above can be posed for problems with such operators, too. It is then natural to ask which generalizations of the Picone inequality are favourable to be applied to the (p, q)-Laplacian. If one tries to use v p u p−1 or v q u q−1 as a test function, then, taking into account (1.1), the following two quantities have two be estimated: There are at least two known generalized Picone inequalities in this regard. The first one was obtained in [11], where its equivalence to two convexity principles for variational integrals is also shown. Its partial form can be stated as follows.
Theorem 1.2 ([11, Proposition 2.9 and Remark 2.10]). Let p, q > 1 and let u, v be differentiable functions in Ω such that u > 0, v ≥ 0. If q ≤ p, then The second generalization of (1.1) was obtained in [17] in the context of study of an equation with indefinite nonlinearity. Later, this result was also applied in [8] to an eigenvalue problem for the (p, q)-Laplacian.
Moreover, the equality in (1.4) is satisfied in Ω if and only if u ≡ kv for some constant k > 0.
Remark 1.4. For convenience of further applications of (1.4), we rewrite it, assuming q ≤ p, as follows: (1.5) Notice that both (1.3) and (1.4) turn to the Picone inequality (1.1) when p = q. Moreover, we emphasize that (1.3) requires q ≤ p, while (1.4) asks for p ≤ q. Our main result, Theorem 1.7 below, posits the fact that the inequality (1.4) remains valid for some p > q, although the set of feasible values of p and q is not of a trivial structure. This set is defined and characterized in the following lemma. Lemma 1.5. Let q > 1 be fixed. Let the function g : [0, +∞) × (1, +∞) → R be defined as and set p = sup{p > 1 : p ∈ I(q)}. Then max{2, q} < p < q + 1 and the following assertions hold: (ii) if q < 2, then there exist p * ∈ (q, p] and p * ∈ (q, 2) such that [q, p * ] ⊂ I(q) and [p * , p] ⊂ I(q); In particular, each of the following two assumptions is sufficient to guarantee that p ∈ I(q): (II) 2 ≤ p < q + 1 and (q + 1 − p) p−2 q ≥ (p − q) p−1 .
Remark 1.6. A numerical investigation of the function g indicates the existence of a threshold value q = 1.051633991... with the following property: if q < q, then q < p * < p * < 2 and p * , p * can be chosen such that (p * , p * ) ∩ I(q) = ∅, while if q ≥ q, then p * = p, i.e., (1, p] ⊂ I(q).
Now we are ready to state our main result.
Theorem 1.7. Let p, q > 1 and let u, v be differentiable functions in Ω such that u > 0, v > 0. Assume that one of the following assumptions is satisfied: (i) p ∈ I(q), where I(q) is given by Lemma 1.5; (ii) p ≤ q + 1 and ∇u∇v ≥ 0. Then Moreover, if p < q + 1 and ∇u∇v ≥ 0, then the equality in (1.6) is satisfied in Ω if and only if u ≡ kv for some constant k > 0. Furthermore, the assumptions (i) and (ii) are optimal in the following sense: (I) if p ∈ I(q), then there exist u, v and a point x ∈ Ω such that (1.6) is violated at x; (II) if p > q + 1, then there exist u, v with ∇u∇v ≥ 0 and a point x ∈ Ω such that (1.6) is violated at x.
A closer look at the proof of Theorem 1.7 (ii) reveals that the inequality (1.6) remains valid under the assumption (ii) also for q = 1. In fact, even the following stronger result, which reduces to the commutativity of the scalar product in W 1,2 (Ω) at p = 2, can be obtained by the same method of proof. Proposition 1.8. Let u, v be differentiable functions in Ω such that u > 0, v > 0, and ∇u∇v ≥ 0. Then the following assertions hold: Moreover, if p = 2, then the equality in (1.7) or (1.8) is satisfied in Ω if and only if u ≡ kv for some constant k > 0.
Apart from the choice of v p u p−1 or v q u q−1 as a test function, one could also consider more general test functions of the form v p f (u) or v q f (u) . In this direction, the following partial case of a generalized Picone inequality obtained in [23] by applying an inequality from [18, Lemma 2.1] can be effectively used.
Moreover, the equality in (1.9) is satisfied in Ω if and only if f (u) ≡ kv p−1 for some constant k > 0.
Remark 1.10. Let q > 1. Since v p = (v p/q ) q , we get from (1.9) the complementary inequality Notice that the term |∇(v p/q )| is well-defined if either q ≤ p and v ≥ 0, or q = p and v > 0.
In particular, under any of these assumptions, taking f (s) = s p−1 , we obtain Evidently, (1.11) reduces to the Picone inequality (1.1) if q = p.
As a complementary fact, we provide the following optimal refinement of a generalized Picone inequality obtained in [8,Proposition 8], by analysing the right-hand sides of the inequalities (1.9) and (1.10). Proposition 1.11. Let p, q > 1, α, β > 0, and let u, v be differentiable functions in Ω such that u > 0, v ≥ 0. If q < p, then Finally, let us note that the Picone inequality (1.1) can be used to derive the Díaz-Saa inequality [15,Lemma 2]: , assuming that Ω is smooth and bounded. The inequality (1.12) appeared to be a useful tool in the study of uniqueness of positive solutions to boundary value problems with the p-Laplacian. Its generalization to the (p, q)-Laplacian, together with the corresponding applications, was obtained in [16]. Under the same assumptions on w 1 , w 2 and Ω as above, it can be stated as follows, see [16, Lemma 2.1]. If 1 < q < p and µ > 0, then The inequality (1.13) can be established by applying the generalized Picone inequality (1.3).
The rest of the article is organized as follows. In Section 2, we prove Theorem 1.7 and Lemma 1.5. In Section 3, we provide several applications of Theorem 1.7, as well as of Theorems 1.2 and 1.3, to problems with the (p, q)-Laplacian.

Proofs of Theorem 1.7 and Lemma 1.5
Proof of Theorem 1.7. Since the case p ≤ q is covered by Theorem 1.3, we will assume hereinafter that p > q. Moreover, under any of the assumptions (i) and (ii), p has the upper bound p ≤ q + 1 (see Lemma 1.5 in the case of the assumption (i)).
By straightforward calculations we get and We see from (2.1) and (2.2) that the desired inequality (1.6) is equivalent to Recalling that q − p + 1 ≥ 0, we see that (2.4) is satisfied if its left-hand side is nonpositive. Therefore, let us assume that the left-hand side of (2.4) is positive. In particular, we have ∇u∇v = 0, and hence |∇u|, |∇v| > 0. We consider two separate cases.

1) Suppose that
∇u∇v > 0 and q |∇u| u In this case, in order to validate (2.4) it is sufficient to prove that Let us show that (2.6) is satisfied. We have Combining this strict convexity of f with the facts that and the equality f (s) = 0 for such s happens if and only if s = 1.
In particular, f (s) ≥ 0 for all and the equality f (s) = 0 for such s happens if and only if s = 0 and p = q + 1. Thus, we have derived that f (s) ≥ 0 for all s ≥ 0 provided p ≤ q + 1. In particular, this implies that (1.6) is satisfied under the assumption (ii). Moreover, we have shown that if p < q + 1, then f (s) = 0 if and only if s = 1. Therefore, if p < q + 1, ∇u∇v ≥ 0, and the equality in (1.6) is satisfied in Ω, then we conclude that ∇u∇v = |∇u||∇v| and |∇u| 2) Suppose that ∇u∇v < 0 and q |∇u| u To establish (2.4) under the assumption (2.7), it is sufficient to show that (2.8) Introducing again the notation s = |∇u| Applying Lemma 1.5, we deduce that (2.9) is satisfied whenever p ∈ I(q).
Combining the cases 1) and 2), we conclude that (1.6) holds under the assumption (i), which finishes the proof of the first part of the theorem.
Let us now obtain the optimality of the assumptions (i) and (ii) stated in (I) and (II), respectively. Assume first that p ∈ I(q) and let s 0 ≥ 0 be such that g(s 0 ; p) < 0. |∇v(0)| = α and taking α = s 0 , we conclude that the violation of (2.9) at s 0 implies the violation of (2.8) at x = 0. On the other hand, we have ∇u∇v = −|∇u||∇v|. Thus, the violation of (2.8) at x = 0 is equivalent to the violation of (2.4) at x = 0, which, in its turn, is equivalent to the violation of (1.6) at x = 0. This establishes the case (I).
Assume now that p > q + 1. Set u ≡ const > 0 and let v > 0 be any differentiable function not identically equal to a constant. We readily see that ∇u∇v ≡ 0 and (2.3) is violated at points where |∇v| > 0, which establishes the case (II). Now we provide the proof of Lemma 1.5.
Proof of Lemma 1.5. We start by noting that the assertion (i) follows trivially since p ≤ q implies g(s; p) ≥ (q − 1)s p + qs p−1 + 1 > 0 for all s ≥ 0.
Let us now finish the proof of the assertion (ii) by obtaining the existence of p * . Suppose, by contradiction, that there exists q ∈ (1, 2) such that for any n ∈ N one can find p n ∈ (q, p] and s n > 0 satisfying g(s n ; p n ) < 0, and p n → q as n → +∞. Since the term (p n − q)s n is the only term in g(s n ; p n ) with negative sign, we conclude that s n → +∞ as n → +∞. But then we deduce that 0 > g(s n ; p n ) ≥ (q − 1)s q n + qs q−1 n − (2 − q)s n + (q − 1) > 0 for all sufficiently large n, since (q − 1)s q n is the leading term as s n → +∞. This is a contradiction, and hence the proof of the assertion (ii) is complete.
Finally, we justify the sufficient assumptions (I) and (II). (I) Let 1 < q < p ≤ 2. Considering the sum of the second and third terms of g(s; p), we see that if s 2−p ≤ q p−q , then g(s; p) ≥ 0. Thus, let us assume that s 2−p > q p−q and p < 2. Then we have where the last inequality is satisfied if and only if p ≤ q + q p−1 (q − 1) 2−p .
(II) Let 2 ≤ p < q + 1. As in the previous case, we see that if s p−2 ≥ p−q q , then g(s; p) ≥ 0. Hence, we assume that s p−2 < p−q q and p > 2. Then we have where the last inequality is satisfied if and only if (q + 1 − p) p−2 q ≥ (p − q) p−1 .

Applications to (p, q)-Laplace equations
Throughout this section, we always assume that 1 < q < p and that Ω ⊂ R N is a smooth bounded domain with the boundary ∂Ω, N ≥ 2.
Denote by · r the standard norm of L r (Ω), 1 ≤ r ≤ +∞. Let λ 1 (r) with 1 < r < +∞ stand for the first eigenvalue of the Dirichlet r-Laplacian in Ω, and let ϕ r be the corresponding first eigenfunction which we assume to be positive and normalized as ∇ϕ r r = 1. That is, and λ 1 (r) ϕ r r r = ∇ϕ r r r = 1.

General problem with (p, q)-Laplacian
Consider the boundary value problem where the function f µ (x, s, ξ) : Ω × R × R N → R is sufficiently regular in order that (3.2) possesses a weak formulation with respect to W 1,p 0 (Ω), and satisfies the following assumption: for all µ ∈ M , s > 0, ξ ∈ R N , and a.e. x ∈ Ω, where β m * is given by (3.1).

Eigenvalue-type problem
In the partial case f µ (x, s, ξ) = λ 1 (p)|s| p−2 s + µ|s| q−2 s, (3.2) can be seen as an eigenvalue problem for the (p, q)-Laplacian: see, e.g., [8,9,14,21]. Notice that any nonzero and nonnegative solution of (3.3) belongs to int C 1 0 (Ω) + , see, for instance, [8,Remark 1] or [21,Section 2.4]. Although in the works [8,9] by the present authors the structure of the set of positive solutions to a general version of (3.3) with two parameters has been comprehensively studied, we were not able to characterize completely the range of values of µ for which (3.2) possesses a positive solution. Thanks to our generalized Picone inequality (1.6), as well as to the inequalities (1.3) and (1.4), we can provide additional information in this regard.
First, the same reasoning as in Theorem 3.1 allows to show the following result.
Theorem 3.2. Assume that one of the following assumptions is satisfied: (ii) p ≤ q + 1 and Ω is an N -ball.
Second, we provide the following general result without restrictions on p and q apart from the default assumption 1 < q < p, whose proof is based on a nontrivial application of Picone's inequalities (1.3) and (1.4), and on the usage of results from [8,9]. In order to prove Theorem 3.3, we need the following auxiliary information on the behaviour of positive solutions. Proof. We start with the observation that (3.3) has no nonzero solution for µ ≤ λ 1 (q), see [8,Proposition 1] and [9,Proposition 13]. Thus, throughout the proof, we will assume that µ n > λ 1 (q) for all n ∈ N. In particular, we have lim inf n→+∞ µ n ≥ λ 1 (q).
(i) Let ∇u n p → +∞ as n → +∞. Note first that lim inf n→+∞ µ n > λ 1 (q). Indeed, suppose, by contradiction, that µ n → λ 1 (q), up to a subsequence. Setting v n = un ∇un p and taking u n as a test function for (3.3) with µ = µ n , we have where the inequality follows from the definition of λ 1 (p). Since q < p, ∇u n p → +∞, and µ n → λ 1 (q), we conclude that, simultaneously, v n → ϕ p and v n → kϕ q strongly in L q (Ω), up to a subsequence, where k > 0 is some constant. However, this contradicts the linear independence of ϕ p and ϕ q , see [9,Proposition 13], and hence lim inf n→+∞ µ n > λ 1 (q). Now we prove the convergence (3.4). Let v 0 ∈ W 1,p 0 (Ω) be such that v n → v 0 weakly in W 1,p 0 (Ω) and strongly in L p (Ω), up to a subsequence. First, we show that v 0 ≡ 0 in Ω. Suppose, by contradiction, that v 0 ≡ 0 in Ω. Then, by Egorov's theorem, v n converges to 0 uniformly on a subset of Ω of positive measure. In particular, we have Ω v q−p n ϕ p q dx → +∞ as n → +∞. (3.5) Using now the Picone inequalities (1.1) and (1.5), we get from (3.3) with µ = µ n that This implies that and hence, since lim inf n→+∞ µ n > λ 1 (q), there exists a constant C > 0 independent of n such that On the other hand, choosing u n as a test function for (3.3) with µ = µ n , we get Since we suppose that v 0 ≡ 0 in Ω, we have v n → 0 strongly in L p (Ω) and L q (Ω), which yields 2µ n Ω |v n | q dx ≥ ∇u n p−q p for sufficiently large n ∈ N. (3.9) Combining (3.8) and (3.9), we obtain 2C Ω |v n | q dx ≥ Ω v q−p n ϕ p q dx for sufficiently large n ∈ N, which gives a contradiction to (3.5) and the strong convergence v n → 0 in L q (Ω). Therefore, v 0 ≡ 0 in Ω. Second, we show that v 0 = ϕ p . Since u n is a solution of (3.3) with µ = µ n , we see that v n satisfies Ω |∇v n | p−2 ∇v n ∇ϕ dx + 1 ∇u n p−q p Ω |∇v n | q−2 ∇v n ∇ϕ dx = λ 1 (p) Ω |v n | p−2 v n ϕ dx + µ n ∇u n p−q p Ω |v n | q−2 v n ϕ dx for any ϕ ∈ W 1,p 0 (Ω). (3.10) Taking ϕ = v n and recalling that ∇v n p = 1 and that v n converges in L p (Ω) to a nonzero function v 0 , we conclude that there exists a constant B ≥ 0 such that B n := µ n ∇u n p−q p → B as n → +∞.
Taking now ϕ = v n − v 0 in (3.10), we see that the boundedness of B n implies lim n→+∞ Ω |∇v n | p−2 ∇v n (∇v n − ∇v 0 ) dx = 0, which guarantees that v n → v 0 strongly in W 1,p 0 (Ω) by the (S + )-property of the p-Laplacian. Passing to the limit in (3.10), we deduce that v 0 is a nonzero and nonnegative solution of the problem −∆ p u = λ 1 (p)|u| p−2 u + B|u| q−2 u in Ω, The standard regularity result [19] and the strong maximum principle yield v 0 ∈ int C 1 0 (Ω) + . Hence, Thus, applying the Picone inequality (1.1), we get which yields B = 0, and hence v 0 ≡ ϕ p in Ω. Now we are ready to prove that v n → ϕ p in C 1 0 (Ω). Thanks to the boundedness of B n , using the Moser iteration process in (3.10), we can find M 1 > 0 independent of n such that v n ∞ ≤ M 1 for all n. Thus, since 1/ ∇u n p−q p is also bounded, applying to the equation (3.10) the regularity results [20, Theorem 1.7] and [19], we derive the existence of θ ∈ (0, 1) and M 2 > 0, both independent of n, such that v n ∈ C 1,θ 0 (Ω) and v n C 1,θ 0 (Ω) ≤ M 2 for every sufficiently large n. Since C 1,θ 0 (Ω) is compactly embedded into C 1 0 (Ω), we conclude that v n → ϕ p in C 1 0 (Ω), up to a subsequence. Finally, let us show that lim n→+∞ µ n = β * . First, let {µ n k } be a subsequence such that lim k→+∞ µ n k = lim inf n→+∞ µ n . Taking u n k as a test function for (3.3) with µ = µ n k , we get and hence the convergence of v n k to ϕ p along a sub-subsequence yields Second, we choose a subsequence {µ n k } such that lim (µ n − λ 1 (q)) Ω u q−p n ϕ p q dx ≤ Ω |∇ϕ q | p dx − λ 1 (p) Ω ϕ p q dx < +∞, and therefore lim sup n→+∞ µ n ≤ λ 1 (q). Recalling now that lim inf n→+∞ µ n ≥ λ 1 (q), we finish the proof of the assertion (ii).