We obtain a generalization of the Picone inequality which, in combination with the classical Picone inequality, appears to be useful for problems with the -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 in a bounded domain under certain assumptions on the nonlinearity and with a special attention to the resonance case , where is the first eigenvalue of the p-Laplacian.
1 Picone inequalities
Throughout this section, we denote by a nonempty connected open set in , . The current classical version of the Picone inequality (also commonly referred to as the Picone identity) for the p-Laplacian can be stated as follows.
[1, Theorem 1.1] Let and let be differentiable functions in such that , . Then
Moreover, the equality in (1.1) is satisfied in if and only if for some constant .
In the linear case , inequality (1.1) is a direct consequence of the simple identity
whose one-dimensional version was used by M. Picone in [2, Section 2] to prove the Sturm comparison theorem. Subsequently, due to the nontrivial and convenient choice of the test function , identity (1.2) and 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 in [3,4,5,6,7,8,9,10,11], 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 , the so-called -Laplacian. The motivation for corresponding studies comes from both the intrinsic mathematical interest and applications in natural sciences, see, for instance, [12,13,14,15,16,17,18,19] 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 -Laplacian. If one tries to use or as a test function, then, taking into account (1.1), the following two quantities have to be estimated:
There are at least two known generalized Picone inequalities in this regard. The first one was obtained in , where its equivalence to two convexity principles for variational integrals is also shown. Its particular form can be stated as follows.
[7, Proposition 2.9 and Remark 2.10] Let and let be differentiable functions in such that , . If , then
Although the case of equality in (1.3) is not discussed in , one can show that if , then the equality in (1.3) is satisfied in if and only if .
The second generalization of (1.1) was obtained in  in the context of study of an equation with indefinite nonlinearity. Later, this result was also applied in  to an eigenvalue problem for the -Laplacian.
[8, Lemma 1] Let and let be differentiable functions in such that , . If , then
Moreover, the equality in (1.4) is satisfied in if and only if for some constant .
For convenience of further applications of (1.4), we rewrite it, assuming , as follows:
Note that both (1.3) and (1.4) turn to the Picone inequality (1.1) when . Moreover, we emphasize that (1.3) requires , while (1.4) asks for . Our main result, Theorem 1.8, posits the fact that inequality (1.4) remains valid for some , although the set of feasible values of p and q is not of a trivial structure (Figure 1). This set is defined and characterized in the following lemma.
Let be fixed. Let the function be defined as
and set . Then . Moreover, there exists such that the following assertions hold:
if , then there exist and satisfying such that and ;
if , then .
Furthermore, if and , then , i.e., .
In particular, each of the following two explicit assumptions is sufficient to guarantee that :
Assertions (i) and (ii) of Lemma 1.6 yield for any . A numerical investigation of the function g indicates that and that in assertion (i) can be chosen such that , that is, if , then (Figure 1).
Now we are ready to state our main result.
Let and let be differentiable functions in such that , . Assume that one of the following assumptions is satisfied:
, where is given by Lemma 1.6;
Moreover, if and , then the equality in (1.6) is satisfied in if and only if for some constant .
Furthermore, assumptions (i) and (ii) are optimal in the following sense:
A closer look at the proof of Theorem 1.8(ii) reveals that inequality (1.6) remains valid under assumption (ii) also for . In fact, even the following stronger result, which reduces to the commutativity of the scalar product in at , can be obtained by the same method of proof.
Let be differentiable functions in such that , , and . Then the following assertions hold:
if , then(1.7)
if , then(1.8)
Moreover, if , then the equality in (1.7) or (1.8) is satisfied in if and only if for some constant .
Apart from the choice of or as a test function, one could also consider more general test functions of the form or . In this direction, the following partial case of a generalized Picone inequality obtained in  by applying an inequality from [9, Lemma 2.1] can be effectively used.
[10, Theorem 2.2] Let and let be differentiable functions in such that , . Assume that satisfies for all . Then
Moreover, the equality in (1.9) is satisfied in if and only if for some constant .
Let . Since , we get from (1.9) the complementary inequality
Note that the term is well-defined if either and , or and . In particular, under any of these assumptions, taking , we obtain
Moreover, under the assumption , the application of Young’s inequality gives
Evidently, (1.11) and (1.12) reduce to the Picone inequality (1.1) if .
As a complementary fact, we provide the following optimal refinement of a generalized Picone inequality obtained in [5, Proposition 8], by analysing the right-hand sides of inequalities (1.9) and (1.10).
Let , , and let be differentiable functions in such that , . If , then
where if , and if .
Finally, let us note that the Picone inequality (1.1) can be used to derive the Díaz-Saa inequality :
which, in particular, holds (in the sense of distributions) for all such that a.e. in , and , assuming that is smooth and bounded, see [21, Theorem 2.5]. Inequality (1.13) 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 -Laplacian, together with the corresponding applications, was obtained in , see also [3,21]. Under the same assumptions on and as above, it can be stated as follows. If and , then
Inequality (1.14) 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 1.8 and Lemma 1.6. In Section 3, we provide several applications of Theorem 1.8, as well as of Theorems 1.2 and 1.4, to problems with the -Laplacian.
2 Proofs of Theorem 1.8 and Lemma 1.6
Proof of Theorem 1.8
Since the case is covered by Theorem 1.4, we will assume hereinafter that . Moreover, under any of the assumptions (i) and (ii), p has the upper bound (see Lemma 1.6 in the case of assumption (i)).
By straightforward calculations, we get
We see from (2.1) and (2.2) that the desired inequality (1.6) is equivalent to
Dividing by , we reduce (2.3) to
Recalling that , 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 , and hence . We consider two separate cases.
In this case, in order to validate (2.4) it is sufficient to prove that(2.5)
Denoting , we see that (2.5) holds provided(2.6)
Let us show that (2.6) is satisfied. We have
Combining this strict convexity of f with the facts that and , we see that
and the equality for such s happens if and only if . In particular, for all provided . Assume that . Since f is concave on , , for , and for , we conclude that
and the equality for such s happens if and only if and . Thus, we have derived that for all provided . In particular, this implies that (1.6) is satisfied under assumption (ii). Moreover, we have shown that if , then if and only if . Therefore, if , , and the equality in (1.6) is satisfied in , then we conclude that and , which yields in , that is, for some constant .
To establish (2.4) under assumption (2.7), it is sufficient to show that
Introducing again the notation , we see that inequality (2.8) holds if
Applying Lemma 1.6, we deduce that (2.9) is satisfied whenever .
Combining cases (1) and (2), we conclude that (1.6) holds under assumption (i), which finishes the proof of the first part of the theorem.
Let us now obtain the optimality of assumptions (i) and (ii) stated in (I) and (II), respectively. Assume first that and let be such that . Consider
for some . Noting that and taking , we conclude that the violation of (2.9) at implies the violation of (2.8) at . On the other hand, we have . Thus, the violation of (2.8) at is equivalent to the violation of (2.4) at , which, in its turn, is equivalent to the violation of (1.6) at . This establishes case (I).
Assume now that . Set and let be any differentiable function not identically equal to a constant. We readily see that and (2.3) is violated at points where , which establishes case (II).□
Now we provide the proof of Lemma 1.6.
Proof of Lemma 1.6
To prove that , we first note that
for all . This yields , and hence . Second, we have for any , which implies that . Third, we see that
Therefore, since g is uniformly continuous on compact subsets of , we conclude that . Moreover, the continuity of g gives .
Let us prove assertions (i) and (ii). To this end, we note that
In view of this monotonicity, inequalities (2.10) and (2.11) yield . At the same time, we see that if , then
which shows, in particular, that . Thus, we deduce that provided . Let us now define the critical value
Clearly, . Since q is presented in the function g only as a positive coefficient, we see that implies provided , i.e., . Combining this fact with the already obtained inclusion , we conclude that if , then . Note that if q is sufficiently close to 1, then , and so . Indeed, choosing, for instance, , we have , which yields , and hence . This implies, by the continuity of g, that also for , and this completes the proof of assertion (ii).
Assume now that . In particular, recalling that , we have . We start by showing that, in addition to (2.11), there exists with the property that for all and . Indeed, suppose, by contradiction, that for any one can find and such that , and as . Then must be bounded, since otherwise as . Therefore, passing to the limit along appropriate subsequences of and , we get a contradiction to (2.11). Thus, the monotonicity (2.12) in combination with (2.10) and (2.11) yields , which establishes the existence of from assertion (i).
Now, we show the existence of such that for all and . Suppose, by contradiction, that for any one can find and satisfying , and as . Since the term is the only term in with negative sign, we conclude that as . But then we deduce that
for all sufficiently large n, since is the leading term as , which is impossible. Recalling that , we conclude that . Since , we deduce that and , which completes the proof of assertion (i).
Finally, we justify the sufficient assumptions (I) and (II).
Let . Considering the sum of the second and third terms of , we see that if , then . Thus, let us assume that and . Then we have
where the last inequality is satisfied if and only if .
Let . As in the previous case, we see that if , then . Hence, we assume that and . Then we have
where the last inequality is satisfied if and only if .□
3 Applications to (p,q)-Laplace equations
Throughout this section, we always assume that and that is a smooth bounded domain with boundary , .
Denote by the standard norm of , . Let with stand for the first eigenvalue of the Dirichlet r-Laplacian in , and let be the corresponding first eigenfunction which we assume to be positive and normalized as . That is,
Note that is simple and , where
and is the unit exterior normal vector to . Finally, for a weight function satisfying , we define
We remark that , which follows from the simplicity of and linear independence of and , see [22, Proposition 13].
3.1 General problem with (p,q)-Laplacian
Consider the boundary value problem
where the function is sufficiently regular in order that (3.2) possesses a weak formulation with respect to and satisfies the following assumption:
(A) there exist and satisfying such that
for all , , , and a.e. , where is given by (3.1).
We obtain the following nonexistence result in the class of -solutions.
Let , where is defined in Lemma 1.6, and let (A) be satisfied. If , then (3.2) has no solution in .
Suppose, by contradiction, that (3.2) possesses a solution for some . Noting that since , we choose as a test function for (3.2). Applying Theorems 1.1 and 1.8, we get
which is impossible.□
Note that we cannot claim that each weak solution of (3.2) belongs to since we do not impose suitable “good” assumptions on the function apart from those stated above; see, e.g.,  for a related discussion. We also remark that neither of the generalized Picone inequalities (1.3), (1.4), and (1.9) can be used (at least, as directly as (1.6)) to establish Theorem 3.1.
3.2 Eigenvalue-type problem
In the special case , (3.2) can be seen as an eigenvalue problem for the -Laplacian:
see, e.g., [5,18,22,24]. Note that any nonzero and nonnegative solution of (3.3) belongs to , see, for instance, [5, Remark 1] or [18, Section 2.4]. Although in the works in [5,22] 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 for which (3.2) possesses a positive solution. Thanks to our generalized Picone inequality (1.6), as well as to inequalities (1.3) and (1.4), we can provide additional information in this regard.
First, the same reasoning as in Theorem 3.1 allows us to show the following result.
Assume that one of the following assumptions is satisfied:
, where is defined in Lemma 1.6;
and is an N-ball.
Then (3.3) has no positive solution for , where is given by (3.1). Moreover, if and is an N-ball, then (3.3) has no positive solution also for .
Let u be a positive solution of (3.3). Recall that . Moreover, under assumption (ii), both u and are radially symmetric with respect to the centre of and nonincreasing in the corresponding radial direction (see [25, Theorem 3.10]), which yields in . Clearly, satisfies (A) with (so ) and . Therefore, applying Theorem 1.8 as in the proof of Theorem 3.1, we obtain the desired nonexistence for . On the other hand, if , , and is an N-ball, then Theorem 1.8 yields for some . Since u is a solution of (3.3), we see that u must also be an eigenfunction of the q-Laplacian associated with the eigenvalue , which is impossible in view of [22, Proposition 13].□
Note that Theorem 3.2 is optimal for the considered range of p and q in the sense that for any problem (3.3) possesses a positive solution, see [22, Theorem 2.5 (i)]. Nevertheless, we are not aware of the corresponding existence result for under assumption (i) of Theorem 3.2, or if and is an N-ball.
Second, we provide the following general result without restrictions on p and q apart from the default assumption , whose proof is based on a nontrivial application of Picone’s inequalities (1.3) and (1.4), and on the usage of results from [5,22].
Then . Moreover, (3.3) has at least one positive solution if , and no positive solution if or . Furthermore, if , then (3.3) has at least one positive solution if and only if .
In order to prove Theorem 3.3, we need the following auxiliary information on the behaviour of positive solutions.
Let be a sequence, and let be a positive solution of (3.3) with , . Then the following assertions hold:
if , then and, up to a subsequence,(3.4)
if , then .
We start with the observation that (3.3) has no nonzero solution for , see [5, Proposition 1] and [22, Proposition 13]. Thus, throughout the proof, we will assume that for all . In particular, we have .
Let as . Note first that . Indeed, suppose, by contradiction, that , up to a subsequence. Setting and taking as a test function for (3.3) with , we have
where the inequality follows from the definition of . Since , , and , we conclude that, simultaneously, and strongly in , up to a subsequence, where is some constant. However, this contradicts the linear independence of and , see [22, Proposition 13], and hence .
Now we prove the convergence (3.4). Let be such that weakly in and strongly in , up to a subsequence. First, we show that in . Suppose, by contradiction, that in . Then, by Egorov’s theorem, converges to 0 uniformly on a subset of of positive measure. In particular, we have(3.5)
Note that the integral in (3.5) is well-defined since due to the -regularity of and . Using now the Picone inequalities (1.1) and (1.5), we get from (3.3) with and the test function that(3.6)
This implies that(3.7)
and hence, since , there exists a constant independent of n such that(3.8)
On the other hand, choosing as a test function for (3.3) with , we get
Since we suppose that in , we have strongly in and , which yields(3.9)
Combining (3.8) and (3.9), we obtain
which gives a contradiction to (3.5) and the strong convergence in . Therefore, in .
Second, we show that . Since is a solution of (3.3) with , we see that satisfies(3.10)
Taking and recalling that and that converges in to a nonzero function , we conclude that there exists a constant such that
Taking now in (3.10), we see that the boundedness of implies
which guarantees that strongly in , see, e.g., [26, Lemma 5.9.14]. Passing to the limit in (3.10), we deduce that is a nonzero and nonnegative solution of the problem
The standard regularity result  and the strong maximum principle yield , and so . Thus, applying the Picone inequality (1.1), we get
which yields , and hence in .
Now we are ready to prove that in . Thanks to the boundedness of , using the Moser iteration process in (3.10), we can find independent of n such that for all n. Thus, since is also bounded, applying to Eq. (3.10) the regularity results [28, Theorem 1.7] and , we derive the existence of and , both independent of n, such that and for every sufficiently large n. Since is compactly embedded into , we conclude that in , up to a subsequence.
Finally, let us show that . First, let be a subsequence such that . Taking as a test function for (3.3) with , we get
and hence the convergence of to along a sub-subsequence yields
Second, we choose a subsequence such that and denote it, for simplicity, as . Using Picone’s inequalities (1.1) and (1.5) with , we get from (3.3) with that
which implies that(3.11)
Note that both sides of (3.11) share the same homogeneity with respect to , and hence we can replace by the normalized function :(3.12)
The convergence in along a sub-subsequence yields the existence of a constant such that for all and all sufficiently large . Therefore, since in , and pointwise in , the Lebesgue dominated convergence theorem guarantees
Here, the latter convergence can be easily seen from the expansion