Abstract

We establish an estimate for the ratio of eigenvalues of the Dirichlet eigenvalue problem for the equation with one-dimensional -Laplacian involving a nonnegative unimodal (single-well) potential.

1. Introduction

We consider the eigenvalue problem for the equation with the one-dimensional -Laplacian , a nonnegative differentiable function , and the Dirichlet boundary condition where . Equation (1.1) is also frequently called half-linear equation, since its solution space is homogeneous but not additive, that is, it has just one half of the properties which characterize linearity. We refer to the books in [1, 2] for the presentation of the essentials of the qualitative theory of differential equations with the one-dimensional -Laplacian. Our research is motivated by [3], where the linear case in (1.1), (1.2) is investigated under the assumption that is a nonnegative unimodal function (an alternative terminology is the single-well potential). Concerning the history of the problem of the ratio of eigenvalues in the linear case, we refer to the papers [4–7] and the reference given therein. For estimates of the ratio of eigenvalues of BVP's involving -Laplacian see, for example, [8, 9]. Similarly to the linear case treated in [3], throughout the paper we suppose that Under this assumption it is shown in [3] that the eigenvalues of satisfy Moreover, if the equality holds in (1.5) for a pair of different integers, then in . In our paper we show that this statement can be extended in a natural way to (1.1), (1.2). We show that (1.5) holds true for the half-linear case if in (1.5) the power 2 by integers is replaced by the power . As we will see, some arguments used in [3] can be extended directly to (1.1), while others have to be “properly half linearized”.

The investigation of BVP (1.1), (1.2) is closely related to the half-linear trigonometric functions and to the half-linear Prüfer transformation. Consider the equation and its solution given by the initial condition . This solution is a periodic odd function, we denote it by , see [2, 10]. If , it reduces to the classical sine function. The derivative defines the half-linear cosine function and for these functions the Pythagorian identity can be formulated as the identity We will also use the half-linear tangent and cotangent functions By a direct computation, (1.6) can be written in the form and using (1.9) we have Like for , for and for , which is equivalent to for , . A similar formula to (1.10) for is related to the Riccati equation associated with (1.1). Namely, if is a solution of (1.1) in some interval , then the function solves the Riccati equation In particular, from  (1.6)

Let be a nontrivial solution of (1.1) and consider the half-linear Prüfer transformation (see [2, 10]) Then using the same procedure as in case of the classical linear Prüfer transformation one can verify that and are solutions of From (1.15), at the points where , that is, where , . Also, solutions of (1.15) behave similarly as in the linear case which means that the eigenvalues of (1.1), (1.2) are simple, form an increasing sequence and the corresponding eigenfunction has exactly zeros in . Moreover, if , then with the associated eigenfunction .

2. Preliminary Computations

To prove our main result, we will use the half-linear Prüfer transformation in a modified form. Therefore, we rewrite (1.1) into the form with . Note that due to the fact that , all eigenvalues of (1.1), (1.2) are positive. Let be a nontrivial solution of (2.1) for which . For this solution we introduce the Prüfer angle and radius by Differentiating the first equation and comparing it with the second one we obtain Equation (2.1) can be written as and similarly one can rewrite (1.6) as . Differentiating the second equation in (2.2) and substituting into (2.4) we have Multiplying (2.3) by , (2.5) by , adding the resulting equations and dividing them by we get By a similar computation, we get the equation for the radius Concerning the dependence of on the eigenvalue parameter , we have from (2.6) Sometimes, we will skip the argument of and when its value is not important or it is clear what value we mean. The last equation can be regarded as a first-order linear (nonhomogeneous) differential equation for .Multiplying this equation by the integration factor we have (since for and hence )

The dependence of the function on plays a crucial role in the proof of our main statement. Applying (2.10) and (2.6), we have

3. Ratio of Eigenvalues

In the previous section we have prepared computations which we now use in the proof of our main result which reads as follows.

Theorem 3.1. Suppose that is a nonnegative differentiable function such that (1.3) holds. Then one has for eigenvalues of (1.1), (1.2) If for two different integers the equality holds, then on.

Proof. Let be a nontrivial solution of (2.1) for which , and let , be its Prüfer radius and angle given by (2.2) with . A value corresponds to an eigenvalue of (1.1), (1.2) if and only if . As noted below (1.16), it follows from (2.2) that when , . Using the same argument as in the linear case (see, e.g., [6]) holds.
Let be given by (2.11). Suppose that we have already proved that for and when the equality happens for some , then on . Like in [3], we investigate (2.1) on the interval using the reflection argument. Let . Then, for the value plays the same role as for , in particular, the function satisfies (1.3) when is replaced by . Further, let be the eigenfunction of (2.1), (1.2) corresponding to the eigenvalue , that is, . Define where is the Prüfer angle of for which . Then and , hence is an eigenfunction of (2.1), (1.2) when is replaced by . Moreover, we have , and and hence Similarly , that is, is the Prüfer angle corresponding to . Denote . The function plays the same role for (2.1), (1.2) with replaced by , as for the original eigenvalue problem. Hence let us also suppose that we have proved that with the equality only if on . Now, consider the function Under the monotonicity assumptions on and , the function is nondecreasing and when for two different values , then on and on . Let and be the values of the eigenvalue parameter corresponding to the eigenvalues , . Then Similarly, . Consequently, and therefore In case of the equality in (3.8), we have on and on , altogether on .
Now let us turn our attention to the monotonicity property of . First consider the case . In this case the inequality (with equality implying on ) follows immediately from (2.12) since the integrand in this expression is nonnegative by (1.11). So suppose that for some and some nonnegative integer . Then we need some preliminary computations. Suppose that we already know that the function is strictly increasing with respect to for . In this case we may split the integral below as where we have denoted the integrand in (3.9) by . As soon as we show that each integral is nonnegative and equals zero only if on the corresponding interval, the monotonicity of will be proved.
First we will show the strict monotonicity of with respect to . Fix and suppose, by contradiction, that for some . This implies by (2.6) that for using (1.3) in the last inequality. Hence that is, is convex and strictly increasing for , that is, and hence by (2.2) . By (2.6) also , and implies the existence of such that and for . Fix any and consider the function . This function is a solution of Riccati equation (1.12) and from (1.13) Hence for , which means that in and equality happens for , that is Recall that is the conjugate pair of and is the inverse function of . Let and denote for a moment . We have (suppressing the integration argument) In the last inequality we have used that for by (1.3). Denote and consider the function This function is bounded when its argument is bounded as it can be verified by computing its limit for . But is bounded since for . Consequently, the last integral is bounded below as , while the integral in (3.15) equals since at (3.14) holds. This contradiction shows that for and .
Now we will deal with integrals in (3.9). The first one over the interval is nonnegative since its integrand is nonnegative in this interval by (1.11) and equals 0 only if . Concerning the integrals under the summation sign, first observe that the value of the functions and does not change if we replace by with any integer . Hence, using the substitution which moves to (where (1.11) holds) we have Here we have again used (1.11) since this inequality can be applied in view of the transformation . The last result leads to the investigation of the monotonicity properties (with respect to ) of the radius . We will use the fact that the function has the same monotonicity as . From (2.7) it immediately follows that is increasing for while it is decreasing for . Taking the integral of (2.7) in view of (2.6) and substituting , one gets The function is negative between and , while the denominator of the last fraction is positive by (3.10). Consequently, if we replace by its minimum in this interval, we obtain Using the same argument in the interval where the function is positive, so if we replace by its maximum, we have Summing the last two results Consequently, we have and this inequality shows that each integral in the sum in (3.9) is nonnegative and equals 0 only if . We handle the last integral in (3.9) over in a similar way (suppressing the integration variable ) because of the monotonicity property of and since , so the integrand containing this argument is nonnegative by (1.11).
Therefore, each integral in (3.9) is nonnegative and we have proved the required statement concerning monotonicity (with respect to ) of the function . Finally, since the function plays the same role as , the above used arguments prove also monotonicity with respect to of . This means that the function given in (3.5) is monotone and the proof is complete.