Sturm comparison theorems via Picone-type inequalities for some nonlinear elliptic type equations with damped terms

In this paper, we establish a Picone-type inequality for a class of some nonlinear elliptic type equations with damped terms, and obtain Sturmian comparison theorems using the Picone-type inequality. As an application by using comparison theorem oscillation result and Wirtinger-type inequality are given.


Introduction
Since the pioneering work of Sturm [27] in 1836, Sturmian comparison theorems have been derived for differential equations of various types.In order to obtain Sturmian comparison theorems for ordinary differential equations of second order, Picone [25] established an identity, known as the Picone identity.In the latter years, Jaroš and Kusano [15] derived a Picone-type identity for half-linear differential equations of second order.They also developed Sturmian theory for both forced and unforced half-linear and quasilinear equations based on this identity.Since Picone identities play an important role in the study of qualitative theory of differential equations, establishing Picone identities has become a popular research topic.We refer the reader to Kreith [20,21], Swanson [28,29] for Picone identities and Sturmian comparison theorems for linear elliptic equations and to Allegretto [3], Allegretto and Huang [4,5], Bognár and Došlý [9], Dunninger [12], Kusano, Jaroš and Yoshida [22], Yoshida [32,31,30] for Picone identities, Sturmian comparison and/or oscillation theorems for half-linear elliptic equations.In particular, we mention the paper [12] by Dunninger which seems to be the first paper dealing with Sturmian comparison theorems for half-linear elliptic equations.

(u)
where a k (x), A k (x) are matrices and Most of the work in the literature deals with the Sturmian comparison results for elliptic equations that contain undamped terms.In this paper, we establish Sturmian comparison theorems for a pair of damped elliptic operators p and P defined by where T , (the superscript T denotes the transpose).It is assumed that To the best of our knowledge, damped elliptic operators such as p(u) and P(v) defined as above have not been studied.Note that the principal part of (1.1) and (1.2) are reduced to the p-Laplacian ∇ • |∇u| p−2 ∇u , (p = α + 1).We know that a variety of physical phenomena are modeled by equations involving the p-Laplacian [2,7,8,23,24,26].We refer the reader to Diaz [11] for detailed references on physical background of the p-Laplacian.
We organize this paper as follows.In Section 2, we establish a Picone-type inequality.In Section 3, we present comparison results for the equations p(u) = 0 and P(v) = 0 and in Section 4, as an application we conclude some oscillation results and give a Wirtinger-type inequality.

Picone-type inequalities
In this section, we establish a Picone-type inequality for the coupled operators p and P defined by (1.1) and (1.2) respectively.Let G be a bounded domain in R n with piecewise smooth boundary ∂G, and assume that a The domain D p (G) of p is defined to be the set of all functions u of class C 1 ( Ḡ, R) with the property that a(x Let N = min{ , m} and We will need the following lemmas, in order to prove our results. holds for any function u ∈ C 1 (G, R) and any n-vector function w , then the following Picone-type inequality holds: Proof.We easily see that We observe that the following identity holds: (2.3) A. Tiryaki and S. Şahiner We combine (2.2) with (2.3) to obtain the following: (2.4) Using Young's inequality we have, We combine (2.5)-(2.7)with (2.4) to obtain the desired inequality (2.1).
Theorem 2.4.If v ∈ D P (G), and v = 0 in G, then the following inequality holds for any u ∈ C 1 (G, R): where ϕ(s), Φ(ξ) and C 1 (x) are defined as in Theorem 2.3.

Sturmian comparison theorems
In this section we present some Sturmian comparison results on the basis of the Picone-type inequality obtained in Section 2.
then every solution v ∈ D P (G) of P(v) = 0 must vanish at some point of Ḡ.
Proof.Suppose that, contrary to our claim there exists a solution v ∈ D P (G) of P(v) = 0 satisfying v = 0 on Ḡ.We integrate (2.1) over G and then apply the divergence theorem to obtain and therefore From Lemma 2.1, we see that then it follows from a result of Jaroš, Kusano and Yoshida [17] that for some constant C 0 and some continuous function α(x).Since u = 0 on ∂G, we see that C 0 = 0, which contradicts the fact that u is nontrivial.The proof is complete. and in G.If there exists a nontrivial solution u ∈ D p (G) of p(u) = 0 such that u = 0 on ∂G, then every solution v ∈ D P (G) of P(v) = 0 must vanish at some point of Ḡ.
then every solution v ∈ D P (G) of P(v) = 0 must vanish at some point of G unless u = C 0 e α(x) v, where C 0 = 0 is a constant and ∇α(x) = B(x) A(x) in G. Proof.Suppose that there exists a solution v ∈ D P (G R) and u = 0 on ∂G, we find that u belongs to the Sobolev space W 1,α+1 0 (G) which is the closure in the norm of the class C ∞ 0 (G) of infinitely differentiable functions with compact supports in G [1,13].Then there is a sequence u k of functions in C ∞ 0 (G) converging to u in the norm (3.9).Integrating (2.8) with u = u k over G, then applying the divergence theorem, we have We first claim that lim k→+∞ M(u k ) = M(u) = 0. Since A(x), C(x), D(x) and E(x) are bounded on Ḡ, there exists a constant K 1 > 0 such that (3.12) From the mean value theorem we see that Since also B(x) is bounded on Ḡ, then there is a constant K 2 such that

|B(x)|
A(x) ≤ K 2 on Ḡ.Let us take K 3 = max{1, K 2 }.From the above inequality we have Sturm-Picone theorems for elliptic damped equations 7 Using (3.13) and applying Hölder's inequality, we get for some positive constant K 4 = K 4 (K 1 , K 2 , K 3 ) and so that lim k→+∞ M(u k ) = M(u).We get from (3.10) that M(u) ≥ 0 which together with (3.8) implies M(u) = 0. Let B be an arbitrary ball with B ⊂ G and define where Q G (u k ) denotes the right-hand side of (3.17) with w = u k and with B replaced by G.
A simple calculation yields where q = α+1 α , the constants K 5 , K 6 and K 7 are independent of k and the subscript B indicates the integrals involved in the norm (3.9) are to be taken over B instead of G.It is known that the Nemitski operator ϕ : L α+1 (G) → L q (G) is continuous [6] and it is clear that A. Tiryaki and S. Şahiner Hence we observe that u v = C 0 e α(x) in B for some constant C 0 and some continuous function α(x) as in the proof of Theorem 3.1.Since B is an arbitrary ball with B ⊂ G, we conclude that u v = C 0 e α(x) in G where C 0 = 0.
Hence the result follows from Theorem 3.3.
Remark 3.5.When we take α = 1, b(x) ≡ B(x) ≡ 0 and D i (x) ≡ E i (x) ≡ 0, (i = 1, 2, . . ., , j = 1, 2, . . ., m) that is, in the linear elliptic equation case, and b(x) ≡ B(x) ≡ 0 and D i (x) ≡ E i (x) ≡ 0, (i = 1, 2, . . ., , j = 1, 2, . . ., m) that is, in the half-linear elliptic equation case, our results cannot be reduced to the well-known results.Hence our results are indeed a partial extension of the results that are given in the literature.Improvement of our results is left as an open problem to the researchers.

Applications
Let Ω be an exterior domain in R n , that is, Ω ⊃ {x ∈ R n : |x| ≥ r 0 } for some r 0 > 0. We consider the following equations: and where the operators p and P are defined in Section 1 and The domain D p (Ω) of p is defined to be the set of all functions u of class C 1 (Ω, R) with the property that a(x)|∇u| α−1 ∇u ∈ C 1 (Ω, R n ).The domain D P (Ω) of P is defined similarly.
A solution u ∈ D p (Ω) of (4.1) (or v ∈ D P (Ω) of (4.2)) is said to be oscillatory in Ω if it has a zero in Ω r for any r > 0, where A bounded domain G with Ḡ ⊂ Ω is said to be a nodal domain for the equation (4.1), if there exists a nontrivial function u ∈ D p (G) such that p(u) = 0 in G and u = 0 on ∂G.The equation (4.1) is called nodally oscillatory in Ω, if (4.1) has a nodal domain contained in Ω r for any r > 0.
Proof.Since (4.1) in nodally oscillatory in Ω, there exist a nodal domain G ⊂ Ω r for any r > 0, and hence there exists a nontrivial function u ∈ D p (G) such that p(u) = 0 in G and u = 0 on ∂G.The conditions (4.3) and (4.4) ensures that V(u) ≥ 0 is satisfied.From Corollary 3.2 it follows that every solution v ∈ D P (Ω) of (4.2) vanishes at some point of Ḡ, that is, v must have a zero in Ω r for any r > 0. This implies that v is oscillatory in Ω.
The following is an immediate consequence of Various criteria for nodal oscillation can be found in [32].For example for linear elliptic equations of the form u + c(x)u = 0, x ∈ R 2 , (4.6) c(x) being a continuous function in R 2 , have been given by Kreith and Travis [19].They showed that (4.6) is nodally oscillatory if Applying this result to the equation (4.5) with α = 1, a(x) ≡ 1 we have the following result.When we take α = 1, m = 1, a(x) ≡ 1, C(x) ≡ 0, Corollaries 4.2-4.3reduce to Corollaries 3-4 given in [16], respectively.Inequality (2.8) is utilized to establish Wirtinger-type inequality concerning the elliptic type nonlinear equation P(v) = 0. We know that a typical Wirtinger inequality is the following.