Forced oscillation criteria for quasilinear elliptic inequalities with p(x)-Laplacian are derived by using the Riccati inequality. The approach used is to reduce forced oscillation problems for quasilinear elliptic inequalities with p(x)-Laplacian to one-dimensional oscillation problems for Riccati inequalities with variable exponents.More general quasilinear elliptic inequalities with mixed nonlinearities are also investigated

Yoshida Norio

Forced oscillation criteria for quasilinear elliptic inequalities with p(x)-Laplacian are derived by using the Riccati inequality. The approach used is to reduce forced oscillation problems for quasilinear elliptic inequalities with p(x)-Laplacian to one-dimensional oscillation problems for Riccati inequalities with variable exponents.
More general quasilinear elliptic inequalities with mixed nonlinearities are also investigated.ArticleToyama mathematical journal，vol.34，Page 93-106departmental bulletin pape

Yoshida, Norio

University of Toyama Repository

Toyama Math. J.Vol. 34(2011), 93-106Forced oscillation criteria for quasilinearelliptic inequalities with p(x)-Laplacianvia Riccati method¤Norio YoshidaAbstract. Forced oscillation criteria for quasilinear elliptic inequal-ities with p(x)-Laplacian are derived by using the Riccati inequality.The approach used is to reduce forced oscillation problems for quasi-linear elliptic inequalities with p(x)-Laplacian to one-dimensional os-cillation problems for Riccati inequalities with variable exponents.More general quasilinear elliptic inequalities with mixed nonlineari-ties are also investigated.1. IntroductionThere is much current interest in studying oscillations of half-linear el-liptic equations with p-Laplacian(p = ®+ 1) of the formr ¢ ¡a(x)jruj®¡1ru¢+ c(x)juj®¡1u = 0;where ® > 0 is a constant, r = (@=@x1; :::; @=@xn) and the dot ¢ denotesthe scalar product (cf. [1, 3{6, 11, 16]). The operator r ¢ ¡jrujp(x)¡2ru¢is said to be p(x)-Laplacian, and becomes p-Laplacian r ¢ ¡jrujp¡2ru¢ if2000 Mathematics Subject Classi¯cation. 35B05, 35J92.Key words and phrases. forced oscillation, elliptic inequalities, half-linear, quasilin-ear, p(x)-Laplacian, Riccati method.¤This research was partially supported by Grant-in-Aid for Scienti¯c Research (C)(No.20540159), the Ministry of Education, Culture, Sports, Science and Technology, Japan.9394 Norio Yoshidap(x) = p (constant). The study of various mathematical problems withvariable exponent growth condition has been received considerable atten-tion in recent years (see [7]). These problems arise from nonlinear elasticitytheory, electrorheological °uids (cf. [14,20]). In 2007 Zhang [19] studied os-cillation problems for the p(t)-Laplacian equation¡ju0jp(t)¡2u0¢0 + t¡µ(t)g(t; u) = 0; t > 0:It is noted that the elliptic equation with p(x)-Laplacian (p(x) = ®(x)+ 1)r ¢ ¡A(x)jrvj®(x)¡1rv¢+ C(x)jvj®(x)¡1v = 0is not half-linear if ®(x) is not a constant. However, it is shown that theelliptic inequality with p(x)-Laplacian (p(x) = ®(x) + 1)vQ0[v] · 0 (1)is half-linear in the sense that a constant multiple of a solution v of (1) isalso a solution of (1), whereQ0[v] := r ¢¡A(x)jrvj®(x)¡1rv¢¡A(x)(log jvj)jrvj®(x)¡1r®(x) ¢ rv+jrvj®(x)¡1B(x) ¢ rv + C(x)jvj®(x)¡1v: (2)In fact, it can be shown that(kv)Q0[kv] = jkj®(x)+1vQ[v] (k 2 R)(cf. Yoshida [17, Proposition 2.1]). We refer to Allegretto [2] and Yoshida[18] for Picone identity arguments for elliptic operators with p(x)-Laplacian.The objective of this paper is to investigate oscillatory behavior of so-lutions of the quasilinear elliptic inequality vQ[v] · 0 with p(x)-Laplacian(p(x) = ®(x) + 1) and a forcing term, whereQ[v] := r ¢ ¡A(x)jrvj®(x)¡1rv¢¡A(x)(log jvj)jrvj®(x)¡1r®(x) ¢ rv+jrvj®(x)¡1B(x) ¢ rv+C(x)jvj®(x)¡1v +D(x)jvj¯(x)¡1v + E(x)jvj°(x)¡1v ¡ f(x): (3)We note that log jvj in (2), (3) has singularities at zeros of v, but v log jvjbecomes continuous at the zeros of v if we de¯ne v log jvj = 0 at the zeros, inForced oscillation criteria for quasilinear elliptic inequalities with p(x)-Laplacian 95light of the fact that lim"!+0 " log " = 0. Therefore, we conclude that vQ[v]has no singularities and is continuous in ­. We remark that vQ[v] · 0 isnot half-linear.The approach used is to reduce the multi-dimensional oscillation prob-lems to one-dimensional oscillation problems for Riccati di®erential inequal-ities, and to utilize the Riccati techniques.In Section 2 we establish Riccati inequality for vQ[v] · 0, and in Section 3we give oscillation results for vQ[v] · 0 on the basis of the Riccati inequalityobtained in Section 2.2. Riccati inequalityLet ­ be an exterior domain in Rn, that is, ­ includes the domain fx 2Rn; jxj ¸ r0g for some r0 > 0. We assume that A(x) 2 C(­; (0;1)), B(x) 2C(­;Rn), C(x) 2 C(­;R), D(x) 2 C(­; [0;1)), E(x) 2 C(­; [0;1)),f(x) 2 C(­;R), and that ®(x) 2 C1(­; (0;1)), ¯(x) 2 C(­; (0;1)),°(x) 2 C(­; (0;1)), and that ¯(x) > ®(x) > °(x) > 0.The domain DQ(­) of Q is de¯ned to be the set of all functions v of classC1(­;R) such that A(x)jrvj®(x)¡1rv 2 C1(­;Rn).A solution v 2 DQ(­) of vQ[v] · 0 is said to be oscillatory in ­ if it hasa zero in ­r for any r > 0, where­r = ­ \ fx 2 Rn; jxj > rg:We use the notation:A(r;1) = fx 2 Rn; jxj > rg;A[r;1) = fx 2 Rn; jxj ¸ rg:Since ­ is an exterior domain in Rn, we see that­r1 = A(r1;1)for some large r1 ¸ r0.96 Norio YoshidaLemma 1. If v 2 DQ(­) and jv(x)j ¸ ¸ in A[r2;1) for some ¸ > 0 andsome r2 > r1, then we obtain the following:¡r ¢ÃA(x)jrvj®(x)¡1rvjvj®(x)¡1v!¸ C(x) + F (¯(x); ®(x); °(x);D(x); E(x))¡ ¸¡®(x)jf(x)j+®(x)A(x)¯¯¯¯rvv¯¯¯¯®(x)+1+B(x) ¢Ãjrvj®(x)¡1rvjvj®(x)¡1v!¡ vQ[v]jvj®(x)+1 (4)in A[r2;1), whereF (¯(x); ®(x); °(x);D(x); E(x))=µ¯(x)¡ °(x)®(x)¡ °(x)¶µ¯(x)¡ ®(x)®(x)¡ °(x)¶®(x)¡¯(x)¯(x)¡°(x)D(x)®(x)¡°(x)¯(x)¡°(x)E(x)¯(x)¡®(x)¯(x)¡°(x) :Proof. Letting'(v) := jvj®(x)¡1v = jv(x)j®(x)¡1v(x);we derive the following:¡r ¢ÃA(x)jrvj®(x)¡1rv'(v)!= ¡A(x)jrvj®(x)¡1rµ1'(v)¶¢ rv¡ 1'(v)r ¢ ¡A(x)jrvj®(x)¡1rv¢ (5)andA(x)jrvj®(x)¡1rµ1'(v)¶¢ rv= ¡®(x)A(x)¯¯¯¯rvv¯¯¯¯®(x)+1¡ 1'(v)A(x)(log jvj)jrvj®(x)¡1r®(x) ¢ rv (6)(see Yoshida [17, (2.5) and (2.7) in the proof of Lemma 2.1]). Combining(5) with (6) yields¡r ¢ÃA(x)jrvj®(x)¡1rv'(v)!= ®(x)A(x)¯¯¯¯rvv¯¯¯¯®(x)+1¡ 1'(v)hr ¢ ¡A(x)jrvj®(x)¡1rv¢¡A(x)(log jvj)jrvj®(x)¡1r®(x) ¢ rvi: (7)Forced oscillation criteria for quasilinear elliptic inequalities with p(x)-Laplacian 97Using (3) we see that1'(v)hr ¢ ¡A(x)jrvj®(x)¡1rv¢¡A(x)(log jvj)jrvj®(x)¡1r®(x) ¢ rvi=1'(v)hQ[v]¡ jrvj®(x)¡1B(x) ¢ rv ¡ C(x)jvj®(x)¡1v¡D(x)jvj¯(x)¡1v ¡ E(x)jvj°(x)¡1v + f(x)i=vQ[v]jvj®(x)+1 ¡B(x) ¢Ãjrvj®(x)¡1rvjvj®(x)¡1v!¡ C(x)¡ 1'(v)hD(x)jvj¯(x)¡1v + E(x)jvj°(x)¡1vi+1'(v)f(x): (8)Applying Young's inequality, we obtain1'(v)hD(x)jvj¯(x)¡1v + E(x)jvj°(x)¡1vi= D(x)jvj¯(x)¡®(x) + E(x)jvj°(x)¡®(x)¸ F (¯(x); ®(x); °(x);D(x); E(x)) (9)(cf. Jaro·s, Kusano and Yoshida [8, p.717]). It can be shown that1'(v)f(x) · jf(x)jjvj®(x) · ¸¡®(x)jf(x)j: (10)Combining (7){(10), we arrive at the desired inequality (4).Lemma 2. If v 2 DQ(­), vQ[v] · 0 in ­ and jv(x)j ¸ ¸ in A[r2;1) forsome ¸ > 0 and some r2 > r1, then we obtain the Riccati inequality:r ¢W (x) + F¸(x) + ®(x)A(x)¡1=®(x)jW (x)j1+(1=®(x))+hW (x); A(x)¡1B(x)i · 0 in A[r2;1); (11)where hU; V i denotes the scalar product of U; V 2 Rn,W (x) =A(x)jrvj®(x)¡1rvjvj®(x)¡1vandF¸(x) = C(x) + F (¯(x); ®(x); °(x);D(x); E(x))¡ ¸¡®(x)jf(x)j:98 Norio YoshidaProof. SincejW (x)j = A(x)¯¯¯¯rvv¯¯¯¯®(x);we easily see that¯¯¯¯rvv¯¯¯¯®(x)+1=µ jW (x)jA(x)¶®(x)+1®(x)=jW (x)j1+(1=®(x))A(x)1+(1=®(x))and hence®(x)A(x)¯¯¯¯rvv¯¯¯¯®(x)+1= ®(x)A(x)¡1=®(x)jW (x)j1+(1=®(x)): (12)It is clear thatB(x) ¢Ãjrvj®(x)¡1rvjvj®(x)¡1v!= hW (x); A(x)¡1B(x)i: (13)Combining (4), (12), (13), and taking account of vQ[v] · 0, we arrive atthe desired inequality (11).Lemma 3. If v 2 DQ(­), vQ[v] · 0 in ­ and jv(x)j ¸ ¸ in A[r2;1) forsome ¸ > 0 and some r2 > r1, then we obtainr ¢ ¡Ã(x)W (x)¢+ Ã(x)F¸(x) + ®(x)Ã(x)A(x)¡1=®(x)jW (x)j1+(1=®(x))+hW (x); Ã(x)A(x)¡1B(x)¡rÃ(x)i · 0 (14)in A[r2;1) for any Ã(x) 2 C1(A[r2;1);R).Proof. It is easy to see thatr ¢ ¡Ã(x)W (x)¢ = Ã(x)r ¢W (x) + hW (x);rÃ(x)i: (15)Combining (11) with (15) yields the desired inequality (14).The following two lemmas follow by the same arguments as were used inYoshida [17, Lemmas 2.4 and 2.5], and will be omitted.Forced oscillation criteria for quasilinear elliptic inequalities with p(x)-Laplacian 99Lemma 4. If v 2 DQ(­), vQ[v] · 0 in ­ and jv(x)j ¸ ¸ in A[r2;1) forsome ¸ > 0 and some r2 > r1, then we derive the Riccati inequality:r ¢ ¡Ã(x)W (x)¢+ F^¸(x) + ®(x)®(x) + 1g(x)jW (x)j1+(1=®(x)) · 0 (16)in A[r2;1) for any Ã(x) 2 C1(A[r2;1); (0;1)), whereg(x) =®(x) + 12Ã(x)A(x)¡1=®(x);F^¸(x) = Ã(x)F¸(x)¡ 1®(x) + 1g(x)¡®(x)Ã(x)®(x)+1¯¯¯¯B(x)A(x)¡ rÃ(x)Ã(x)¯¯¯¯®(x)+1:Lemma 5. Assume that the following hypothesis holds:(H) ®(x) ´ ®(jxj) in A[r1;1).If v 2 DQ(­), vQ[v] · 0 in ­ and jv(x)j ¸ ¸ in A[r2;1) for some ¸ > 0and some r2 > r1, then we have the Riccati inequality:Y 0(r) +ZSrF^¸(x) dS +®(r)®(r) + 1ª(r)¡1=®(r)jY (r)j1+(1=®(r)) · 0 (17)for r ¸ r2, whereSr = fx 2 Rn; jxj = rg;ª(r) =ZSrg(x)¡®(r)Ã(x)®(r)+1 dS;Y (r) =ZSrÃ(x)hW (x); º(x)i dS;º(x) being the unit exterior normal vector x=r on Sr.3. Oscillation resultsIn this section we present oscillation results for vQ[v] · 0 by using Riccatiinequality in Section 2.Theorem 1. Assume that the hypothesis (H) is satis¯ed, and that thereexists a function Ã(x) 2 C1(A[r1;1); (0;1)) such that the Riccati inequal-ity (17) has no solution on [r;1) for all large r and any ¸ > 0. Then,100 Norio Yoshidafor every solution v 2 DQ(­) of vQ[v] · 0, either v is oscillatory in ­ orsatis¯es the conditionlim infjxj!1jv(x)j = 0: (18)Proof. Suppose to the contrary that there is a nonoscillatory solutionv 2 DQ(­) of vQ[v] · 0 such that lim inf jxj!1 jv(x)j > 0. Then, there is anumber r2 > r1 such that jv(x)j ¸ ¸ in A[r2;1) for some ¸ > 0. Lemma 5implies that (17) has a solution Y (r) on [r2;1) for some r2 > r1 and some¸ > 0. This contradicts the hypothesis and completes the proof.Now we need to investigate su±cient conditions for Riccati inequality(17) to have no solution on [r;1) for all large r and any ¹ > 0.LetD = f(r; s) 2 R2; r ¸ s ¸ r1g;D0 = f(r; s) 2 R2; r > s ¸ r1gand we consider the kernel function H(r; s), which is de¯ned, continuousand su±ciently smooth on D, so that the following conditions are satis¯ed:(K1) H(r; s) ¸ 0 and H(r; r) = 0 for r ¸ s ¸ r1;(K2) there exists a constant k0 > 0 such thatlimr!1H(r; s)H(r; r1)= k0 for all s ¸ r1;(K3)@H@s(r; s) · 0, ¡@H@s(r; s) = h(r; s)H(r; s) for (r; s) 2 D0, whereh(r; s) 2 C(D0;R)(cf. Kong [10], Philos [13]).We let ½(s) 2 C1([r1;1); (0;1)) and de¯ne an integral operator A½¿ byA½¿ (y; r) =Z r¿H(r; s)y(s)½(s) ds; r ¸ ¿ ¸ r1;where y 2 C([¿;1);R). It is easily seen that A½¿ is linear and positive, andin fact satis¯es the following:(A1) A½¿ (k1y1 + k2y2; r) = k1A½¿ (y1; r) + k2A½¿ (y2; r) for k1; k2 2 R;Forced oscillation criteria for quasilinear elliptic inequalities with p(x)-Laplacian 101(A2) A½¿ (y; r) ¸ 0 for y ¸ 0;(A3) A½¿ (y0; r) = ¡H(r; ¿)y(¿)½(¿) +A½¿ ([h¡ ½¡1½0]y; r)(see Wong [15]).Theorem 2. Assume that the hypothesis (H) of Lemma 5 holds. If thereexist functions Ã(x) 2 C1(A[r1;1); (0;1)) and ½(s) 2 C1([r1;1); (0;1))such that for any ¸ > 0lim supr!11H(r; r1)A½r1ÃZSrF^¸(x) dS ¡ 1®(r) + 1¯¯¯¯h¡ ½0½¯¯¯¯®(r)+1ª; r!=1;then, for every solution v 2 DQ(­) of vQ[v] · 0, either v is oscillatory in­ or satis¯es the condition (18).Proof. The proof follows by using exactly the same arguments as inYoshida [17, Theorem 3.2], and will be omitted.Following the classical idea of Kamenev [9], we de¯ne H(r; s) and ½(s)byH(r; s) = (r ¡ s)¹; ¹ > 1; (r; s) 2 D;½(s) = sº ; º 2 R:Then we obtain the following corollary.Corollary. Assume that the hypothesis (H) of Lemma 5 is satis¯ed, and,moreover, that ¹ > 1 and º is a real number. If there exists a functionÃ(x) 2 C1(A[r1;1); (0;1)) such that for any ¸ > 0lim supr!11r¹Z rr1h!nsº+n¡1(r ¡ s)¹M [F^¸](s)¡ 1®(s) + 1sº¡®(s)+1jºr ¡ (¹+ º)sj®(s)+1(r ¡ s)¹¡®(s)¡1ª(s)ids =1;then, for every solution v 2 DQ(­) of vQ[v] · 0, either v is oscillatory in­ or satis¯es the condition (18), where !n denotes the surface area of theunit sphere S1 and M [F^¸](r) denotes the spherical mean of F^¸(x) over thesphere Sr.In addition to the hypotheses (K1){(K3) we suppose the following:102 Norio Yoshida(K4)@H@r(r; s) = ~h(r; s)H(r; s) for (r; s) 2 D0, where ~h(r; s) 2 C(D0;R).Theorem 3. Assume that the hypothesis (H) of Lemma 5 holds. If thereare functions Ã(x) 2 C1(A[r1;1); (0;1)) and ½(s) 2 C1([r1;1); (0;1))such that for each » ¸ r1 and for any ¸ > 0lim supr!1A½»ÃZSrF^¸(x) dS ¡ 1®(r) + 1¯¯¯¯h¡ ½0½¯¯¯¯®(r)+1ª; r!> 0;lim supr!1~A½»ÃZSrF^¸(x) dS ¡ 1®(r) + 1¯¯¯¯~h+½0½¯¯¯¯®(r)+1ª; r!> 0;then, for every solution v 2 DQ(­) of vQ[v] · 0, either v is oscillatory in­ or satis¯es the condition (18).Proof. The proof is quite similar to that of Yoshida [17, Theorem 3.4],and hence is omitted.Remark. In the case where n = 1, Li and Li [12] established oscillationresults for the second-order nonlinear di®erential equation£a(t)jy0(t)j¾¡1y0(t)¤0 + q(t)f(y(t)) = r(t)which are related to Theorem 1.Example. We consider the quasi-linear elliptic inequalityv³r ¢ ¡A(x)jrvj®(x)¡1rv¢¡A(x)(log jvj)jrvj®(x)¡1r®(x) ¢ rv+C(x)jvj®(x)¡1v +D(x)jvj¯(x)¡1v + E(x)jvj°(x)¡1v ¡ f(x)´· 0 (19)in an exterior domain ­, where®(x) = ®(jxj) = 1 + e¡jxj; ¯(x) = 2 + e¡jxj; °(x) = e¡jxj;A(x) =(®(x) + 1)®(x)+13 ¢ 6®(x)jxjn+1 ;C(x) = jxj1¡n; D(x) = jxj3¡3n; E(x) = jxjn¡1;f(x) =~f(x)jxjn¡1+± (~f(x) is a bounded function in ­; 0 < ± < 1):Forced oscillation criteria for quasilinear elliptic inequalities with p(x)-Laplacian 103A simple computation shows thatF (¯(x); ®(x); °(x);D(x); E(x)) = 2D(x)1=2E(x)1=2= 2jxj1¡nand thatF¸(x) = C(x) + F (¯(x); ®(x); °(x);D(x); E(x))¡ ¸¡®(x)jf(x)j= 3jxj1¡n ¡ e¡(1+e¡jxj) log ¸jf(x)j:Choosing Ã(x) = 1, we observe that F^¸(x) = F¸(x) andM [F^¸](r) = 3r1¡n ¡ e¡(1+e¡r) log ¸M [jf(x)j](r):Sinceg(x) =®(x) + 12A(x)¡1=®(x);we easily obtainª(r) = !n2®(r)rn¡1(®(r) + 1)®(r)M [A](r):Letting ¹ = 3 and º = 0, we see that1r¹Z rr1·!nsº+n¡1(r ¡ s)¹M [F^¸](s)¡ 1®(s) + 1sº¡®(s)+1jºr ¡ (¹+ º)sj®(s)+1(r ¡ s)¹¡®(s)¡1ª(s)¸ds=1r3Z rr1·!nsn¡1(r ¡ s)3M [F^¸](s)¡ 1®(s) + 1s¡®(s)+1j3sj®(s)+1(r ¡ s)2¡®(s)ª(s)¸ds=!nr3Z rr1·(r ¡ s)3M [F^¸](s)sn¡1¡(r ¡ s)2¡®(s)M [A](s) 3 ¢ 6®(s)sn+1(®(s) + 1)®(s)+1¸ds=!nr3Z rr1h3(r ¡ s)3 ¡ (r ¡ s)1¡e¡sids¡!nr3Z rr1(r ¡ s)3sn¡1e¡(1+e¡s) log ¸M [jf(x)j](s) ds: (20)104 Norio YoshidaIt can be shown thatZ rr1(r ¡ s)3ds = (r ¡ r1)44= O¡r4¢(r !1); (21)Z rr1(r ¡ s)1¡e¡sds = O¡r2¢ (r !1) (22)(see Yoshida [17, Example 4.1]). It is easy to check that e¡(1+e¡s) log ¸ · K¸andM [jf(x)j](r) · Krn¡1+±;whereK¸ = maxf1; e¡2 log ¸g andK is a positive constant. Hence, it followsthat for any ¸ > 0Z rr1(r ¡ s)3sn¡1e¡(1+e¡s) log ¸M [jf(x)j](s) ds = K¸O¡r4¡±¢(r !1): (23)Combining (20){(23) yieldslimr!1(!nr3Z rr1h3(r ¡ s)3 ¡ (r ¡ s)1¡e¡sids¡!nr3Z rr1(r ¡ s)3sn¡1e¡(1+e¡s) log ¸M [jf(x)j](s) ds)=1 (24)for any ¸ > 0. Since all the hypotheses of Corollary are satis¯ed in view of(20) and (24), it follows from Corollary that for every solution v 2 DQ(­)of (19), either v is oscillatory in ­ or satis¯es the condition (18)References[1] W. Allegretto, Sturm theorems for degenerate elliptic equations, Proc.Amer. Math. Soc., 129 (2001), 3031{3035.[2] W. Allegretto, Form estimates for the p(x)-Laplacian, Proc. Amer.Math. Soc., 135 (2007), 2177{2185.[3] W. Allegretto and Y.X. Huang, A Picone's identity for the p-Laplacianand applications, Nonlinear Anal., 32 (1998), 819{830.[4] W. Allegretto and Y.X. Huang, Principal eigenvalues and Sturm com-parison via Picone's identity, J. Di®erential Equations, 156 (1999),427{438.Forced oscillation criteria for quasilinear elliptic inequalities with p(x)-Laplacian 105[5] O. Do·sl¶y and P. ·Reh¶ak, Half-linear Di®erential Equations, North-Holland Mathematics Studies, 202, Elsevier Science B.V., Amsterdam,2005.[6] D. R. Dunninger, A Sturm comparison theorem for some degeneratequasilinear elliptic operators, Boll. Un. Mat. Ital. A (7), 9 (1995),117{121.[7] P. Harjulehto, P. HÄastÄo, ¶U. Le^ and M. Nuortio, Overview of di®eren-tial equations with non-standard growth, Nonlinear Anal., 72 (2010),4551{4574.[8] J. Jaro·s, T. Kusano and N. Yoshida, Picone-type inequalities for half-linear elliptic equations and their applications, Adv. Math. Sci. Appl.,12 (2002), 709{724.[9] I. V. Kamenev, An integral criterion for oscillation of linear di®erentialequations of second order, Math. Notes, 23 (1978), 136{138: Math.Zamtki, 23 (1978), 249{251.[10] Q. Kong, Interval criteria for oscillation of second-order linear ordinarydi®erential equations, J. Math. Anal. Appl., 229 (1999), 258{270.[11] T. Kusano, J. Jaro·s and N. Yoshida, A Picone-type identity and Stur-mian comparison and oscillation theorems for a class of half-linear par-tial di®erential equations of second order, Nonlinear Anal., 40 (2000),381{395.[12] W. T. Li and X. Li, Oscillation criteria for second-order nonlineardi®erential equations with integrable coe±cients, Appl. Math. Lett.,13 (2000), 1{6.[13] Ch. G. Philos, Oscillation theorems for linear di®erential equations ofsecond order, Arch. Math. (Basel), 53 (1989), 482{492.[14] M. Rºu·zi·cka, Electrorheological Fluids: Modeling and MathematicalTheory, Lecture Notes in Mathematics, Vol. 1748, Springer-Verlag,Berlin, 2000.106 Norio Yoshida[15] J. S. W. Wong, On Kamenev-type oscillation theorems for second-order di®erential equations with damping, J. Math. Anal. Appl., 258(2001), 244{257.[16] N. Yoshida, A Picone identity for half-linear elliptic equations and itsapplications to oscillation theory, Nonlinear Anal., 71 (2009), 4935{4951.[17] N. Yoshida, Oscillation criteria for half-linear elliptic inequalities withp(x)-Laplacians via Riccati method, Nonlinear Anal., 74 (2011), 2563{2575.[18] N. Yoshida, Picone identities for half-linear elliptic operators withp(x)-Laplacians and applications to Sturmian comparison theory, Non-linear Anal., 74 (2011), 5631{5642.[19] Q. H. Zhang, Oscillatory property of solutions for p(t)-Laplacian equa-tions, J. Inequal. Appl., 2007, Art. ID 58548, 8 pp.[20] V. V. Zhikov, Averaging of functionals of the calculus of variationsand elasticity theory, Math. USSR Izv., 29 (1987), 33{66.Norio YoshidaDepartment of MathematicsUniversity of ToyamaToyama, 930-8555Japan(Received December 15, 2011)

Forced oscillation criteria for quasilinear elliptic inequalities with p(x)-Laplacian via Riccati method

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Forced oscillation criteria for quasilinear elliptic inequalities with p(x)-Laplacian via Riccati method

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