Liouville theorems for a singular parabolic differential inequality with a gradient term

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1 Fang and Xu Jounal of Inequalities and Applications 214, 214:62 R E S E A R C H Open Access Liouville theoems fo a singula paabolic diffeential inequality with a gadient tem Zhong Bo Fang * and Lijun Xu * Coespondence: fangzb7777@hotmail.com School of Mathematical Sciences, Ocean Univesity of China, Qingdao, 2661, P.R. China Abstact In this pape, we study poofs of some new Liouville theoems fo a stongly p-coecive quasi-linea paabolic type diffeential inequality with a gadient tem and singula vaiable coefficients. The poofs ae based on the test function method developed by Mitidiei and Pohozaev. 1 Intoduction We conside a singula quasi-linea paabolic diffeential inequality with a gadient tem u t Lu a(x)u q b(x) u s, x, t >, (1.1) and the initial condition is given by u(x,)=u, x, (1.2) whee q, s >, R N (N 1) is a bounded domain with sufficiently smooth bounday o = R N, the initial function u L 1 loc () is nonnegative, and the coefficients a(x)andb(x) ae positive and singula on the bounday o at the oigin. Let A : R R N R be a Caathéodoy function and define an opeato L as Lu = div ( A(x, u, u) u ), fo any u W 1,1 loc ( (, )) such that A(x, u, u) L1 loc ( (, )). The opeato L is said to be stongly p-coecive (S-p-C), if thee exist thee constants c 1, c 2 >andp >1 such that the inequality c 1 η p 2 A(x, u, η) c 2 η p 2 (1.3) holds fo all (x, u, η) R R N. Fo example, the Laplace opeato is S-2-C and the p-laplace opeato is S-p-C. Inequality (1.1) appeas in many fields, such as fluid mechanics, biological species, and population dynamics, see [1 3]. Fom the pespective of fluid mechanics, (1.1) descibes the non-newtonian filtation phenomenon and the flow of gas in a poous medium, in which a(x)u q is called the souce tem and b(x) u s is called the gadient absoption tem. 214 Fang and Xu; licensee Spinge. This is an Open Access aticle distibuted unde the tems of the Ceative Commons Attibution License ( which pemits unesticted use, distibution, and epoduction in any medium, povided the oiginal wok is popely cited.

2 Fang and Xu Jounal of Inequalities and Applications 214, 214:62 Page 2 of 12 Thee have been many esults on the nonexistence of nonnegative nontivial global solutions fo nonlinea paabolic type equations o inequalities (and systems), see [4 15] and efeences theein. In 1966, Fujita [4] studied the following Cauchy poblem of the semilinea heat equation: t u u = u p, x R N, t >, and he obtained the citical exponent q c =1+2/N on the existence vesus nonexistence of nonnegative nontivial global solution. Fom then on, thee have been a numbe of extensions of Fujita s esults in many diections, and many scholas pay close attention to his esults. The nonexistence of a global solution fo nonlinea patial diffeential equations, which aises fom the Liouville type theoem of hamonic function, is a nonlinea Liouville type theoem. Thus one can pove some popeties of solutions in a bounded domain and this is a kind of essential eflection of blow-up o singulaity theoy as well, see [5] and efeences theein. In the whole space, Katsatos [6] studied a Liouville type theoem on the solution fo a quasi-linea paabolic equation without singula vaiable coefficient and gadient tem. Fo a moe geneal evolution and quasi-linea paabolic type inequality, see [7, 8] and efeences theein. By using the test function method, Mitidiei and Pohozaev [9, 1] studied the Liouville type theoems fo elliptic and paabolic equations without gadient souce tem. Aftewads, by using the same method, Wei [11] obtained the nonexistence of the nonnegative nontivial global weak solution fo a semilinea paabolic inequality with a weight function a(x, t) ( x 2 + t) γ, but without a gadient tem. Note that the gadient tem in the ight-hand side of (1.1)willbingsubstantial difficulty to the eseach. In the whole space, Mitidiei and Pohozaev [12]putfowadan impotant esult fo a diffeential inequality with a gadient nonlineaity; they consideed the following quasi-linea elliptic inequality: p u u q u s, x R N, (1.4) and they poved that it has only constant solutions if 1 ,s, q + s > p 1, and q+ (N 1)s N 2 < N(p 1). Late, Kaisti and Filippucci did futhe eseach on geneal foms of N p (1.4)(cf. [13, 14]). Galakhov [15] discussed a stongly p-coecive elliptic, paabolic, and hypebolic type diffeential inequality with a gadient nonlineaity of constant coefficient in bounded and unbounded domains. Moeove, they obtained a Liouville theoem by making use of the test function method. Recently, Li and Li [16, 17] extended thei conclusions to the case of an elliptic type inequality with singula vaiable coefficients and obtained the nonexistence of solutions to that inequality and to the Hanack inequality. Fo singula paabolic poblems in conical domains, efe to [18 2] and the efeences theein. Besides the woks mentioned above, thee ae few studies on nonlinea Liouville theoems fo a paabolic type diffeential inequality (1.1) with singula vaiable coefficients and a gadient tem. The pupose of this pape is to investigate the influence of an S- p-c opeato, the exponent of singula coefficients, and the gadient nonlineaity on the nonexistence of a nonnegative nontivial global weak solution. The main difficulty lies in choosing a suitable test function accoding to diffeent singulaities of a(x), b(x), and inequality (1.1) with gadient nonlineaity. We will use the test function technique, developed by Mitidiei and Pohozaev [9, 1, 12], to show some Liouville theoems on the weak

3 Fang and Xu Jounal of Inequalities and Applications 214, 214:62 Page 3 of 12 solution fo poblem (1.1)-(1.2) with bounded and unbounded domains. In paticula, we ecall that no use of compaison esults and no conditions at infinity on the solution ae equied. This pape is oganized as follows: In Section 2, some peliminaies, such as some basic definitions and assumptions, and ou main esults ae given. We pove the main esults in Sections 3 and 4. 2 Peliminaies and main esults Since p > 1 and inequality (1.1) is degeneate o singula, poblem (1.1)-(1.2) has no classical solution in geneal, and so it is easonable to conside othe solutions. We state the two definitions of a weak solution and of a solution with a positive lowe bound. Definition 1 A nonnegative function u(x, t) is calleda weak solution of poblem (1.1)-(1.2) in S = (, ), if the following two conditions ae satisfied: (i) A(x, u, u) u L 1 1,p loc (S) and u Wloc (S) Lq loc (S), (ii) fo any nonnegative function ϕ C 1(S),wehave S a(x)u q ϕ + S u ϕ(x,)dx u t ϕ S A(x, u, u) u ϕ + S b(x) u s ϕ. (2.1) Definition 2 A nonnegative function u W 1,p loc (S) Lq loc (S) suchthata(x, u, u) u L 1 loc (S) is said to be a solution with positive lowe bound in S, iftheeexistsc >such that u(x, t) c a.e. in S. We conside two types of singulaities. Type 1: Both a(x) and b(x) ae singula on the bounday, i.e., theeexistconstants c, d >and α, β R such that a(x) c ρ(x) α, b(x) d ρ(x) β, x ε, (2.2) whee ρ(x)=dist(x, ) and ε = {x : ρ(x) ε} fo sufficiently small ε >. Type 2: Both a(x) and b(x) ae singula at the oigin in the case that = R N, i.e.,theeexist constants c 1, d 1 >and α, β R such that a(x) c 1 x α, b(x) d 1 x β, x R N \{}. (2.3) To establish apioiestimates of the solutions, we define some test functions which have the fom of a sepaation of the vaiables, and those functions will be widely used in the sequel. Fo the space vaiable and singula Type 1, we intoduce a cut-off function ξ ε C 1 (; [, 1]) that satisfies ξ ε (x) 1, ξ ε (x)=1 in\ 2ε, ξ ε (x)= in ε,

4 Fang and Xu Jounal of Inequalities and Applications 214, 214:62 Page 4 of 12 and ξ ε cε 1, x, (2.4) fo some constant c >. Fo the space vaiable and singula Type 2, we conside a cut-off function ξ R C 1 (RN ; [, 1]) that satisfies and ξ R (x) 1, ξ R (x)=1 inb R (), ξ R (x)= inr N \B 2R (), ξ R čr 1, x, (2.5) fo some constant č >. Set χ(x)=ξ λ, (2.6) whee λ > is a paamete to be chosen late accoding to the natue of poblem (1.1)-(1.2). Fo the time vaiable, assume that η(t) C ([, )) and η(t) 1, η(t) = 1 in [, 1], η(t)= in[2, ], C η (t). Then, by appopiate combinations of the above functions, one can choose suitable test functions and obtain the following nonlinea Liouville theoems. s pq Theoem 1 Let p >1,q > max{1, p 1, }, and let s (, ). Suppose that A is S-p-C and p s q+1 a, b : R + (o a, b : R N \{} R + ) ae continuous functions that satisfy (2.2)(o (2.3)). (I) When both a(x) and b(x) ae singula on, if max{, q 1} < θ < q, α > θp θ, q(p s) s + α[(p s)(θ q)+s] β <, θq then any nonnegative weak solution to poblem (1.1)-(1.2) is tivial, i.e., u(x, t) =a.e. in S. (II) When both a(x) and b(x) ae singula at, we conside the following two conditions: (i) { max, q 1, N } N(1 q) < θ < q, α ; θq (ii) { max{, q 1} < θ < min q, N } { } N(1 q) θp N, α < min,, p θ +1 q θ N[q(p s) s]+α[(p s)(θ q)+s] β >. θq

5 Fang and Xu Jounal of Inequalities and Applications 214, 214:62 Page 5 of 12 If one of the conditions (i) and (ii) holds, then any nonnegative weak solution to poblem (1.1)-(1.2) is tivial, i.e., u(x, t)=a.e. in S, whee θ is a paamete and = q +1 p. Theoem 2 Suppose that the assumptions of Theoem 1 ae satisfied. Then poblem (1.1)- (1.2) has no solution with positive lowe bound in S, if α > p o β < α αs p, when both a(x) and b(x) ae singula on, o if α α αs p, when both a(x) and b(x) ae singula at. Remak 1 If inequality (1.1) is modified with u t + Lu a(x)u q b(x) u s, x, t >, (1.1a) one can obtain the following esults by a simila agument in the poof of Theoem 1. Suppose that the assumptions of Theoem 1 ae satisfied. (I ) When both a(x) and b(x) ae singula on,if θ > q, α > θp θ, q(p s) s + α[(p s)(θ q)+s] β <, θq then any nonnegative weak solution to poblem (1.1a)-(1.2)is tivial,i.e., u(x, t) =a.e. in S. (II ) When both a(x) and b(x) ae singula at, we conside the following two conditions: (i ) (ii ) { θ > max q, N } N(1 q), α N[q(p s) s]+α[(p s)(θ q)+s], θq q < θ < N p, β > { } N(1 q) α < min θp N,, θ +1 q θ N[q(p s) s]+α[(p s)(θ q)+s]. θq If one of the conditions (i )and(ii ) holds, then any nonnegative weak solution to poblem (1.1a)-(1.2)istivial,i.e., u(x, t)=a.e. in S,wheeθ is a paamete and = q+1 p. Remak 2 Since Theoem 2 is independent of the paamete θ, onecanalsoobtainthe same esults fo poblem (1.1a)-(1.2).

6 Fang and Xu Jounal of Inequalities and Applications 214, 214:62 Page 6 of The poof of Theoem 1 Ou poof mainly consists of thee steps. Step 1. Fo T >,define := [, 2T]andletψ(x, t)=η T (t)χ(x)beacut-offfunction, whee χ(x)isdefinedas(2.6)andη T (t)=η(t/t). Clealy, we get C T η T(t) t. Fo k > 1, which will be detemined late, and d = q θ >, multiplying both sides of (1.1) with a test function = ψ k u d > and integating the esult ove,wehave a(x)u q d ψ k u 1 d χ k 1 ψk (x) dx u1 d 1 d t ( ) + A(x, u, u) u u d ψ k ( ) + A(x, u, u) u u d ψ k + b(x) u s u d ψ k. It follows fom (1.3)that a(x)u θ ψ k + u 1 d (x)χ k (x) dx 1 ψk u1 d c 1 d u d 1 u p ψ k d 1 t + c 2 k u d( u p 2 u ) ψ k 1 ψ + b(x) u s u d ψ k. By the definition of ψ,weget a(x)u θ ψ k + u 1 d (x)χ k (x) dx + c 1 d u d 1 u p ψ k C u 1 d ψ k 1 + c 2 k u d u p 1 ψ k 1 ψ T + b(x) u s u d ψ k. (3.1) By applying Young s inequality to the thee tems in the ight-hand side of (3.1), we obtain C u 1 d ψ k 1 T 1 a(x)u θ ψ k T θ a(x) q θ 1 ψ k θ, (3.2) 4 c 2 k u d u p 1 ψ k 1 ψ c 1d 4 u d 1 u p ψ k u θ ψ k p ψ p, (3.3)

7 Fang and Xu Jounal of Inequalities and Applications 214, 214:62 Page 7 of 12 whee = q +1 p, b(x) u s u d ψ k c 1d 4 u d 1 u p ψ k b(x) p p s u (p s)(θ q)+s p s ψ k. (3.4) Since d >,<1 d =1 q + θ < θ,<θ < θ,and (p s)(θ q)+s p s < s,weknowthat max{, q 1} < θ < q, { q > max 1, p 1, } ( s, s, p s ) pq. q +1 Applying Young s inequality to the two tems in the ight-hand side of (3.3) and(3.4) yields C u θ ψ k p ψ p 1 4 C b(x) p a(x)u q d ψ k p s u (p s)(θ q)+s p s ψ k 1 a(x)u θ ψ k 4 a(x) θ a(x) (p s)(θ q) s ψ k pθ ψ θp, (3.5) θp b(x) ψ k. (3.6) Let k be lage enough such that k > max{ θ pθ, p, }. Fom inequalities (3.1)-(3.6), we deduce 1 a(x)u θ ψ k + 4 CT θ u 1 d (x)χ k (x) dx + c 1d u d 1 u p ψ k 2 a(x) θ ψ k pθ ψ θp a(x) q θ 1 ψ k θ a(x) (p s)(θ q) s θp b(x) ψ k. We then have T CT θ a(x)u θ χk T T a(x) q θ 1 χ k θ a(x) (p s)(θ q) s θp T b(x) χ k. a(x) θ χ k pθ χ θp

8 Fang and Xu Jounal of Inequalities and Applications 214, 214:62 Page 8 of 12 Itcanbeseenthat χ θp T CT θ a(x)u θ χk λ θp T CT θ T = λ θp supp ξ T supp ξ supp ξ T T supp ξ supp ξ ξ θp(λ 1) ξ θp,andwetakeλ > max{1, θp k }.Wethenobtain a(x) q θ 1 ξ λ(k θ ) a(x) θ a(x) (p s)(θ q) s ξ λ(k pθ) ξ θp θp b(x) ξ λk a(x) q θ 1 λ θp a(x) (p s)(θ q) s θp T supp ξ a(x) θ ξ θp b(x). (3.7) Step 2. When both a(x) andb(x) ae singula on,wetakeξ = ξ ε in (3.7). Evidently, one has supp ξ = 2ε \ ε = { x : ε ρ(x) 2ε }. (3.8) By (2.2), we get T CT θ a(x)u θ ξ kλ T T ρ(x) α(q θ 1) λ θp 2ε \ 2ε T ρ(x) α( θ) 2ε \ 2ε ξ ε θp ρ(x) α[(p s)(θ q)+s] βθp. (3.9) 2ε \ 2ε By a standad change of vaiables and (2.4), we then obtain T a(x)u θ \ 2ε T a(x)u θ ξ kλ CT θ = CT θ T T λ θp T ρ(x) α(q θ 1) λ θp 2ε \ 2ε ρ(x) α[(p s)(θ q)+s] βθp 2ε \ 2ε {x :ε ρ(x) 2ε} T {x :ε ρ(x) 2ε} ρ(x) α(q θ 1) T ρ(x) α( θ) ξ ε θp ρ(x) α( θ) 2ε \ 2ε ξ ε θp

9 Fang and Xu Jounal of Inequalities and Applications 214, 214:62 Page 9 of 12 whee σ 1 = σ 3 = T CT θ ε α(q θ 1) {x :ε ρ(x) 2ε} T ε α[(p s)(θ q)+s] βθp ρ(x) α[(p s)(θ q)+s] βθp λ θp ε α( θ) θp 2ε \ ε 2ε \ ε T T 2ε \ ε CT q θ 1 ε αθ ()(α 1) Tε (1 α)+θ(α p) Tε α[(p s)(θ q)+s] βθp+ = CT q θ 1 ε σ 1 Tε σ 2 Tε σ 3, (3.1) αθ ()(α 1), σ 2 = q 1 (1 α)+θ(α p), α[(p s)(θ q)+s] βθp + q(p s) s. q(p s) s By the conditions in (I) of Theoem 1, wehaveσ 1 >,σ 2 >,andσ 3 >.Then,letting ε and combining with a andom selection of T yield S a(x)u θ =, (3.11) and hence we get u(x, t) =a.e. ins. Step 3. When both a(x)andb(x)aesingulaat{},take = R N. Choosing ξ = ξ R in (3.7), one has supp ξ = B 2R ()\B R () = { x : R x 2R }. (3.12) It follows fom (2.3)that T R N a(x)u θ χ k CT θ λ θp T T B 2R ()\B R () T B 2R ()\B R () B 2R ()\B R () Combining the above esult with (2.5), we get T B R () T CT θ λ θp a(x)u θ a(x)u θ χ k R N T B 2R ()\B R () T B 2R ()\B R () x α(q θ 1) x α( θ) ξ R θp x α(q θ 1) x α( θ) ξ R θp x α[(p s)(θ q)+s] βθp. (3.13)

10 Fang and Xu Jounal of Inequalities and Applications 214, 214:62 Page 1 of 12 = CT θ T λ θp B 2R ()\B R () T T {x :R x 2R} T CT θ R α(q θ 1) {x :R x 2R} {x :R x 2R} T + R α[(p s)(θ q)+s] βθp x α[(p s)(θ q)+s] βθp x α(q θ 1) x α( θ) ξ R θp x α[(p s)(θ q)+s] βθp B 2R ()\B R () T B 2R ()\B R () R α( θ) θp T CT q θ 1 α(q θ 1) N α( θ)+θp α[(p s)(θ q)+s]+βθp R N N TR TR B 2R ()\B R () = CT θ R σ 4 TR σ 5 TR σ 6, (3.14) whee σ 4 = N σ 6 = N + α(q θ 1), σ 5 = N q 1 α[(p s)(θ q)+s] βθp. q(p s) s α( θ)+θp, If one of the conditions (i) and (ii) in (II) of Theoem 1 holds, we have σ 4 <,σ 5 <, and σ 6 <.Then,lettingR and combining with a andom selection of T, we finally aive at S a(x)u θ =. (3.15) Theefoe, we obtain u(x, t) = a.e. in S, and the poof is completed. 4 The poof of Theoem 2 In this poof, we use the method of contadiction. We take the same test function as in Section 3 and the poof is simila to that of Theoem 1. Supposetheeexistsu such that u c >. (i) Assume that both a(x) andb(x) ae singula on. It follows fom (2.2), (3.7), and (3.1)that T ρ α (x)u θ χ k CT q θ 1 ε σ 1 Tε σ 2 Tε σ 3. 2ε \ ε By the definition of χ and the assumption u c >,wehave CT q θ 1 ε σ 1 Tε σ 2 Tε σ 3 CTε 1 α.

11 Fang and Xu Jounal of Inequalities and Applications 214, 214:62 Page 11 of 12 Then max { T θ ε σ 1, ε σ 2, ε σ 3 } ε 1 α, i.e., one of following inequalities holds: T θ ε 1 α σ 1, ε σ 2 ε 1 α, ε σ 3 ε 1 α. (4.1) Since < ε <1,ifα > p o β < α αs,andt > 1 is lage enough and ε is sufficiently small, p all inequalities in (4.1) do not hold, which is a contadiction. (ii) Assume that both a(x) andb(x) ae singula at. By combining (2.3) and(3.7) with (3.14), we get the inequality T B R () x α u θ CT θ R σ 4 TR σ 5 TR σ 6. Combining this inequality with u c >,wehave CT θ R σ 4 TR σ 5 TR σ 6 CTR N α, i.e., max{t θ R σ 4, R σ 5, R σ 6} R N α. Clealy, we have the inequalities R σ 5 R N α, R σ 6 R N α. (4.2) If α α αs, the inequalities in (4.2) do not hold, which is a contadiction. The p poof is completed. Competing inteests The authos declae that they have no competing inteests. Authos contibutions All authos contibuted equally to the manuscipt and ead and appoved the final manuscipt. Acknowledgements This wok is suppoted by the Natual Science Foundation of Shandong Povince of China (ZR212AM18) and the Fundamental Reseach Funds fo the Cental Univesities (No ). The authos would like to sinceely thank all the eviewes fo thei insightful and constuctive comments. Received: 31 Octobe 213 Accepted: 3 Januay 214 Published: 11 Feb 214 Refeences 1. Bebenes, J, Ebely, D: Mathematical Poblems fom Combustion Theoy. Spinge, New Yok (1989) 2. Samaskii, AA, Galaktionov, VA, Kudyumov, SP, Mikhailov, AP: Blow-up in Quasilinea Paabolic Equations. de Guyte, Belin (1995) 3. Wu, Z, Zhao, J, Yin, J: Nonlinea Diffusion Equations. Wold Scientific, Singapoe (21) 4. Fujita, H: On the blowing up of solutions to the Cauchy poblem fo u t = u + u 1+α.J.Fac.Sci.,Univ.Tokyo,Sect.1A, Math. 13, (1966) 5. Galaktionov, VA, Vazquez, JL: Continuation of blow up solutions of nonlinea heat equations in seveal space dimensions. Commun. Pue Appl. Math. 5(1),1-67 (1997) 6. Katsatos, AG, Kuta, VV: On the citical Fujita exponents fo solutions of fist-ode nonlinea evolution inequalities. J. Math. Anal. Appl. 269, (22) 7. Picciillo, AM, Toscano, L, Toscano, S: Blow-up esults fo a class of fist-ode nonlinea evolution inequalities. Diffe. Equ. 212, (25) 8. Jiang, ZX, Zheng, SM: A Liouville-type theoem fo a doubly degeneate paabolic inequality. J. Math. Phys. 3A(3), (21) 9. Mitidiei, E, Pohozaev, SI: Nonexistence of positive solution fo quasilinea elliptic poblems on R N.Poc.SteklovInst. Math. 227, (1999)

12 Fang and Xu Jounal of Inequalities and Applications 214, 214:62 Page 12 of Mitidiei, E, Pohozaev, SI: Nonexistence of weak solutions fo some degeneate elliptic and paabolic poblems on R N. J. Evol. Equ. 1, (21) 11. Wei, GM: Nonexistence of global solutions fo evolutional p-laplace inequalities with singula coefficients. J. Math. Anal. Appl. 28A(2), (27) 12. Mitidiei, E, Pohozaev, SI: A pioi estimates and the absence of solutions of nonlinea patial diffeential equations and inequalities. Poc. Steklov Inst. Math. 234, (21) 13. Kaisti, G: On the absence of solutions of systems of quasilinea elliptic inequalities. Diffe. Equ. 38, (22) 14. Filippucci, R: Nonexistence of positive weak solutions of elliptic inequalities. Nonlinea Anal. 7(8), (29) 15. Galakhov, EI: Some nonexistence esults fo quasi-linea PDE s. Commun. Pue Appl. Anal. 6, (27) 16. Li, XH, Li, FQ: A pioi estimates fo nonlinea diffeential inequalities and applications. J. Math. Anal. Appl. 378, (211) 17. Li, XH, Li, FQ: Nonexistence of solutions fo singula quasilinea diffeential inequalities with a gadient nonlineaity. Nonlinea Anal. 75, (212) 18. Bandle, C, Essen, M: On positive solutions of Emden equations in cone-like domains. Ach. Ration. Mech. Anal. 4, (199) 19. Laptev, GG: Absence of solutions to semilinea paabolic diffeential inequalities in cones. Mat. Sb. 192, 51-7 (21) 2. Fang, ZB, Chao, F, Zhang, LJ: Liouville theoems of slow diffusion diffeential inequalities with vaiable coefficients in cone.j.koeansoc.ind.appl.math.15(1), (211) / X Cite this aticle as: Fang and Xu: Liouville theoems fo a singula paabolic diffeential inequality with a gadient tem. Jounal of Inequalities and Applications 214, 214:62

On absence of solutions of a semi-linear elliptic equation with biharmonic operator in the exterior of a ball

Tansactions of NAS of Azebaijan, Issue Mathematics, 36, 63-69 016. Seies of Physical-Technical and Mathematical Sciences. On absence of solutions of a semi-linea elliptic euation with bihamonic opeato