Research Article Existence for Elliptic Equation Involving Decaying Cylindrical Potentials with Subcritical and Critical Exponent

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1 International Differential Equations Volume 2015, Article ID , 4 pages Research Article Existence for Elliptic Equation Involving Decaying Cylindrical Potentials with Subcritical and Critical Exponent Mohammed El Mokhtar Ould El Mokhtar Department of Mathematics, College of Science, Qassim University, P.O. Box 6644, Buraidah 51452, Saudi Arabia Correspondence should be addressed to Mohammed El Mokhtar Ould El Mokhtar; med.mokhtar66@yahoo.fr Received 4 July 2015; Accepted 13 October 2015 Academic Editor: Gershon Wolansky Copyright 2015 Mohammed El Mokhtar Ould El Mokhtar. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We consider the existence of nontrivial solutions to elliptic equations with decaying cylindrical potentials and subcritical exponent. We will obtain a local minimizer by using Ekeland s variational principle. 1. Introduction In this paper, we study the existence of nontrivial solutions of the following problem: Δu μ y 2 u= y aγ u γ 2 u(1+λg(x)) in R, y =0, u>0, (P λ,μ ) where y R k,andletk and be integers such that 3 and k belongs to {1,...,}. 2 = 2/( 2) is the critical Sobolev exponent, γ 2, 0 a < 1, g is a continuous function on R,andλ and μ are parameters which we will specify later. We denote point x in R by the pair (y, z) R k R k, D 1,2 0 = D 1,2 0 ((Rk \{0}) R k ),andh μ = H μ ((R k \{0}) R k ),theclosureofc 0 ((Rk \{0}) R k ) with respect to the norms u u =( R 2 1/2 ), u μ ( u =( R 2 μ y 2 u 2 1/2 )dx) with μ<μ k = ((k 2)/2) 2 for k =2. From the Hardy inequality, it is easy to see that the norm u μ is equivalent to u. (1) We define the weighted Sobolev space D := H μ L γ (R, y b dx) L 2 (R, y 2 dx) with b=aγ,whichisa Banach space with respect to the norm defined by (u) := u μ +( R y b u γ dx) 1/γ. My motivation of this study is the fact that such equations arise in the search for solitary waves of nonlinear evolution equations of the Schrödinger or Klein-Gordon type (cf. [1 3]). Roughly speaking, a solitary wave is a nonsingular solution which travels as a localized packet in such a way that the physical quantities corresponding to the invariances of the equation are finite and conserved in time. Accordingly, a solitary wave preserves intrinsic properties of particles such as the energy, the angular momentum, and the charge, whose finiteness is strictly related to the finiteness of the L 2 -norm. Owing to their particle-like behavior, solitary waves can be regarded as a model for extended particles and they arise in many problems of mathematical physics, such as classical and quantum field theory, nonlinear optics, fluid mechanics, and plasma physics (see, e.g., [4]). Several existence and nonexistence results are available in the case k =, and we quote, for example, [5 7] and the references therein. When μ = 0, g(x) 1; problem (P λ,μ ) has been studied in the famous papers by Brézis and irenberg [8] and Xuan [9] which consider the existence and nonexistence of nontrivial solutions to quasilinear Brézis- irenberg-type problems with singular weights. Concerning the existence result in the case k <,we cite [10, 11] and the references therein. As noticed in [10], for

2 2 International Differential Equations μ<0and a=0, Badiale and Rolando have considered the problem (P 0,μ ). They established the existence of nontrivial nonnegative radial solution when β (0, 2) and γ (2 β,2 ) or β (2,+ )and γ (2,2 β ); in addition, if the function f(u) = u γ 1 u is odd, then (P 0,μ ) has infinitely many radial solutions. In [5], Badiale et al. proved the nonexistence of nonzero classical solutions when k and the pair (β, γ) belongs to the light gray region. That is, (β, γ) A = A 1 A 2 A 3,where A 1 := {(β, γ) R 2 :β (0, 2), γ (2 β,2 ), γ 2}\{(2,2 )}, A 2 := {(β, γ) R 2 :β (2, ), γ (2,2 β ), γ 2}, A 3 := {(β, γ) R 2 :β [, + ), γ [2,2 ]}. Since our approach is variational, we define the functional I λ,μ on D by I λ,μ (u) := ( 1 2 ) u 2 μ ( 1 γ ) R y b u γ (1 + λg (x))dx. We say that u D is a weak solution of the problem (P λ,μ ) if it is a nontrivial nonnegative function and satisfies I λ,μ (u), V := R ( u V μ y 2 uv y b u γ 2 uv (1 + λg (x))) = 0, for V D. Throughout this work, we consider the following regions R 1, R 2,suchthat R 1 := {(2, γ) R 2 :γ (2 2 2a,2 )}, R 2 := {(2, γ) R 2 :γ (2,2 2 2a )} with 2 2 2a = 2/( (2 2a)). Concerning the perturbation g, we assume g L (R ), g (x) >0 x R. (2) (3) (4) (5) (G) 2. Preliminaries We list here a few integrals inequalities. The first inequality that we need is the weighted Hardy inequality [13] μ k R y 2 V 2 dx R V 2 dx, V H μ. (6) The starting point for studying (P λ,μ ) is the Hardy-Sobolev- Maz ya inequality that is peculiar to the cylindrical case k< and that was proved by Gazzini and Musina in [14]. It states that there exists positive constant C γ such that C γ V ( R γ 2/γ dx) ( V R 2 μ y 2 V 2 )dx, (7) for μ = 0;equationof(P λ,μ ) is related to a family of inequalities given by Caffarelli et al. [15], for any V C c ((Rk \ {0}) R k ). The embedding H μ L γ (R, y b dx) is compact, where b=aγand L γ (R, y b dx) is the weighted L γ space with respect to the norm u 2 γ,b =( y b V γ 2/γ dx). (8) R Definition 2. Assume 2 k<, 0<μ μ k,and2<γ<2. Then, the infimum S μ,γ defined by S μ,γ =S μ,γ (k, γ) := is achieved on H μ. inf R ( V 2 μ y 2 V 2 )dx V D\{0} ( R y b V γ dx) 2/γ (9) Lemma 3. Let (u n ) D be a Palais-Smale sequence ((PS) δ for short) of I λ,μ such that I λ,μ (u n ) δ, I β,λ,μ (u n) 0 in D (dual of D) as n, for some δ R.Then,u n uin D and I β,λ,μ (u) = 0. Proof. From (10), we have ( 1 2 ) u n 2 μ (1 γ ) R y b u n γ (1 + λg (x))dx (10) In our work, we prove the existence of at least one critical point of I λ,μ by Ekeland s variational principle in [12]. We will state our main result. Theorem 1. Assume that 2<k, μ<μ k, 0<a<1,and (G) hold. If (2, γ) R 1 R 2, then there exists Λ >0such that the problem (P λ,μ ) has at least one nontrivial solution for any λ>λ. =δ+o n (1), u n 2 μ y b u n γ (1 + λg (x))dx=o n (1), R for n large, where o n (1) denotes o n (1) 0 as n.then, δ+o n (1) =I λ,μ (u n ) ( 1 γ ) I β,λ,μ (u n),u n (11) Thispaperisorganizedasfollows.InSection2,wegive some preliminaries. Section 3 is devoted to the proof of Theorem 1. =( (12) ) 2γ u n 2 μ,

3 International Differential Equations 3 and (u n ) is bounded in D. Going if necessary to a subsequence, we can assume that there exists u D such that u n u in D, Then, we deduce that I λ,μ (u n )=I λ,μ (u) +( 1 2 ) V n 2 μ u n u in L γ (R, y b dx), (13) ( 1 γ ) R y b V n γ +o n (1), (21) u n u a.e. in R. Consequently, we get, for all V C 0 ((Rk \{0}) R k ), ( u V μ R y 2 uv (14) y b u γ 2 uv (1 + λg (x))) = 0, I λ,μ (u n),u n = V n 2 μ R y b V n γ +o n (1). From the fact that V n 0in D, we can assume that lim n V n 2 μ = lim n R y b V n γ =α 0. (22) which means that 3. Existence Result I β,λ,μ (u) =0. (15) Assuming that α>0, we have by definition of S μ,γ α S μ,γ l (2/γ), (23) and so α (S μ,γ ) γ/(γ 2). (24) Firstly, we require the following lemmas. Lemma 4. Let (u n ) D be a (PS) δ sequence of I λ,μ for some δ R.Then, and either u n u or δ I λ,μ (u) +( u n u ind (16) 2γ μ,γ ) γ/(γ 2). (17) Proof. We know that (u n ) is bounded in D. Uptoasubsequence if necessary, we have that u n u in D u n u a.e. in R. (18) Denote V n =u n u,andthenv n 0.AsinBrézis and Lieb [16], we have Then, we get δ I λ,μ (u) + ( 2γ μ,γ ) γ/(γ 2). (25) Therefore, if not, we obtain α=0.thatis,u n uin D. Lemma 5. Suppose that 2<k, μ<μ k,and(g) hold. If (2, γ) R 1 R 2, then there exist Λ >0and and ] positive constants such that, for all λ>λ, (i) there exist ω R such that I λ,μ (ω) < 0, (ii) we have I λ,μ (u) ] >0 for u μ = 0. (26) Proof. (i) Let t 0 > 0 where t 0 is small, and φ C 0 ((Rk \ {0}) R k ) such that φ 0.ChoosingΛ = t 0 φ 1 γ,then, if λ>λ large enough, R y b u n γ = R y b V n γ + R y b u γ, u n 2 μ = V n 2 μ + u 2 μ. (19) From Lebesgue theorem and by using the assumption (G),we obtain lim g (x) n R y b u n γ dx = lim n R g (x) y b u γ dx. (20) I λ,μ (t 0 φ) := ( t2 0 2 ) φ 2 μ (tγ 0 γ ) R y b φ γ 1 ( tγ 0 γ ) R y b φ γ λg (x) <( t2 0 2 ) φ 2 μ (tγ 0 γ ) R y b φ γ 1 ( t 0 γ ) y b φ g (x) <0. R Thus, if ω=t 0 φ,weobtainthati λ,μ (ω) < 0. (27)

4 4 International Differential Equations (ii) By the Holder inequality and the definition of S μ,γ and since γ>2,wegetforallu D \{0} I λ,μ (u) := ( 1 2 ) u 2 μ ( 1 γ ) R y b u γ (1 + λg (x))dx ( 1 2 ) u 2 μ (1 γ )S μ,γ u γ μ (1 + λ g ). (28) If λ>λ, then there exist ] >0and 0 >0small enough such that I λ,μ (u) ] >0 for u μ = 0. (29) We also assume that t 0 issmallenoughsuchthat t 0 φ μ < 0. Thus, we have c 1 = inf {I λ,μ (u) :u B 0 }<0, where B 0 ={u D, (u) 0 }. (30) Using Ekeland s variational principle, for the complete metric space B ρ0 with respect to the norm of D, wecanprovethat there exists a (PC) c1 sequence (u n ) B ρ0 such that u n u 1 for some u 1 with (u 1 ) ρ 0. ow, we claim that u n u 1.Ifnot,byLemma4,wehave c 1 I λ,μ (u 1 )+( c 1 +( 2γ μ,γ ) γ/(γ 2) 2γ μ,γ ) γ/(γ 2) >c 1, (31) which is a contradiction. Then, we obtain a critical point u 1 of I λ,μ for all λ>λ large enough satisfying c 1 =( ) 2γ u 1 2 μ >0. (32) Proof of Theorem 1. From Lemmas 4 and 5, we can deduce that there exists at least a nontrivial solution u 1 for our problem (P λ,μ ) with positive energy. Conflict of Interests The author declares that there is no conflict of interests regarding the publication of this paper. [2] V.BenciandD.Fortunato, Solitarywavesinthenonlinearwave equation and in gauge theories, Fixed Point Theory and Applications,vol.1,no.1,pp.61 86,2007. [3] W.A.Strauss, onlinearinvariantwaveequations, ininvariant Wave Equations,vol.73ofLectureotes inphysics,pp , Springer, Berlin, Germany, [4] G. B. Withan, Linear and onlinear Waves, JohnWiley& Sons, [5] M. Badiale, M. Guida, and S. Rolando, Elliptic equations with decaying cylindrical potentials and power-type nonlinearity, Differential Equations,vol.12,no.12,pp , [6] P. Caldiroli and R. Musina, On the existence of extremal functions for a weighted Sobolev embedding with critical exponent, Calculus of Variations and Partial Differential Equations,vol.8, no. 4, pp , [7] F. Catrina and Z. Q. Wang, On the Ca Caffarelli-Kohn- irenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions, Communications on Pure and Applied Mathematics, vol. 54, no. 2, pp , [8] H. Brézis and L. irenberg, Positive solutions of nonlinear elliptic equations involving critical sobolev exponents, Communications on Pure and Applied Mathematics, vol.36,no.4, pp , [9] B. Xuan, The solvability of quasilinear Brezis-irenberg-type problems with singular weights, onlinear Analysis: Theory, Methods and Applications, vol. 62, no. 4, pp , [10] M. Badiale and S. Rolando, A note on nonlinear elliptic problems with singular potentials, supported by MIUR, project, in Variational Methods and onlinear Differential Equations,2005. [11] R. Musina, Existence of extremals for the Maz ya and for the Caffarelli-Kohn-irenberg inequalities, onlinear Analysis, Theory, Methods and Applications, vol.70,no.8,pp , [12] I. Ekeland, On the variational principle, Mathematical Analysis and Applications,vol.47,no.2,pp ,1974. [13] Adimurthi,. Chaudhuri, and M. Ramaswamy, An improved Hardy-Sobolev inequality and its application, Proceedings of the American Mathematical Society, vol.130,no.2,pp , [14] M. Gazzini and R. Musina, Hardy-Sobolev-Maz ya inequalities: symmetry and breaking symmetry of extremal functions, Communications in Contemporary Mathematics, vol.11,no.6, pp , [15] L. Caffarelli, R. Kohn, and L. irenberg, First order interpolation inequalities with weights, Compositio Mathematica,vol. 53,no.3,pp ,1984. [16] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proceedings of the American Mathematical Society, vol.88,no.3,pp , References [1] M. Badiale, V. Benci, and S. Rolando, Solitary waves: physical aspects and mathematical results, Rendiconti del Seminario Matematico,vol.62,no.2,pp ,2004.

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