B Xuan

F Bernis

G Bliss

I Caffarelli

Jihui Zhang

Xiusong Gu

XM He

Yang Yang

YJ Wang

Boundary Value Problems

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Yang et al. Boundary Value Problems 2012, 2012:137http://www.boundaryvalueproblems.com/content/2012/1/137RESEARCH Open AccessExistence of multiple solutions for theBrezis-Nirenberg-type problem with singularcoefﬁcientsYang Yang1*, Jihui Zhang2 and Xiusong Gu3*Correspondence: yynjnu@126.com1School of Science, JiangnanUniversity, Wuxi, 214122, ChinaFull list of author information isavailable at the end of the articleAbstractBy energy estimates and by establishing a local (PS) condition, we obtain themultiplicity of solutions to a class of Brezis-Nirenberg-type problem with singularcoeﬃcients via minimax methods and the Krasnoselskii genus theory.Keywords: Brezis-Nirenberg-type problem; minimax method1 Introduction andmain resultsThis paper is concerned with multiple solutions for the semilinear Brezis-Nirenberg-typeproblem with singular coeﬃcients{–div( Du|x|a ) = λ|u|*–u|x|*b + β|u|q–u|x|α , x ∈ ;u = , x ∈ ∂, ()where  ⊂ Rn is a bounded smooth domain, and  ∈ , –∞ < a < n– , a ≤ b < a + ,* = nn–d , d = a +  – b ∈ (, ],  < q < , α < ( + a)q + n( – q ). β > , λ >  are two realparameters.The starting point of the variational approach to the problem is the Caﬀarelli-Kohn-Nirenberg inequality (see []): There is a constant Ca,b >  such that(∫Rn|x|–*b|u|* dx)/*≤ Ca,b∫Rn|x|–a|Du| dx, ()for all u ∈ C∞ (Rn), where–∞ < a < n –  , a≤ b < a + , * =nn – d , d = a +  – b.LetD,a () be the completion of C∞ (Rn) with respect to the weighted norm ‖ ·‖ deﬁnedby‖u‖ =(∫|x|–a|Du| dx)/.© 2012 Yang et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribu-tion License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in anymedium, provided the original work is properly cited.Yang et al. Boundary Value Problems 2012, 2012:137 Page 2 of 8http://www.boundaryvalueproblems.com/content/2012/1/137From the boundedness of  and the standard approximation arguments, it is easy to seethat () holds for any u ∈D,a () in the sense:(∫|x|–α|u|r dx)/r≤ C∫|x|–a|Du| dx ()for  ≤ r ≤ * = nn– , αr ≤ ( + a) + n( r –  ), that is, the embedding D,a () ↪→ Lr(, |x|–α)is continuous, where Lr(, |x|–α) is the weighted Lr space with the norm‖u‖r,α := ‖u‖Lr (,|x|–α ) =(∫|x|–α|u|r dx)/r.On D,a (), we can deﬁne the energy functionalJ(u) = ∫|x|–a|Du| dx – λ*∫|x|–*b|u|* dx – βq∫|x|–α|u|q dx. ()From (), J is well deﬁned in D,a (), and J ∈ C(D,a (),R). Furthermore, the criticalpoints of J are weak solutions of problem ().Breiz-Nirenberg-type problems have been generalized to many situations such as{–div( Du|x|a ) –μu|x|(a+) =|u|*–u|x|*b + λu|x|(a+)–c , x ∈ ,u = , x ∈ ∂. ()Xuan et al. [] derived the explicit formula for the extremal functions of the best embed-ding constant by applying the Bliss lemma []. They got a nontrivial solution for problem() including the resonant and nonresonant cases by variational methods. He and Zou []studied problem () and obtained the multiplicity of solutions with the aid of a pseudo-index theory. In [], problem () has been extended to the p-Laplace case by Xuan.The purpose of this paper is to study the multiplicity of solutions for the Breiz-Nirenberg-type problem () with the aid of a minimax method. We obtain multiple non-trivial solutions of () by proving the local (PS) condition and energy estimates.Our main results are the following.Theorem . Suppose  < q < , then(i) ∀β > , ∃λ >  such that if  < λ < λ, problem () has a sequence of solutions {um}with J(um) <  and J(um)→  as m→ ∞.(ii) ∀λ > , ∃β >  such that if  < β < β, problem () has a sequence of solutions {um}with J(um) <  and J(um)→  as m→ ∞.2 Preliminary resultsLemma . [] Suppose that  ⊂ Rn is an open bounded domain with C boundary and ∈ , –∞ < a < (n – )/. The embedding D,a () ↪→ Lr(, |x|–α) is compact if  ≤ r < *,α < ( + a)r + n( – r ).Lemma . (Concentration compactness principle []) Let –∞ < a < (n – )/, a ≤ b ≤a + , * = n/(n – d), d =  + a – b ∈ [, ], and M(Rn) be the space of bounded measuresYang et al. Boundary Value Problems 2012, 2012:137 Page 3 of 8http://www.boundaryvalueproblems.com/content/2012/1/137on Rn. Suppose that {um} ⊂D,a (Rn) is a sequence such thatum ⇀ u in D,a(Rn),μm :=∣∣|x|–aDum∣∣ dx⇀ μ in M(Rn),νm :=∣∣|x|–bum∣∣*dx⇀ ν in M(Rn),um → u a.e. on Rn.Then there are the following statements:() There exists some at most countable set I , a family {x(i) : i ∈ I} of distinct points in Rn,and a family {ν(i) : i ∈ I} of positive numbers such thatν =∣∣|x|–bu∣∣* dx +∑i∈Iν(i)δx(i) , ()where δx is the Dirac-mass of mass  concentrated at x ∈ Rn.() The following inequality holdsμ ≥ ∣∣|x|–aDu∣∣ dx +∑i∈Iμ(i)δx(i) ()for some family {μ(i) >  : i ∈ I} satisfyingS(ν(i))/* ≤ μ(i) for all i ∈ I, ()where S := infu∈D,a (Rn)\{} Ea,b(u) to be the best embedding constants, andEa,b(u) =∫Rn |x|–a|Du| dx(∫Rn |x|–*b|u|* dx)/*.In particular,∑i∈I(ν(i))/* <∞.Lemma . Assume {un} is a (PS)c sequence with c < ,  < q < , then() ∀λ > , there exists β >  such that for any  < β < β, {un} has a convergentsubsequence in D,a ().() ∀β > , there exists λ >  such that for any  < λ < λ, {un} has a convergentsubsequence in D,a ().Proof () The boundedness of (PS)c sequence.For {un} is a (PS)c sequence, thenJ(un) =∫|x|–a|Dun| dx – λ*∫|x|–*b|un|* dx – βq∫|x|–α|un|q dx, ()〈J ′(un),un〉=∫|x|–a|Dun| dx – λ∫|x|–*b|un|* dx – β∫|x|–α|un|q dx. ()Yang et al. Boundary Value Problems 2012, 2012:137 Page 4 of 8http://www.boundaryvalueproblems.com/content/2012/1/137So, we geto()( + ‖un‖)+ |c| ≥ J(un) – *〈J ′(un),un〉= dn‖un‖ –( q –*)β∫|un|q|x|α dx≥ dn‖un‖ –( q –*)βCα‖un‖q.We have the boundedness of {un} for  < q < , then there exists a subsequence, we stilldenote it by {un}, such thatun ⇀ u in D,a (),un ⇀ u in L*(, |x|–*b),un → u in Lr(, |x|–α),∀≤ r < nn –  ,α < ( + a)r + n( – r),un → u a.e. on .From the concentration compactness principle, there exist nonnegativemeasuresμ, ν anda countable family {xi} ⊂  such that|x|–*b|un|* dx⇀ ν = |x|–*b|u|* dx +∑i∈Iν(i)δx(i) ,|x|–a|Dun| dx⇀ μ ≥ |x|–a|Du| dx + S∑i∈I(ν(i))/*δx(i) .() Up to a subsequence, un → u in L* (, |x|–*b).Since {un} is bounded in D,a (), we may suppose, without loss of generality, that thereexists T ∈ (L′ (, |x|–a))n such thatDun ⇀ T in(L′(, |x|–a))n.On the other hand, |un|*–un is also bounded in L′* (, |x|–*b) and|un|*–un ⇀ |u|*–u in L′*(, |x|–*b).Note thato()‖ϕ‖ = 〈J ′(un),ϕ〉=∫|x|–aDunDϕ dx – λ∫|x|–*b|un|*–unϕ dx– β∫|x|–α|un|q–unϕ dx, ()taking n→ ∞ in (), we have∫|x|–aTDϕ dx = λ∫|x|–*b|u|*–uϕ dx + β∫|x|–α|u|q–uϕ dx ()Yang et al. Boundary Value Problems 2012, 2012:137 Page 5 of 8http://www.boundaryvalueproblems.com/content/2012/1/137for any ϕ ∈D,a (). Let ϕ =ψun in (), where ψ ∈ C(), then it follows that∫|x|–aDununDψ dx +∫|x|–a|Dun|ψ dx= λ∫|x|–*b|un|*ψ dx + β∫|x|–α|un|qψ dx. ()Taking n→ ∞ in (), we have∫|x|–auTDψ dx +∫ψ dμ = λ∫ψ dν + β∫|x|–α|u|qψ dx. ()Let ϕ =ψu in (), then it follows that∫|x|–aTψudx +∫|x|–aTudψ = λ∫|x|–*b|u|*ψ dx + β∫|x|–α|u|qψ dx. ()Thus, it implies that∫ψ dμ = λ∑i∈Iνiψ(xi) +∫|x|–aTDuψ dx, ()which implies thatS(νi)/* ≤ μi = λνi.Hence, νi ≥ (λ–S)n/d if νi = .On the other hand, > c = limn→∞(J(un) –*〈J ′(un),un〉)= limn→∞(dn‖un‖ – β( q –*)∫|x|–α|un|q dx)≥ dn‖u‖ – βC‖u‖q,then ‖u‖q ≤ Cβq/(–q), so that > c = limn→∞(J(un) –*〈J ′(un),un〉)= limn→∞(dn‖un‖ – β( q –*)∫|x|–α|un|q dx)≥ dnμi – βCβq/(–q)≥ dnSnd(λ–) n–dd –Cβ–q .However, if β >  is given, we can choose λ >  so small that for every  < λ < λ, the lastterm on the right-hand side above is greater than , which is a contradiction. Similarly, ifYang et al. Boundary Value Problems 2012, 2012:137 Page 6 of 8http://www.boundaryvalueproblems.com/content/2012/1/137λ >  is given, we can take β >  so small that for every  < β < β, the last term on theright-hand side above is greater than . Then νi =  for each i.Up to now, we have shown thatlimn→∞∫|x|–*b|un|* dx =∫|x|–*b|u|* dx.So, by the Breiz-Lieb lemma,o()‖un‖ = ‖un‖ – λ∫|x|–*b|un|* dx – β∫|x|–α|un|q dx= ‖un – u‖ – ‖u‖ – λ∫|x|–*b|u|* dx – β∫|x|–α|u|q dx= ‖un – u‖ + o()‖u‖since J ′(u) = . Thus, we prove that {un} strongly converges to u in D,a (). 3 Existence of inﬁnitely many solutionsIn this section, we use the minimax procedure to prove the existence of inﬁnitely manysolutions. Let  be the class of subsets of D,a () \ {}, which are closed and symmetricwith respect to the origin. For A ∈ , we deﬁne the genus γ (A) byγ (A) =min{k ∈N : ∃φ ∈ C(A,Rk \ {}),φ(x) = –φ(–x)}.Assume that  < q < , then we obtainJ(u) = ∫|x|–a|Du| dx – λ*∫|x|–*b|u|* dx – λq∫|x|–α|u|q dx≥ ‖u‖ – Cbλ*‖u‖* – βCαq ‖u‖q.Deﬁneh(t) =  t – λCt* – βCtq.Then, given β > , there exists λ >  so small that for every  < λ < λ, there exists  < T <T such that h(t) <  for  < t < T, h(t) >  for T < t < T, h(t) <  for t > T. Similarly,given λ > , we can choose β >  with the property that T, T as above exist for each < β < β. Clearly, h(T) = h(T) = . Following the same idea as in [–], we consider thetruncated functionalJ˜(u) = ∫|x|–a|Du| dx – λ*ψ(u)∫|x|–*b|u|* dx – λq∫|x|–α|u|q dx,where ψ(u) = τ (‖u‖), and τ : R+ → [, ] is a nonincreasing C∞ function such that τ (t) = if t ≤ T and τ (t) =  if t ≥ T. The main properties of J˜ are the following.Yang et al. Boundary Value Problems 2012, 2012:137 Page 7 of 8http://www.boundaryvalueproblems.com/content/2012/1/137Lemma .() J˜ ∈ C and J˜ is bounded below.() If J˜(u)≤ , then ‖u‖ ≤ T and J˜(u) = J(u).() For any λ > , there exists β =min{β,β} such that if  < β < β and c < , then J˜satisﬁes (PS)c condition.() for any β > , there exists λ =min{λ,λ} such that if  < λ < λ and c < , then J˜satisﬁes (PS)c condition.Proof () and () are immediate. To prove () and (), observe that all (PS)c sequences forJ˜ with c < must be bounded. Similar to the proof of Lemma ., there exists a convergentsubsequence. Lemma . Given m ∈N , there is εm <  such thatγ({u ∈D,a () : J˜(u)≤ εm})≥m.Proof Fix m and let Hm be an m-dimensional subspace of D,a (). Take u ∈ Hm, u = ,write u = rmv with v ∈Hm, ‖v‖ =  and rm = ‖u‖. Thus, for  < rm < T, since all the normsare equivalent, we haveJ˜(u) = J(u) = ∫|x|–a|Du| dx – λ*∫|x|–*b|u|* dx – λq∫|x|–α|u|q dx≤ ‖u‖ – λC*‖u‖* – λCq ‖u‖q=  rm –λC*r*m –λCq rqm := εm.Therefore, we can choose rm ∈ (,T) so small that J˜(u) ≤ εm < . Let Srm = {u ∈ D,a () :‖u‖ = rm}, then Srm ∩Hm ⊂ J˜εm . Hence, γ (J˜εm ) ≥ γ (Srm ∩Hm) =m. Denote m = {A ∈  :γ (A)≥m} and letcm = infA∈msupu∈AJ˜(u).Then –∞ < cm ≤ εm <  because J˜εm ∈ m and J˜ is bounded from below. Lemma . Let λ, β be as in () or () of Lemma .. Then all cm are critical values of J˜ ascm → .Proof It is clear that cm ≤ cm+, cm < . Hence, cm → c ≤ . Moreover, since all cm arecritical values of J˜ , we claim that c = . If c < , because Kc is compact and Kc ∈ , itfollows that γ (Kc) =N < +∞ and there exists δ >  such that γ (Kc) = γ (Nδ(Kc)) = N. Bythe deformation lemma there exist ε >  (c + ε < ) and an odd homeomorphism η suchthatη(J˜ c+ε \Nδ(Kc))⊂ J˜ c–ε .Yang et al. Boundary Value Problems 2012, 2012:137 Page 8 of 8http://www.boundaryvalueproblems.com/content/2012/1/137Since cm is increasing and converges to c, there exists m ∈ N such that cm > c – ε andcm+N ≤ c and there exists A ∈ m+N such that supu∈A J˜(u) < c + ε. By the properties of γ ,we haveγ(A \Nδ(Kc))≥ γ (A) – γ (Nδ(Kc))≥m, γ (A \Nδ(Kc) )≥m.Therefore,η(A \Nδ(Kc)) ∈ m.Consequently,supu∈η(A\Nδ (Kc))J˜(u)≥ cm > c – ε,a contradiction, hence cm → . With Lemma . to Lemma ., we have proved Theorem ..Competing interestsThe authors declare that they have no competing interests.Authors’ contributionsAll authors read and approved the ﬁnal manuscript.Author details1School of Science, Jiangnan University, Wuxi, 214122, China. 2School of Mathematical Science, Nanjing NormalUniversity, Nanjing, 210097, China. 3Institute of Science, PLA University of Science and Technology, Nanjing, 211101,China.AcknowledgementsProject is supported by National Natural Science Foundation of China, Tian Yuan Special Foundation (No. 11226116), theChina Scholarship Council, Natural Science Foundation of Jiangsu Province of China for Young Scholar (No. BK2012109),the Fundamental Research Funds for the Central Universities (No. JUSRP11118, JUSRP211A22) and Foundation for youngteachers of Jiangnan University (No. 2008LQN008).Received: 4 September 2012 Accepted: 19 September 2012 Published: 26 November 2012References1. Caﬀarelli, I, Kohn, R, Nirenberg, L: First order interpolation inequalities with weights. Compos. Math. 53, 259-275 (1984)2. Xuan, B, Su, S, Yan, Y: Existence results for Brezis-Nirenberg problems with Hardy potential and singular coeﬃcients.Nonlinear Anal. 67, 2091-2106 (2007)3. Bliss, G: An integral inequality. J. Lond. Math. Soc. 5, 40-46 (1930)4. He, XM, Zou, WM: Multiple solutions for the Brezis-Nirenberg problem with a Hardy potential and singular coeﬃcients.Comput. Math. Appl. 56, 1025-1031 (2008)5. Xuan, B: The solvability of quasilinear Brezis-Nirenberg-type problems with singular weights. Nonlinear Anal. 62,703-725 (2005)6. Bernis, F, Garcia-Azorero, J, Peral, I: Existence and multiplicity of nontrivial solutions in semilinear critical problems offourth-order. Adv. Diﬀer. Equ. 1, 219-240 (1996)7. Wang, YJ, Shen, YT: Multiple and sign-changing solutions for a class of semilinear biharmonic equation. J. Diﬀer. Equ.246, 3109-3125 (2009)8. Wang, YJ, Yin, YM, Yao, YT: Multiple solutions for quasilinear Schrödinger equations involving critical exponent. Appl.Math. Comput. 216, 849-856 (2010)doi:10.1186/1687-2770-2012-137Cite this article as: Yang et al.: Existence of multiple solutions for the Brezis-Nirenberg-type problem with singularcoefﬁcients. Boundary Value Problems 2012 2012:137.

Existence of multiple solutions for the Brezis-Nirenberg-type problem with singular coefficients

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Existence of multiple solutions for the Brezis-Nirenberg-type problem with singular coefficients

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