In this paper, we study a kind of anisotropic p-Laplacian equations in $R^n$. Nontrivial solutions are obtained using mountain pass theorem given by Ambrosetti-Rabinowitz

ElManouni, S.

Electronic Journal of Qualitative Theory of Differential Equations

El Manouni Said

University of Szeged

Electronic Journal of Qualitative Theory of Differential Equations2010, No. 73, 1-9; http://www.math.u-szeged.hu/ejqtde/Note on an anisotropicp -Laplacian equation in Rn∗Said El ManouniDepartment of Mathematics, Faculty of SciencesAl-Imam UniversityP.O. Box 90950, Riyadh 11623, Saudi ArabiaEmail: samanouni@imamu.edu.sa & manouni@hotmail.comAbstractIn this paper, we study a kind of anisotropic p-Laplacian equationsin Rn. Nontrivial solutions are obtained using mountain pass theoremgiven by Ambrosetti-Rabinowitz [1].Key Words and Phrases: anisotropic p-Laplacian equations, nontrivial so-lutions.MSC2000: Primary 35J20, Secondary 35J60, 35J701 IntroductionConsider the following anisotropic p-Laplacian problem in Rn−n∑i=1∂∂xi(ai∣∣∣∣∂u∂xi∣∣∣∣pi−2 ∂u∂xi)= f(x, u) in Rn (1.1)Each pi ∈ (1,∞), ai = ai(x) are measurable real functions satisfying 0 <a0 < ai(x) ∈ L∞(Rn) and we assume that the nonlinear function f satisfiesthe subcritical growth conditions|f(x, u)| ≤ g(x)|u|r ∀(x, u) ∈ Rn × R (1.2)∗This work is supported by Al-Imam University project No. 281206, Riyadh, KSAEJQTDE, 2010 No. 73, p. 1for some r ∈ (p+−1, p̄∗−1). Herein, p+ = max(p1, . . . , pn),1p̄=1nn∑i=11piandp̄∗ =np̄n − p̄with p̄ < n and g : Rn → R is a nonnegative function satisfyingg ∈ Lω(Rn), ω =p̄∗p̄∗ − (r + 1). (1.3)For (1.1) we assume that F ∈ C1(Rn×R), where F (x, s) =∫ s0f(x, t) dt andthere exists θ > p+ such that0 < θF (x, s) ≤ sf(x, s), ∀x ∈ Rn, ∀s ∈ R \ {0}. (1.4)In the isotropic case, we can refer the reader to the works by [3] and [7] whereexistence and regularity results are obtained. As to anisotropic equationswith different orders of derivations in different directions involving criticalexponents with unbounded nonlinearities, to our knowledge they were notintensively studied before, in passing, we mention the work [2]. Let us men-tion also that in [4] the authors have studied another class of anisotropic el-liptic equations. Via an adaptation of the concentration-compactness lemmaof P.-L. Lions to anisotropic operators, they have obtained the existence ofmultiple nonnegative solutions. Let us point out that in the case of boundeddomains, more work in this direction can be found in [5] where the authorsproved existence and nonexistence results for some anisotropic quasilinearelliptic equations.The purpose of this paper is to obtain nontrivial weak solutions usingmountain pass theorem (see e.g. [1]).Our main result is the following.Theorem 1.1. Assume (1.2), (1.3), and (1.4). Moreover, if g ∈ Lωp1p1−1 (Rn)for some r + 1 < p1 < p̄∗, then problem (1.1) has at least one nontrivialsolution.2 PreliminariesWe let 1 ≤ p1, . . . , pn < ∞ be N real numbers. Denote by p the harmonicmean of these numbers, i.e., 1p= 1n∑ni=11pi, and set p+ = max(p1, . . . , pn),EJQTDE, 2010 No. 73, p. 2p− = min(p1, . . . , pn). We always have p− ≤ p ≤ p+. The Sobolev conjugateof p is denoted by p⋆, i.e., p⋆ = npn−p.Anisotropic Sobolev spaces were introduced and studied by Nikol’skĭı [8],Slobodeckĭı [9], Troisi [10], and later by Trudinger [11] in the framework ofOrlicz spaces.Let (W, ‖ · ‖) be the anisotropic Sobolev space defined byW ={u ∈ W 1,p̄(Rn) : ai1pi∂u∂xi∈ Lpi(Rn), i = 1, . . . , n},with the dual (W ∗, ‖ · ‖∗) and the duality pairing 〈·, ·〉. W is a real reflexiveBanach under the norm‖u‖ =n∑i=1(∫Rnai∣∣∣∣∂u∂xi∣∣∣∣pidx)1pi=n∑i=1∣∣∣∣∣∣∣∣ai1pi∂u∂xi∣∣∣∣∣∣∣∣Lpi(Rn).Let us recall that a weak solution of the equation (1.1) is any u ∈ W suchthatn∑i=1∫Rnai∣∣∣∣∂u∂xi∣∣∣∣pi−2 ∂u∂xi∂v∂xidx =∫Rnf(x, u)v dx, (2.1)for all v ∈ W .They coincide with the critical points of the C1-energy functional corre-sponding to problem (1.1)Φ(u) =n∑i=11pi∫Rnai(x)∣∣∣∣∂u∂xi∣∣∣∣pidx −∫RnF (x, u) dx, (2.2)for all u ∈ W.Remark 2.1. Let us remark that (1.3) implies 1ω+ rp̄∗+ 1p̄∗= 1 which guar-antee that the integral given in right side of (2.1) is well defined.To deal with the functional framework we apply the following mountainpass theorem [1].Theorem 2.2. Let I be a C1-differentiable functional on a Banach space Eand satisfying the Palais-Smale condition (PS), suppose that there exists aneighbourhood U of 0 in E and a positive constant α satisfying the followingconditions:EJQTDE, 2010 No. 73, p. 3(I1) I(0) = 0.(I2) I(u) ≥ α on the boundary of U .(I3) There exists an e ∈ W\U such that I(e) < α.Thenc = infγ∈Γsupy∈[0,1]I(γ(y))is a critical value of I with Γ = {h ∈ C([0, 1]); h(0) = 0, h(1) = e}.Let us recall that the functional I : E → R of class C1 satisfies the Palais-Smale compactness condition (PS) if every sequence (uk)∞k=1 ⊂ E for whichthere exists M > 0 such that: I(uk) ≤ M and I′(uk) → 0 strongly in I⋆ as ngoes to infinity (called a (PS) sequence), has a convergent subsequence.Let us define the functional J : W → R by J(u) =∫RnF (x, u) dx for eachu ∈ W. It follows from (1.2),|F (x, u)| ≤1r + 1g(x)|u|r+1 (2.3)for all (x, u) ∈ Rn ×R. Then by standard argument we have F is in C1(Rn ×R), hence we see that J is well defined and continuously Gâteaux differen-tiable withJ′(u)v =∫Rnf(x, u)v dxfor all u, v ∈ W.Proposition 2.3. J ′ is a compact map from W to W ∗.Proof. Let uk be a sequence in W which converges weakly to u. On onehand, in view of Hölder’s inequality and Sobolev embedding, we obtain forall 0 ≤ R ≤ +∞,∫|x|≥Rf(x, u)v dx ≤(∫|x|≥R|g|ω dx)1ω(C1‖u‖)r‖v‖p̄∗,for all u, v ∈ W. Since g ∈ Lω(Rn), we have limR→+∞∫|x|≥R|g|ω dx = 0. Thisimplies with the fact that uk is a bounded sequence, for any ε, there existsRε > 0 such that∫|x|≥Rεf(x, uk)v dx ≤ ε and∫|x|≥Rεf(x, u)v dx ≤ ε (2.4)EJQTDE, 2010 No. 73, p. 4holds for all k.On the other hand, since g ∈ Lωp1p1−1 (Rn) with r + 1 < p1 < p̄∗, applyingYoung’s inequality, we getfp1p1−1 (x, t) ≤ m(x)p1p1−1 trp1p1−1≤p̄∗ − (r + 1)p̄∗m(x)p1p1−1( p̄∗p̄∗−(r+1))+r + 1p̄∗trp1p1−1p̄∗r+1 ,for all t ∈ R and a.e. x ∈ Bε = {x ∈ Rn; |x| < Rε}. A simple calculationshows that rp1p1−1p̄∗r+1< p̄∗. Using the compact imbedding W 1,p̄(Bε) into Lq(Bε)for all q ∈ [1, p̄∗) and the continuity of the Nemytskii’s operator Nf associatedwith fp1p1−1 from Lrp1p1−1p̄∗r+1 (Bε) in L1(Bε), we conclude that∫|x|<Rεf(x, uk)p1p1−1 dx →∫|x|<Rεf(x, u)p1p1−1 dx,which implies that f(x, uk) converges to f(x, u) in Lp1p1−1 (Bε). Hence sinceLp̄∗(Bε) ⊂ Lp1(Bε), we have f(x, uk)v converges to f(x, u)v in L1(Bε), i.e.∫|x|<Rε(f(x, uk) − f(x, u))v dx → 0, (2.5)for all v ∈ Lp1(Bε). Finally, in view of (2.4) and (2.5), we get∫Rnf(x, uk)v dx →∫Rnf(x, u)v dx,for all v ∈ W. This completes the proof of Proposition 2.3 and consequentlyJ′is compact.3 Proof of Theorem 1.1First we need the following Lemmas to show that the functional Φ satisfiesthe geometric conditions of Theorem 2.2.Lemma 3.1. Suppose (1.2), (1.3) and (1.4) hold, then there exist constantsα and ρ, such that Φ(u) ≥ α for ||u|| = ρ.EJQTDE, 2010 No. 73, p. 5Proof. From (2.3), one can easily deduce thatlimt→0F (x, t)up+= 0 and limt→∞F (x, t)up̄∗= 0.This implies that∣∣∣∣F (x, t)up+∣∣∣∣≤ ε ∀|t| < δ1;∣∣∣∣F (x, t)up̄∗∣∣∣∣≤ ε ∀|t| > δ2,for some δ1, δ2 > 0. Now, using the fact that the functionF (x,t)up̄∗is continuousover [δ1, δ2], we conclude that for all ε > 0 there exists cε > 0 such thatF (x, t) ≤ ε|t|p+ + cε|t|p̄∗ ∀(x, t) ∈ Rn × R. (3.1)Using again (2.3), the relation (3.1) and continuous imbedding of W inLp+(Rn) and Lp̄∗(Rn), we get for ||u|| small enough thatΦ(u) =n∑i=11pi∫Rnai∣∣∣∣∂u∂xi∣∣∣∣pidx −∫R∗F (x, u) dx,≥1p+||u||p+ − Cε ||u||p+ − C ′cε ||u||p̄∗≥ ||u||p+((1p+− εC) − C ′cε ||u||p̄∗−p+),for some constants C, C ′ > 0. Therefore, for 0 < ε < 1Cp+, there exist α and ρsmall enough positive constants such that Φ(u) ≥ α > 0 for all ||u|| = ρ.Lemma 3.2. Suppose (1.2) and (1.4) hold. Then Φ(tu) → −∞ as t → +∞.Proof. We haveΦ(t1/p+u) =n∑i=1tpip+pi∫Rnai∣∣∣∣∂u∂xi∣∣∣∣pidx −∫RnF (x, t1/p+u) dx.Remark that F (x, tu) ≥ tθF (x, u) for any t ≥ 1, this is due to the fact thatthe function F (x,tu)tθis increasing for all t > 0, we obtainΦ(t1/p+u) =n∑i=1tpip+pi∫Rnai∣∣∣∣∂u∂xi∣∣∣∣pidx − tθ/p+∫RnF (x, u) dx.EJQTDE, 2010 No. 73, p. 6ThenΦ(t1/p+u) → −∞ as t → +∞follows immediately since θ > p+.Lemma 3.3. Let uk be a Palais-Smale sequence of Φ. Then uk possesses asubsequence converging strongly to some u ∈ W.Proof. First we claim that the sequence uk is bounded. Indeed, Arguingby contradiction and consider a subsequence still denoted by uk such that||uk|| → ∞. We have in view of (1.4)Φ(uk) =n∑i=11pi∫Rnai∣∣∣∣∂uk∂xi∣∣∣∣pidx −∫RnF (x, uk) dx≥1p+∫Rnai∣∣∣∣∂uk∂xi∣∣∣∣pidx −1θ∫Rnf(x, uk)uk dx=1p+∫Rnai∣∣∣∣∂uk∂xi∣∣∣∣pidx +1θ(〈Φ′(uk); uk〉 −n∑i=1∫Rnai∣∣∣∣∂uk∂xi∣∣∣∣pidx)≥(1p+−1θ) n∑i=1∫Rnai∣∣∣∣∂uk∂xi∣∣∣∣pidx +1θ〈Φ′(uk); uk〉.(3.2)Passing to limit in (3.2) as k → ∞, we obtainM ≥(1p+−1θ) n∑i=1∫Rnai∣∣∣∣∂uk∂xi∣∣∣∣pidx ≥(1p+−1θ)||uk||p− ,where M is the constant of Palais-Smale sequence. This gives a contradictionsince p+ < θ. Hence the sequence uk has a subsequence still denoted by ukwhich converges weakly to some u ∈ W. For any pair integer (n, m) we haven∑i=1∫Rnai(x)(∣∣∣∣∂un∂xi∣∣∣∣pi−2 ∂un∂xi−∣∣∣∣∂um∂xi∣∣∣∣pi−2 ∂um∂xi)(∂un∂xi−∂um∂xi)= 〈Φ′(un) − Φ′(um); (un − um)〉 +∫Rn(f(x, un) − f(x, um))(un − um) dx.(3.3)EJQTDE, 2010 No. 73, p. 7By Palais-Smale condition and Proposition 2.3, it is easy to see that the rightside of (3.3) approaches zero. Finally, using the following algebraic relation|ξ1 − ξ2|r ≤ ((|ξ1|r−2ξ1 − |ξ2|r−2ξ2)(ξ1 − ξ2))ρ/2(|ξ1|r + |ξ2|r)1−ρ/2,with ρ = r if 1 < r ≤ 2 and ρ = 2 if 2 < r, the monotonicity of the anisotropicoperator of problem (1.1) now gives the result. This concludes the proof ofLemma 3.3.Therefore lemmas 3.1, 3.2 and 3.3 fit into conditions setting of Theorem2.2 of section 2, this guarantees the existence of at least a nontrivial weaksolution for (1.1).Remark 3.4.1- One can prove that each solution u of problem (1.1) satisfies u ∈ Lσ(Rn)with p̄∗ ≤ σ ≤ ∞. This regularity result is based on an iterative proceduregiven in the works [3] where similar results are obtained for the case of de-generate isotropic p-Laplacian problems.2- Let us also mention that since ω + ε < ωp1p1−1, with 0 < ε < 1 small enough,the restrictive integrability condition g ∈ Lω+ε(Rn), suffices to proceed withthe iterating method and to bound the maximal norm of the solution.References[1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical pointtheory and applications, Funct. Anal. 14 (1973), 349–381.[2] M. Bendahmane and S. El Manouni, Anisotropic Elliptic Equations in RN :Existence and Regularity Results, Top. Meth. Nonl. Anal. 36(1) (2010), 129–153.[3] P. Drabek, Nonlinear Eigenvalues for p-Laplacian in RN , Math. Nachr. 173(1995), 131–139.[4] A. El Hamidi and J. M. Rakotoson. Extremal functions for the anisotropicSobolev inequalities. Ann. I. H. Poincaré-AN 24 (2007), 741–756.[5] I. Fragalà, F. Gazzola and B. Kawohl. Existence and nonexistence results foranisotropic quasi- linear equations. Ann. I. H. Poincaré-AN 21 (2004), 715–734.EJQTDE, 2010 No. 73, p. 8[6] S. El Manouni, The study of nonlinear problems for p-Laplacian in Rn via aRicceri’s principle, submitted.[7] Lao Sen Yu, Nonlinear p-Laplacian problems on unbounded domains, Proceed-ing of A. M. S. 115, 4 (1992), 1037–1045.[8] S. M. Nikol’skĭı, An imbedding theorem for functions with partial derivativesconsidered in different metrics. Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958),321–336. English translation: Amer. Math. Soc. Transl. 90 (1970), 27–44.[9] L. N. Slobodeckĭı, Generalized Sobolev spaces and their application to bound-ary problems for partial differential equations. Leningrad. Gos. Ped. Inst. Učen.Zap. 197 (1958), 54–112. English translation: Amer. Math. Soc. Transl. (2) 57(1966), 207–275.[10] M. Troisi, Teoremi di inclusione per spazi di sobolev non isotropi. Ricerche.Mat., 18 (1969), 3–24.[11] N. S. Trudinger, An imbedding theorem for H0(G, Ω) spaces. Studia Math.,50 (1974), 17–30.(Received October 12, 2010)EJQTDE, 2010 No. 73, p. 9

Note on an anisotropic p-Laplacian equation in Rn

S. El Manouni

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Electronic journal of qualitative theory of differential equations

Note on an anisotropic p-Laplacian equation in $R^n$

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Electronic Journal of Qualitative Theory of Differential Equations2010, No. 73, 1-9; http://www.math.u-szeged.hu/ejqtde/Note on an anisotropicp -Laplacian equation in Rn∗Said El ManouniDepartment of Mathematics, Faculty of SciencesAl-Imam UniversityP.O. Box 90950, Riyadh 11623, Saudi ArabiaEmail: samanouni@imamu.edu.sa & manouni@hotmail.comAbstractIn this paper, we study a kind of anisotropic p-Laplacian equationsin Rn. Nontrivial solutions are obtained using mountain pass theoremgiven by Ambrosetti-Rabinowitz [1].Key Words and Phrases: anisotropic p-Laplacian equations, nontrivial so-lutions.MSC2000: Primary 35J20, Secondary 35J60, 35J701 IntroductionConsider the following anisotropic p-Laplacian problem in Rn−n∑i=1∂∂xi(ai∣∣∣∣ ∂u∂xi∣∣∣∣pi−2 ∂u∂xi)= f(x, u) in Rn (1.1)Each pi ∈ (1,∞), ai = ai(x) are measurable real functions satisfying 0 <a0 < ai(x) ∈ L∞(Rn) and we assume that the nonlinear function f satisfiesthe subcritical growth conditions|f(x, u)| ≤ g(x)|u|r ∀(x, u) ∈ Rn ×R (1.2)∗This work is supported by Al-Imam University project No. 281206, Riyadh, KSAEJQTDE, 2010 No. 73, p. 1for some r ∈ (p+−1, p¯∗−1). Herein, p+ = max(p1, . . . , pn),1p¯=1nn∑i=11piandp¯∗ =np¯n− p¯with p¯ < n and g : Rn → R is a nonnegative function satisfyingg ∈ Lω(Rn), ω =p¯∗p¯∗ − (r + 1). (1.3)For (1.1) we assume that F ∈ C1(Rn×R), where F (x, s) =∫ s0f(x, t) dt andthere exists θ > p+ such that0 < θF (x, s) ≤ sf(x, s), ∀x ∈ Rn, ∀s ∈ R \ {0}. (1.4)In the isotropic case, we can refer the reader to the works by [3] and [7] whereexistence and regularity results are obtained. As to anisotropic equationswith different orders of derivations in different directions involving criticalexponents with unbounded nonlinearities, to our knowledge they were notintensively studied before, in passing, we mention the work [2]. Let us men-tion also that in [4] the authors have studied another class of anisotropic el-liptic equations. Via an adaptation of the concentration-compactness lemmaof P.-L. Lions to anisotropic operators, they have obtained the existence ofmultiple nonnegative solutions. Let us point out that in the case of boundeddomains, more work in this direction can be found in [5] where the authorsproved existence and nonexistence results for some anisotropic quasilinearelliptic equations.The purpose of this paper is to obtain nontrivial weak solutions usingmountain pass theorem (see e.g. [1]).Our main result is the following.Theorem 1.1. Assume (1.2), (1.3), and (1.4). Moreover, if g ∈ Lωp1p1−1 (Rn)for some r + 1 < p1 < p¯∗, then problem (1.1) has at least one nontrivialsolution.2 PreliminariesWe let 1 ≤ p1, . . . , pn < ∞ be N real numbers. Denote by p the harmonicmean of these numbers, i.e., 1p= 1n∑ni=11pi, and set p+ = max(p1, . . . , pn),EJQTDE, 2010 No. 73, p. 2p− = min(p1, . . . , pn). We always have p− ≤ p ≤ p+. The Sobolev conjugateof p is denoted by p⋆, i.e., p⋆ = npn−p.Anisotropic Sobolev spaces were introduced and studied by Nikol’ski˘ı [8],Slobodecki˘ı [9], Troisi [10], and later by Trudinger [11] in the framework ofOrlicz spaces.Let (W, ‖ · ‖) be the anisotropic Sobolev space defined byW ={u ∈W 1,p¯(Rn) : ai1pi∂u∂xi∈ Lpi(Rn), i = 1, . . . , n},with the dual (W ∗, ‖ · ‖∗) and the duality pairing 〈·, ·〉. W is a real reflexiveBanach under the norm‖u‖ =n∑i=1(∫Rnai∣∣∣∣ ∂u∂xi∣∣∣∣pidx) 1pi=n∑i=1∣∣∣∣∣∣∣∣ai 1pi ∂u∂xi∣∣∣∣∣∣∣∣Lpi(Rn).Let us recall that a weak solution of the equation (1.1) is any u ∈W suchthatn∑i=1∫Rnai∣∣∣∣ ∂u∂xi∣∣∣∣pi−2 ∂u∂xi∂v∂xidx =∫Rnf(x, u)v dx, (2.1)for all v ∈W .They coincide with the critical points of the C1-energy functional corre-sponding to problem (1.1)Φ(u) =n∑i=11pi∫Rnai(x)∣∣∣∣ ∂u∂xi∣∣∣∣pidx−∫RnF (x, u) dx, (2.2)for all u ∈W.Remark 2.1. Let us remark that (1.3) implies 1ω+ rp¯∗+ 1p¯∗= 1 which guar-antee that the integral given in right side of (2.1) is well defined.To deal with the functional framework we apply the following mountainpass theorem [1].Theorem 2.2. Let I be a C1-differentiable functional on a Banach space Eand satisfying the Palais-Smale condition (PS), suppose that there exists aneighbourhood U of 0 in E and a positive constant α satisfying the followingconditions:EJQTDE, 2010 No. 73, p. 3(I1) I(0) = 0.(I2) I(u) ≥ α on the boundary of U .(I3) There exists an e ∈W\U such that I(e) < α.Thenc = infγ∈Γsupy∈[0,1]I(γ(y))is a critical value of I with Γ = {h ∈ C([0, 1]); h(0) = 0, h(1) = e}.Let us recall that the functional I : E → R of class C1 satisfies the Palais-Smale compactness condition (PS) if every sequence (uk)∞k=1 ⊂ E for whichthere exists M > 0 such that: I(uk) ≤M and I′(uk)→ 0 strongly in I⋆ as ngoes to infinity (called a (PS) sequence), has a convergent subsequence.Let us define the functional J : W → R by J(u) =∫RnF (x, u) dx for eachu ∈W. It follows from (1.2),|F (x, u)| ≤1r + 1g(x)|u|r+1 (2.3)for all (x, u) ∈ Rn×R. Then by standard argument we have F is in C1(Rn×R), hence we see that J is well defined and continuously Gaˆteaux differen-tiable withJ′(u)v =∫Rnf(x, u)v dxfor all u, v ∈W.Proposition 2.3. J ′ is a compact map from W to W ∗.Proof. Let uk be a sequence in W which converges weakly to u. On onehand, in view of Ho¨lder’s inequality and Sobolev embedding, we obtain forall 0 ≤ R ≤ +∞,∫|x|≥Rf(x, u)v dx ≤(∫|x|≥R|g|ω dx) 1ω(C1‖u‖)r‖v‖p¯∗,for all u, v ∈ W. Since g ∈ Lω(Rn), we have limR→+∞∫|x|≥R|g|ω dx = 0. Thisimplies with the fact that uk is a bounded sequence, for any ε, there existsRε > 0 such that∫|x|≥Rεf(x, uk)v dx ≤ ε and∫|x|≥Rεf(x, u)v dx ≤ ε (2.4)EJQTDE, 2010 No. 73, p. 4holds for all k.On the other hand, since g ∈ Lωp1p1−1 (Rn) with r + 1 < p1 < p¯∗, applyingYoung’s inequality, we getfp1p1−1 (x, t) ≤ m(x)p1p1−1 trp1p1−1≤p¯∗ − (r + 1)p¯∗m(x)p1p1−1( p¯∗p¯∗−(r+1))+r + 1p¯∗trp1p1−1p¯∗r+1 ,for all t ∈ R and a.e. x ∈ Bε = {x ∈ Rn; |x| < Rε}. A simple calculationshows that rp1p1−1p¯∗r+1< p¯∗. Using the compact imbeddingW 1,p¯(Bε) into Lq(Bε)for all q ∈ [1, p¯∗) and the continuity of the Nemytskii’s operatorNf associatedwith fp1p1−1 from Lrp1p1−1p¯∗r+1 (Bε) in L1(Bε), we conclude that∫|x|<Rεf(x, uk)p1p1−1 dx→∫|x|<Rεf(x, u)p1p1−1 dx,which implies that f(x, uk) converges to f(x, u) in Lp1p1−1 (Bε). Hence sinceLp¯∗(Bε) ⊂ Lp1(Bε), we have f(x, uk)v converges to f(x, u)v in L1(Bε), i.e.∫|x|<Rε(f(x, uk)− f(x, u))v dx→ 0, (2.5)for all v ∈ Lp1(Bε). Finally, in view of (2.4) and (2.5), we get∫Rnf(x, uk)v dx→∫Rnf(x, u)v dx,for all v ∈ W. This completes the proof of Proposition 2.3 and consequentlyJ′is compact.3 Proof of Theorem 1.1First we need the following Lemmas to show that the functional Φ satisfiesthe geometric conditions of Theorem 2.2.Lemma 3.1. Suppose (1.2), (1.3) and (1.4) hold, then there exist constantsα and ρ, such that Φ(u) ≥ α for ||u|| = ρ.EJQTDE, 2010 No. 73, p. 5Proof. From (2.3), one can easily deduce thatlimt→0F (x, t)up+= 0 and limt→∞F (x, t)up¯∗= 0.This implies that∣∣∣∣F (x, t)up+∣∣∣∣ ≤ ε ∀|t| < δ1;∣∣∣∣F (x, t)up¯∗∣∣∣∣ ≤ ε ∀|t| > δ2,for some δ1, δ2 > 0. Now, using the fact that the functionF (x,t)up¯∗is continuousover [δ1, δ2], we conclude that for all ε > 0 there exists cε > 0 such thatF (x, t) ≤ ε|t|p+ + cε|t|p¯∗ ∀(x, t) ∈ Rn × R. (3.1)Using again (2.3), the relation (3.1) and continuous imbedding of W inLp+(Rn) and Lp¯∗(Rn), we get for ||u|| small enough thatΦ(u) =n∑i=11pi∫Rnai∣∣∣∣ ∂u∂xi∣∣∣∣pidx−∫R∗F (x, u) dx,≥1p+||u||p+ − Cε ||u||p+ − C ′cε ||u||p¯∗≥ ||u||p+((1p+− εC)− C ′cε ||u||p¯∗−p+),for some constants C,C ′ > 0. Therefore, for 0 < ε < 1Cp+, there exist α and ρsmall enough positive constants such that Φ(u) ≥ α > 0 for all ||u|| = ρ.Lemma 3.2. Suppose (1.2) and (1.4) hold. Then Φ(tu)→ −∞ as t→ +∞.Proof. We haveΦ(t1/p+u) =n∑i=1tpip+pi∫Rnai∣∣∣∣ ∂u∂xi∣∣∣∣pidx−∫RnF (x, t1/p+u) dx.Remark that F (x, tu) ≥ tθF (x, u) for any t ≥ 1, this is due to the fact thatthe function F (x,tu)tθis increasing for all t > 0, we obtainΦ(t1/p+u) =n∑i=1tpip+pi∫Rnai∣∣∣∣ ∂u∂xi∣∣∣∣pidx− tθ/p+∫RnF (x, u) dx.EJQTDE, 2010 No. 73, p. 6ThenΦ(t1/p+u)→ −∞ as t→ +∞follows immediately since θ > p+.Lemma 3.3. Let uk be a Palais-Smale sequence of Φ. Then uk possesses asubsequence converging strongly to some u ∈ W.Proof. First we claim that the sequence uk is bounded. Indeed, Arguingby contradiction and consider a subsequence still denoted by uk such that||uk|| → ∞. We have in view of (1.4)Φ(uk) =n∑i=11pi∫Rnai∣∣∣∣∂uk∂xi∣∣∣∣pidx−∫RnF (x, uk) dx≥1p+∫Rnai∣∣∣∣∂uk∂xi∣∣∣∣pidx−1θ∫Rnf(x, uk)uk dx=1p+∫Rnai∣∣∣∣∂uk∂xi∣∣∣∣pidx+1θ(〈Φ′(uk); uk〉 −n∑i=1∫Rnai∣∣∣∣∂uk∂xi∣∣∣∣pidx)≥(1p+−1θ) n∑i=1∫Rnai∣∣∣∣∂uk∂xi∣∣∣∣pidx+1θ〈Φ′(uk); uk〉.(3.2)Passing to limit in (3.2) as k →∞, we obtainM ≥(1p+−1θ) n∑i=1∫Rnai∣∣∣∣∂uk∂xi∣∣∣∣pidx ≥(1p+−1θ)||uk||p− ,whereM is the constant of Palais-Smale sequence. This gives a contradictionsince p+ < θ. Hence the sequence uk has a subsequence still denoted by ukwhich converges weakly to some u ∈W. For any pair integer (n,m) we haven∑i=1∫Rnai(x)(∣∣∣∣∂un∂xi∣∣∣∣pi−2 ∂un∂xi−∣∣∣∣∂um∂xi∣∣∣∣pi−2 ∂um∂xi)(∂un∂xi−∂um∂xi)= 〈Φ′(un)− Φ′(um); (un − um)〉+∫Rn(f(x, un)− f(x, um))(un − um) dx.(3.3)EJQTDE, 2010 No. 73, p. 7By Palais-Smale condition and Proposition 2.3, it is easy to see that the rightside of (3.3) approaches zero. Finally, using the following algebraic relation|ξ1 − ξ2|r ≤ ((|ξ1|r−2ξ1 − |ξ2|r−2ξ2)(ξ1 − ξ2))ρ/2(|ξ1|r + |ξ2|r)1−ρ/2,with ρ = r if 1 < r ≤ 2 and ρ = 2 if 2 < r, the monotonicity of the anisotropicoperator of problem (1.1) now gives the result. This concludes the proof ofLemma 3.3.Therefore lemmas 3.1, 3.2 and 3.3 fit into conditions setting of Theorem2.2 of section 2, this guarantees the existence of at least a nontrivial weaksolution for (1.1).Remark 3.4.1- One can prove that each solution u of problem (1.1) satisfies u ∈ Lσ(Rn)with p¯∗ ≤ σ ≤ ∞. This regularity result is based on an iterative proceduregiven in the works [3] where similar results are obtained for the case of de-generate isotropic p-Laplacian problems.2- Let us also mention that since ω+ ε < ωp1p1−1, with 0 < ε < 1 small enough,the restrictive integrability condition g ∈ Lω+ε(Rn), suffices to proceed withthe iterating method and to bound the maximal norm of the solution.References[1] A. Ambrosetti and P. H. Rabinowitz, Dual variational methods in critical pointtheory and applications, Funct. Anal. 14 (1973), 349–381.[2] M. Bendahmane and S. El Manouni, Anisotropic Elliptic Equations in RN :Existence and Regularity Results, Top. Meth. Nonl. Anal. 36(1) (2010), 129–153.[3] P. Drabek, Nonlinear Eigenvalues for p-Laplacian in RN , Math. Nachr. 173(1995), 131–139.[4] A. El Hamidi and J. M. Rakotoson. Extremal functions for the anisotropicSobolev inequalities. Ann. I. H. Poincare´-AN 24 (2007), 741–756.[5] I. Fragala`, F. Gazzola and B. Kawohl. Existence and nonexistence results foranisotropic quasi- linear equations. Ann. I. H. Poincare´-AN 21 (2004), 715–734.EJQTDE, 2010 No. 73, p. 8[6] S. El Manouni, The study of nonlinear problems for p-Laplacian in Rn via aRicceri’s principle, submitted.[7] Lao Sen Yu, Nonlinear p-Laplacian problems on unbounded domains, Proceed-ing of A. M. S. 115, 4 (1992), 1037–1045.[8] S. M. Nikol’ski˘ı, An imbedding theorem for functions with partial derivativesconsidered in different metrics. Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958),321–336. English translation: Amer. Math. Soc. Transl. 90 (1970), 27–44.[9] L. N. Slobodecki˘ı, Generalized Sobolev spaces and their application to bound-ary problems for partial differential equations. Leningrad. Gos. Ped. Inst. Ucˇen.Zap. 197 (1958), 54–112. English translation: Amer. Math. Soc. Transl. (2) 57(1966), 207–275.[10] M. Troisi, Teoremi di inclusione per spazi di sobolev non isotropi. Ricerche.Mat., 18 (1969), 3–24.[11] N. S. Trudinger, An imbedding theorem for H0(G, Ω) spaces. Studia Math.,50 (1974), 17–30.(Received October 12, 2010)EJQTDE, 2010 No. 73, p. 9

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Note on an anisotropic p-Laplacian equation in $R^n$

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