DSpace at VNU: Multiple solutions for a class of N-Kirchhoff type equations via variational methods tài liệu, giáo án, b...
RACSAM DOI 10.1007/s13398-014-0177-3 ORIGINAL PAPER Multiple solutions for a class of N-Kirchhoff type equations via variational methods Nguyen Thanh Chung · Hoang Quoc Toan Received: March 2014 / Accepted: June 2014 © Springer-Verlag Italia 2014 Abstract In this article, we consider the following N -Kirchhoff type problem ⎧ ⎪ ⎨ −M |∇u| N d x N u = λ f (x, u) + μg(x, u) in , ⎪ ⎩ u = on ∂ , where is a bounded smooth domain of R N , N ≥ 2, M : R+ → R is a continuous function, N −2 ∇u), f, g : × R → R are two Carathéodory functions and λ, μ are N u = div(|∇u| positive parameters Using variational method, we show the existence of at least three weak solutions for the problem Keywords N -Kirchhoff type equations · Multiple solutions · Variational method Mathematics Subject Classifications (2010) 35J55 · 35J65 Introduction In this article, we consider the following N -Kirchhoff type problem ⎛ ⎞ ⎧ ⎪ ⎪ ⎨ −M ⎝ |∇u| N d x ⎠ N u = λ f (x, u) + μg(x, u) ⎪ ⎪ ⎩ u=0 in , (1.1) on ∂ , N T Chung (B) Department of Mathematics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam e-mail: ntchung82@yahoo.com H Q Toan Department of Mathematics, Hanoi University of Science, 334 Nguyen Trai, Thanh Xuan, Hanoi, Vietnam e-mail: hq_toan@yahoo.com N T Chung, H Q Toan where is a bounded smooth domain of R N , N ≥ 2, N u = div(|∇u| N −2 ∇u), f, g : × R → R are two Carathéodory functions and λ, μ are positive parameters Throughout this paper, we assume that M : R+ → R is a continuous and nondecreasing function, and there exist m > and α > such that (M0 ) M(t) ≥ m t α−1 for all t ∈ R+ := [0, ∞) According to the definition given by Adimurthi in [1,2], we say that a function ϕ has a subcritical growth if, for every δ > 0, lim sup |t|→∞ x∈ |ϕ(x, t)| N exp(δ|t| N −1 ) = (1.2) Such a notion of criticality is motivated by the Trudinger-Moser inequality after the celebrated papers [11,20] In the present paper we will assume that the nonlinearities f, g belong to the class A of the Carathéodory functions ϕ : × R → R having a subcritical growth and satisfying the following condition: for every L > 0, sup |ϕ(x, t)| ∈ L ∞ ( ) |t|≤L (1.3) Problems of the above type arise for instance in conformal geometry When N = 2, the prescribed Gaussian curvature equation is a subcritical equation on a two dimensional manifold M with metric tensor g of the type − u = k(x) exp (2u) + h(x), (1.4) where is the Laplace-Beltrami operator associated to the metric g, k is the Gaussian curvature and h is a given Hölder function The solvability of nonlinear boundary value problems in the presence of an exponential nonlinearity is an interesting topic, we refer to recent papers [9,12,16,19] In this paper, motivated by the ideas introduced in [16] and the multiplicity result for Kirchhoff type problems in [3,18], we want to handle problem (1.1) with exponential nonlinearities To our best knowledge, this is the first contribution to nonlocal problems in this direction Further information on nonlocal problem involving p-Laplace operators and p(x)-Laplace operators, we refer to [4–8,10,13–15] Definition 1.1 Let X be a real Banach space We denote by W X the class of functionals A : X → R possessing the following property: if {u m } is a sequence in X weakly converging to u ∈ X and lim inf m→∞ A(u m ) ≤ A(u), then {u m } has a subsequence strongly converging to u The key in our argument is the following result which was presented in the paper by Ricceri [17] Proposition 1.2 (see [17]) Let X be a separable and reflexive real Banach space; : X → R a coercive, sequentially weakly lower semicontinuous C functional, belonging to W X , bounded on each bounded subset of X and whose derivative admits a continuous inverse on X ∗ ; J : X → R a C functional with compact derivative Assume that has a strict local minimum u , with (u ) = J (u ) = Finally, assume that max lim sup u →+∞ J (u) J (u) , lim sup (u) u→u (u) ≤0 N -Kirchhoff type equations and that sup min{ (u), J (u)} > u∈X Set θ ∗ := inf (u) : u ∈ X, min{ (u), J (u)} > J (u) Then, for each compact interval ⊂ (θ ∗ , +∞), there exists a number σ > with the following property: for every λ ∈ and every C functional : X → R with compact ∗ derivative, there exists μ > such that for each μ ∈ [0, μ∗ ], the equation (u) = λJ (u) + μ (u) has at least three solutions whose norms are less than σ Main result In this section, we will discuss the existence of weak solutions for problem (1.1) When there is no misunderstanding, we always use c to denote a positive constant For simplicity, we denote by X the Sobolev space W01,N ( ) endowed the norm ⎞1 ⎛ u =⎝ N |∇u| d x ⎠ N From the Rellich Kondrachov theorem, X is continuously embedded in L r ( ) for every r ≥ 1, that is there exists a constant Sr > such that |u|r ≤ Sr u (2.1) |u|r d x r We also denote by |u|r the norm of u in the space L r ( ), i.e., |u|r = It is also well-known that if L φ ( ) is the Orlicz-Lebesgue space generated by the function N φ(t) = exp(|t| N −1 ) − 1, i.e., ⎧ ⎨ L φ ( ) := u : ⎩ → R, measurable : ⎫ ⎬ φ(u(x)) d x < ∞ , ⎭ and L φ ∗ ( ) is the linear hull of L φ ( ) equipped with the norm ⎫ ⎧ ⎬ ⎨ u(x) dx ≤ , φ u L φ ∗ ( ) = inf λ > : ⎭ ⎩ λ X is continuously embedded in L φ ∗ ( ) Throughout the sequel we will use the following facts: N (T M1 ) For every u ∈ X , for every δ > 0, exp(δ|u(.)| N −1 ) ∈ L ( ); N T Chung, H Q Toan (T M2 ) There exists a constant C N depending on N and on the measure of such that, for N −1 every < δ ≤ α N , where α N = N w N −1 , being w N −1 the measure of the (N − 1) dimensional surface of the unit sphere in R N , N exp(δ|u(x)| N −1 ) d x ≤ C N sup u ≤1 Due to the assumption (M0 ) and the above results, problem (1.1) can be treated variationally in X , i.e., solutions of problem (1.1) will be obtained as critical points of a suitable functional defined on X If f ∈ A, we put t F(x, t) = f (x, s) ds, (x, t) ∈ ×R and t M(s) ds, t ≥ M(t) = Furthermore, let us define the functionals ρ, , J : X = W01,N ( ) → R by ρ(u) = |∇u| N d x, (u) = J (u) = F(x, u) d x, u ∈ X M ρ(u) , N Some simple computations show that the functionals their derivatives are given by ⎛ ⎞ (u)(v) = M ⎝ J (u)(v) = |∇u| N d x ⎠ f (x, u)v d x (2.2) and J are of class C in X , and |∇u| N −2 ∇u∇v d x, for all u, v ∈ X Moreover, since f ∈ A, the mapping J : X → X is compact, see [16, Lemma 3] Definition 2.1 We say that u ∈ X is a weak solution of problem (1.1) if ⎛ ⎞ M⎝ |∇u| N d x ⎠ |∇u| N −2 ∇u∇v d x − λ for all v ∈ X Theorem 2.2 Let f ∈ A be such that (F1 ) supu∈X F(x, u(x)) d x > 0; f (x, u)v d x − μ g(x, u)v d x = N -Kirchhoff type equations (F2 ) lim supt→0 supx∈ F(x,t) |t| N α (F3 ) lim sup|t|→+∞ ≤ 0; supx∈ F(x,t) |t| N α ≤ 0, where F(x, t) = t f (x, s) ds Under such hypotheses, if we set ⎧ ⎪ ⎪ ⎪ M |∇u| N d x ⎪ ⎨ ∗ : u ∈ X, inf θ := ⎪ N F(x, u(x)) d x ⎪ ⎪ ⎪ ⎩ ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ F(x, u(x)) d x > , ⎪ ⎪ ⎪ ⎪ ⎭ then for each compact interval ⊂ (θ ∗ , +∞), there exists a number σ > with the following property: for every λ ∈ and every g ∈ A there exists μ∗ > such that, for each μ ∈ [0, μ∗ ], problem (1.1) has at least three weak solutions whose norms are less than σ The following result is useful in proving the main result of this paper Lemma 2.3 (i) The functional (ii) belongs to the class W X is sequentially weakly lower semicontinuous; Proof (i) Let {u m } ⊂ X be a sequence that converges weakly to u in X Then we have |∇u| N d x ≤ lim inf m→∞ |∇u m | N d x (2.3) Combining (2.3) with the continuity and monotonicity of the function t → M(t), we get ⎞ ⎛ lim inf M ⎝ |∇u m | N d x ⎠ lim inf (u m ) = m→∞ N m→∞ ⎞ ⎛ ⎝ ≥ M lim inf |∇u m | N d x ⎠ m→∞ N ⎞ ⎛ ⎝ N ≥ M |∇u| d x ⎠ N = (u) (2.4) Thus, the functional is sequentially weakly lower semicontinuous (ii) It is well known that ρ(u) belongs to W X Since M is continuous and strictly increasing we deduce that belongs to W X Proof of Theorem 2.2 We wish to apply Proposition 1.2 by taking X = W01,N ( ), and J are as before In view of [16, Lemma 3], J is well defined and continuously Gâteaux differentiable with compact derivative J given by J (u)(v) = f (x, u)v d x N T Chung, H Q Toan for all u, v ∈ X Moreover, by Lemma 2.3, is a sequentially weakly lower semicontinuous and C functional belonging to W X , and a simple computation shows that it is also coercive In fact, by (M0 ), we have ⎞ ⎛ ⎝ (u) = M |∇u| N d x ⎠ N ⎞α ⎛ m0 ⎝ |∇u| N d x ⎠ ≥ Nα = m0 u Nα Nα (2.5) from which we have the coercivity of It is evident that u = is the only global minimum of and that (u ) = J (u ) = Moreover, it is easy to see that, if u ≤ r then (u) ≤ N1 M(r N ) and so is bounded on each bounded subset of X Now, let us show that the operator : X → X ∗ is invertible on X On account of the well-known Minty-Browder theorem (see [21, Theorem 26.A(d)]), it suffices to prove that is strictly convex, hemicontinuous and coercive in the sense of monotone operators So, let u, v ∈ X with u = v and λ, μ ∈ [0, 1] with λ + μ = Since the operator ρ : X → X ∗ given by ρ (u)(v) = |∇u| N −2 ∇u∇v d x is strictly montone, by [21, Proposition 25.10], ρ is strictly convex Moreover, since M is nondecreasing the function M is convex in [0, +∞) Thus, we have M(ρ(λu + μv)) < M(λρ(u) + μρ(v)) ≤ λ M(ρ(u)) + μ M(ρ(u)) This shows that is strictly monotone For any u ∈ X , by (M0 ), one has M(ρ(u)) |∇u| N d x (u)(u) = u u (ρ(u))α ≥ m0 u = m0 u N α−1 , from which we have the coercivity of Standard arguments ensure that is hemicontinuous Thus, in view of [21, Theorem 26.A(d)] there exists −1 : X ∗ → X and it is bounded Let us prove that −1 is continuous by showing that it is sequentially continuous Let {wm } ⊂ X ∗ be a sequence strongly converging to w ∈ X ∗ and let u m = −1 (wm ), m = 1, 2, , and u = −1 (w) Then, {u m } is bounded in X and without loss of generality, we can assume that it converges weakly to a certain u ∈ X Since {wm } converges strongly to w, it is easy to see that lim m→∞ (u m )(u m − u ) = lim wm (u m − u ) = m→∞ N -Kirchhoff type equations ⎛ ⎞ lim M ⎝ |∇u m | N d x ⎠ or m→∞ |∇u m | N −2 ∇u m (∇u m − ∇u ) d x = (2.6) Since {u m } is bounded in X , passing to a subsequence, if necessary, we may assume that |∇u m | N d x → t0 ≥ as m → ∞ If t0 = then {u m } converges strongly to u = in X and the proof is finished because of the continuity and injectivity of If t0 > 0, it follows from the continuity of the function M that ⎛ ⎞ M⎝ |∇u m | N d x ⎠ → M(t0 ) as m → ∞ Thus, by (M0 ), for sufficiently large m, we have ⎞ ⎛ |∇u m | d x ⎠ ≥ C4 > (2.7) |∇u m | N −2 ∇u m (∇u m − ∇u ) d x = (2.8) M⎝ N From (2.6) and (2.7), it follows that lim m→∞ From (2.8) and the fact that {u m } converges weakly to u in X we deduce that {u m } converges strongly to u in X The continuity and injectivity of imply that {u m } converges strongly to u, so −1 is continuous Let us prove that J (u) lim sup ≤ (2.9) (u) u→0 By the assumption (F2 ), for every every x ∈ and |t| ≤ η1 , > there exists some constant η1 > such that, for F(x, t) ≤ |t| N α (2.10) As f belongs to A, for fixed δ > and q > N α, there exists c > such that, for every x ∈ and |t| ≥ η1 , N (2.11) F(x, t) ≤ c|t|q exp δ|t| N −1 Then, for every x ∈ and t ∈ R, one has N F(x, t) ≤ |t| N α + c exp δ|t| N −1 |t|q (2.12) N T Chung, H Q Toan After choosing p > 1, we apply Hölder’s inequality to get N exp(δ|u(x)| N −1 )|u(x)|q d x ⎡ ≤⎣ exp pδ u |u(x)| u N N −1 N N −1 ⎞1 ⎤1 ⎛ p p |u(x)| p q d x ⎠ dx⎦ ⎝ , (2.13) where p is the conjugate of p By combining the previous inequality with (T M2 ) and bearing in mind that X is contin- J (u) ≤ (S N α ) N α u ≤ (S N α ) N α Nα Nα m0 N −1 N αN pδ uously embedded in L r ( ) for every r ≥ 1, for u ≤ + c(S p q )q C Np u (u) + c(S p q )q C Np , one has q Nα m0 (u) q Nα (2.14) Therefore, J (u) Nα + c(S p q )q C Np ≤ (S N α ) N α (u) m0 Since q > N α and the fact that Let us prove now that Nα m0 q Nα ( (u)) q−N α Nα (u) → as u → 0, claim (2.9) immediately follows lim sup u →∞ By the assumption (F3 ), for every every x ∈ and |t| > η2 , J (u) ≤ (u) (2.15) > 0, there exists some constant η2 > such that for F(x, t) ≤ |t| N α (2.16) From condition (1.3), there exists some constant K > such that for every x ∈ sup | f (x, t)| ≤ K |t|≤η2 Then, for every x ∈ , (2.17) and t ∈ R, F(x, t) ≤ K η2 + |t| N α and so J (u) ≤ K η2 meas( ) + |u| N α d x (2.18) Since X is continuously embedded into L N α ( ), we get |u| N α d x J (u) Nα N α K η2 meas( ) · + · ≤ (u) m0 u Nα m0 u Nα N α K η2 meas( ) Nα ≤ · + · S NNαα m0 u Nα m0 and claim (2.15) follows at once (2.19) N -Kirchhoff type equations In view of (2.9) and (2.15), we get max lim sup u →∞ J (u) J (u) , lim sup (u) u→0 (u) ≤0 and all the assumptions of Proposition 1.2 are satisfied Notice that, if G : ×R → R is defined by G(x, t) = g(x, s) ds, then the functional (u) = G(x, u) d x is continuously Gâteaux differentiable in X , it has compact derivative, and for every u, v ∈ X , (u)(v) = g(x, u)v d x Then, Proposition 1.2 applies and there exists σ > such that for every λ ∈ , it is possible to find μ∗ > verifying the following condition: for each μ ∈ [0, μ∗ ], the functional − λJ − μ has at least three critical points whose norms are less than σ It is clear that critical points of − λJ − μ are precisely weak solutions of problem (1.1) The proof is completed Acknowledgments The authors would like to thank the referees for their suggestions and helpful comments which improved the presentation of the original manuscript This work is supported by Vietnam National Foundation for Science and Technology Development (Grant N.101.02.2014.03) References Adimurthi: Positive solutions of the semilinear Dirichlet problem with critical growth in the unit disc in R2 Proc Indian Acad Sci Math Sci 99, 49–73 (1989) Adimurthi: Existence of positive solutions of the semilinear Dirichlet problem with critical growth for the N -Laplacian Ann Sc Norm Sup Pisa Cl Sci 17, 393–413 (1990) Cammaroto, F., Vilasi, L.: Multiple solutions for a Kirchhoff-type problem involving the p(x)-Laplacian operator Nonlinear Anal 74, 1841–1852 (2011) Chen, C.Y., Kuo, Y.C., Wu, T.F.: The Nehari manifold for a Kirchhoff type problem involving signchanging weight functions J Differ Equ 250, 1876–1908 (2011) Chung, N.T.: Multiple solutions for a p(x)-Kirchhoff-type equation with sign-changing nonlinearities Complex Var Elliptic Equ 58(12), 1637–1646 (2013) Chung, N.T.: Multiplicity results for a class of p(x)-Kirchhoff type equations with combined nonlinearities Electron J Qual Theory Differ Equ 2012(42), 1–13 (2012) Dai, G., Hao, R.: Existence of solutions for a p(x)-Kirchhoff-type equation J Math Anal Appl 359, 275–284 (2009) Fan, X.L.: On nonlocal p(x)-Laplacian Dirichlet problems Nonlinear Anal 72, 3314–3323 (2010) de Freitas, L.R.: Multiplicity of solutions for a class of quasilinear equations with exponential critical growth Nonlinear Anal (TMA) 95, 607–624 (2014) 10 Kirchhoff, G.: Mechanik Teubner, Leipzig (1883) 11 Moser, J.: A sharp form of an equality by N Trudinger Indiana Univ Math J 20, 1077–1092 (1971) 12 Goyal, S., Sreenadh, K.: Lack of coercivity for N -Laplace equation with critical exponential nonlinearities in a bounded domain Electron J Differ Equ 2014(15), 1–22 (2014) 13 Bisci, G.M.; R˘adulescu, V.: Mountain pass solutions for nonlocal equations Ann Acad Sci Fennicae (in press) 14 Han, X., Dai, G.: On the sub-supersolution method for p(x)-Kirchhoff type equations J Inequal Appl 2012, 283 (2012) 15 Ma, T.F.: Remarks on an elliptic equation of Kirchhoff type Nonlinear Anal 63, 1967–1977 (2005) 16 El Manouni, S., Faraci, F.: Multiplicity results for some elliptic problems of N -Laplace type Taiwan J Math 16(3), 901–911 (2012) 17 Ricceri, B.: A further three critical points theorem Nonlinear Anal 71, 4151–4157 (2009) 18 Ricceri, B.: On an elliptic Kirchhoff-type problem depending on two parameters J Global Optim 46(4), 543–549 (2010) N T Chung, H Q Toan 19 de Souza, M.: Existence of solutions to equations of N -Laplacian type with Trudinger-Moser nonlinearities Appl Anal (2013) (to appear) 20 Trudinger, N.S.: On imbedding into Orlicz spaces and some applications J Math Mech 17, 473–483 (1967) 21 Zeidler, E.: Nonlinear functional analysis and applications Nonlinear Monotone Operators, vol II/B Springer, New York (1990) ... p(x)-Laplacian Dirichlet problems Nonlinear Anal 72, 3314–3323 (2010) de Freitas, L.R.: Multiplicity of solutions for a class of quasilinear equations with exponential critical growth Nonlinear Anal (TMA)... Qual Theory Differ Equ 2012(42), 1–13 (2012) Dai, G., Hao, R.: Existence of solutions for a p(x)-Kirchhoff -type equation J Math Anal Appl 359, 275–284 (2009) Fan, X.L.: On nonlocal p(x)-Laplacian... Gaussian curvature equation is a subcritical equation on a two dimensional manifold M with metric tensor g of the type − u = k(x) exp (2u) + h(x), (1.4) where is the Laplace-Beltrami operator associated