DSpace at VNU: A MULTIPLICITY RESULT FOR A CLASS OF EQUATIONS OF p-LAPLACIAN TYPE WITH SIGN-CHANGING NONLINEARITIES

Glasgow Math J 51 (2009) 513–524 C 2009 Glasgow Mathematical Journal Trust doi:10.1017/S001708950900514X Printed in the United Kingdom A MULTIPLICITY RESULT FOR A CLASS OF EQUATIONS OF p-LAPLACIAN TYPE WITH SIGN-CHANGING NONLINEARITIES NGUYEN THANH CHUNG Department of Mathematics and Informatics, Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Quang Binh, Vietnam e-mail: ntchung82@yahoo.com ´ˆ C ANH NGO ˆ and QUO Department of Mathematics, College of Science, Vietnam National University, Hanoi, Vietnam Department of Mathematics, National University of Singapore, Science Drive 2, Singapore 117543 e-mail: bookworm vn@yahoo.com (Received 11 July 2008; accepted 22 December 2008) Abstract Using variational arguments we study the non-existence and multiplicity of non-negative solutions for a class equations of the form −div(a(x, ∇u)) = λf (x, u) in , where is a bounded domain in ‫ޒ‬N , N 3, f is a sign-changing Carath´eodory function on × [0, +∞) and λ is a positive parameter 2002 Mathematics Subject Classification 35J20, 35J60, 35J65, 58E05 Introduction This paper deals with the non-existence and multiplicity of non-negative, non-trivial solutions to the following problem, −div(a(x, ∇u)) = λf (x, u) in u = on ∂ , where , (1.1) (1.2) is a bounded domain in ‫ޒ‬N , function a satisfies |a(x, ξ )| c0 (h0 (x) + h1 (x)|ξ |p−1 ) for all ξ ∈ ‫ޒ‬N , a.e x ∈ , h0 (x) 0, h1 (x) for a.e x ∈ , p and λ > is a parameter When h0 and h1 belong to L∞ ( ), the problem has been studied by many p authors (see [2, 8, 10] for details) Here we study the situation that h0 ∈ L p−1 ( ) and h1 ∈ L1loc ( ) Then problem (1.1)–(1.2) now may be non-uniform in the sense that the functional associated to the problem may be infinity for some u We point out the fact that in [4, 13], D M Duc and N T Vu have studied the following Dirichlet elliptic problem, −div(a(x, ∇u)) = f (x, u) in u = on ∂ , , (1.3) (1.4) where the nonlinear term f verifies the so-called Ambrosetti–Rabinowitz condition The authors obtained the existence of a weak solution by using a variant of the ´ˆ C ANH NGO ˆ NGUYEN THANH CHUNG AND QUO 514 Mountain pass theorem introduced in [3] Then, H Q Toan and Q.-A Ngoˆ [12] gave some multiplicity results in the case when f (x, u) = h(x)|u|r−1 u + g(x)|u|s−1 u Using the Mountain pass theorem in [3] combined with Ekeland’s variational principle in [5] they proved that problem (1.3)–(1.4) has at least two weak solutions Motivated by K Perera [10] and M Mih˘ailescu and V R˘adulescu [7], the goal of this work is to investigate the problem (1.1)–(1.2) with positive parameter λ and the sign-changing nonlinearity f We also not require that the nonlinear term f verifies the Ambrosetti–Rabinowitz condition as in [4, 12] In order to state our main result, let us introduce the following hypotheses on problem (1.1)–(1.2) Assume that N and p < N Let be a bounded domain with a smooth boundary ∂ Consider that a : × ‫ޒ‬N → ‫ޒ‬N , a = a(x, ξ ), is the continuous derivative with respect to ξ of the continuous function A : × ‫ޒ‬N → ‫ޒ‬, A = A(x, ξ ), that is, a(x, ξ ) = ∂A(x, ξ )/∂ξ and A(x, 0) = for a.e x ∈ Assume that there are positive constant c0 and two non-negative measurable functions h0 , h1 such that h0 ∈ Lp/p−1 ( ), h1 ∈ L1loc ( ), h1 (x) 1, a.e x ∈ Suppose that a and A satisfy the following hypotheses (A1 ) |a(x, ξ )| c0 (h0 (x) + h1 (x)|ξ |p−1 ) for all ξ ∈ ‫ޒ‬N , a.e x ∈ (A2 ) The following inequality holds (a(x, ξ ) − a(x, ψ)) · (ξ − ψ) for all ξ, ψ ∈ ‫ޒ‬N , a.e x ∈ , with equality if and only if ξ = ψ (A3 ) There exists a positive constant k0 such that A x, ξ +ψ 1 A(x, ξ ) + A(x, ψ) − k0 h1 (x)|ξ − ψ|p 2 for all ξ, ψ ∈ ‫ޒ‬N , a.e x ∈ , that is, A is p-uniformly convex (A4 ) There exists a constant k1 > such that the following inequalities hold true k1 h1 (x)|ξ |p for all ξ ∈ ‫ޒ‬N , a.e x ∈ a(x, ξ ) · ξ pA(x, ξ ) EXAMPLE 1 Let A(x, ξ ) = h(x) p |ξ | , p a(x, ξ ) = h(x)|ξ |p−2 ξ, with p and h ∈ L1loc ( ) Then we get the operator div(h(x)|∇u|p−2 ∇u), and if h(x) ≡ in we conclude the well-known p-Laplacian operator pu := div(|∇u|p−2 ∇u) as in [8, 10] Let A(x, ξ ) = p h(x) (1 + |ξ |2 ) − p MULTIPLICITY RESULT FOR A CLASS OF p-LAPLACIAN-TYPE EQUATIONS with p 515 p 2, h ∈ L p−1 ( ) Then a(x, ξ ) = h(x)(1 + |ξ |2 ) p−2 ξ We obtain the generalised mean curvature operator div h(x)(1 + |∇u|2 ) p−2 ∇u It should be observed that the above examples have not been considered in [2, 8, 10] yet For more information and connection on these operators, the reader may consult either [2] or [8] and the references therein As in [10], we assume that function f : × [0, +∞) → ‫ ޒ‬is a sign-changing Carath´eodory function and satisfies the following hypotheses: (F1 ) f (x, 0) = 0, |f (x, t)| Ctp−1 for all t ∈ [0 + ∞), a.e x ∈ , and for some constant C > (F2 ) There exist two positive constants t0 , t1 > such that F(x, t) for t t0 and F(x, t1 ) > t uniformly in x, where F(x, t) = f (x, s) ds (F3 ) lim sup F(x,t) t→∞ Let W 1,p ( ) be the usual Sobolev space and W0 ( ) be the closure of C0∞ ( ) under the norm 1,p |∇u| dx u = p p 1,p We now consider the following subspace of W0 ( ): 1,p H := u ∈ W0 ( ) : h1 (x)|∇u|p dx < +∞ Then H is an infinite dimensional Banach space with respect to the norm (see [4]) u We define the functional λ H = p p h1 (x)|∇u| dx : H → ‫ ޒ‬by λ (u) = (u) − I(u), (1.5) where (u) = A(x, ∇u) dx , I(u) = λ F(x, u) dx , u ∈ H (1.6) 1,p Since h0 ∈ Lp/p−1 ( ), then the value λ (u) may be infinity for some u ∈ W0 ( ), that 1,p is, the functional may not be defined throughout W0 ( ) In order to overcome this 1,p difficulty, we choose the subspace H of W0 ( ) 516 ´ˆ C ANH NGO ˆ NGUYEN THANH CHUNG AND QUO DEFINITION We say that u ∈ H is a weak solution of problem (1.1)–(1.2) if and only if a(x, ∇u)∇ϕ dx −λ f (x, u)ϕ dx = (1.7) for all ϕ ∈ C0∞ ( ) Then we have the following remark which plays an important role in our arguments REMARK (i) By (A4 ) and (i) in Proposition 2, it is easy to see that 1,p 1,p H = u ∈ W0 ( ) : (u) < ∞ = u ∈ W0 ( ) : (ii) Since h1 (x) 1, a.e x ∈ continuous embeddings , we have u 1,p H → W0 ( ) → Li ( ), u p H λ (u) and η ∈ C([0, +∞) × X, X) satisfying η(1, F c+ε ) ⊂ F c−ε This is a contradiction since F c−ε = ∅ due to the fact that c = inf F REMARK By Corollary 2.1.1 in [6], if F : X → ‫ ޒ‬is a locally Lipschitz, bounded from below function and it satisfies the (C) condition, then F is coercive This leads us to state the following lemma ´ˆ C ANH NGO ˆ NGUYEN THANH CHUNG AND QUO 518 LEMMA If F : X → ‫ ޒ‬is a locally Lipschitz, bounded from below function and it satisfies the (C) condition then it satisfies the (PS) condition Proof Let {un }n ⊂ X be a sequence such that F(un ) is bounded and DF(un ) → in X By Remark 3, F is coercive, and this helps us to deduce that {un }n is bounded in X Hence also (1 + un )DF(un ) → in X , and because F satisfies the (C) condition, it follows that {un }n has a strongly convergent subsequence This completes the proof Similar to Theorem 3, we have the following new result THEOREM Let F be continuous on X and be of class Cw1 (X) where X is a Banach space Assume that (i) F is bounded from below, c = inf F, (ii) F satisfies the (C) condition Then c is a critical value of F (i.e there exists a critical point u0 ∈ X such that F(u0 ) = c) The proof of Theorem follows from Lemma 1, so we omit it Next we provide a variant Mountain pass theorem due to Duc [3] PROPOSITION (see [3]) Let F ∈ Cw1 (X) where X is a Banach space and satisfies (PS) condition Assume that F(0) = and there exist a positive constant ρ and z0 ∈ X such that (i) z0 X > ρ and F(z0 ) (ii) α = inf {F(u) : u ∈ X, u X = ρ} > Assume that the set G = {ϕ ∈ C([0, 1], X) : ϕ(0) = 0, ϕ(1) = z0 } is not empty Put β := inf {max F(ϕ([0, 1])) : ϕ ∈ G} Then β α and β is a critical value of F For the use of Proposition 1, we refer the reader to [3, 12, 13] We end this section by studying some certain properties of the functional λ given by (1.5) but we first recall some results which will be used throughout this work PROPOSITION (see [4]) (i) A verifies the growth condition |A(x, ξ )| c0 (h0 (x)|ξ | + h1 (x)|ξ |p ) for all ξ ∈ ‫ޒ‬N , a.e x ∈ (ii) A(x, ξ ) is convex with respect to ξ Moreover, by (A3 ) for all u, v ∈ H we have u+v 1 (u) + (v) − k0 u − v 2 p H (2.1) Using the method as in [4] with some simple computations we obtain the following proposition which concerns the smoothness of the functional λ MULTIPLICITY RESULT FOR A CLASS OF p-LAPLACIAN-TYPE EQUATIONS 519 PROPOSITION 1,p (i) If {um } is a sequence weakly converging to u in W0 ( ), then (u) lim inf (um ) m→∞ and lim I(um ) = I(u) m→∞ (ii) The functionals and I are continuous on H (iii) Functional λ is weakly continuously differentiable on H and we have D λ (u)(ϕ) = a(x, ∇u)∇ϕ dx −λ f (x, u)ϕ dx for all u, ϕ ∈ H Proofs of the theorems Proof of Theorem Let us denote by S the best constant in the Sobolev embedding ) → Lp ( ), i.e 1,p W0 ( p |∇u|p dx S= inf 1,p W0 ( )\{0} |u| dx p p (3.1) Then, if u is a weak solution of problem (1.1)–(1.2), multiplying (1.1) by u and integrating by parts combined with conditions (A4 ) and (F1 ) gives k1 |∇u|p dx k1 h1 (x)|∇u|p dx (3.2) a(x, ∇u)∇u dx = λ f (x, u)u dx Cλ |u|p dx Hence, choosing λ = k1 S/C, where S is given by (3.1), we conclude the proof We will prove Theorem by using critical point theory Set f (x, t) = for all t < and consider the energy functional λ : H → ‫ ޒ‬which is given by (1.5) LEMMA If u is a critical point of λ then u is non-negative in Proof Observe that if u is a critical point of of u, i.e u− (x) = {u(x), 0} we have 0=D − λ (u)(u ) λ, denoting by u− the negative part a(x, ∇u)∇u− dx −λ = f (x, u)u− dx (3.3) k1 h1 (x)|∇u− |p dx = k1 u− p H, which yields that u for a.e x in Thus, non-trivial critical points of the functional λ are non-negative, non-trivial solutions of problem (1.1)–(1.2) ´ˆ C ANH NGO ˆ NGUYEN THANH CHUNG AND QUO 520 The following lemma shows that the functional λ satisfies all of the assumptions of Theorem Then problem (1.1)–(1.2) admits a weak solution u1 ∈ H as a global minimiser and u1 LEMMA The functional on H λ is bounded from below and satisfies the (PS) condition Proof By conditions (F1 ) and (F3 ), there exists a constant Cλ = C(λ) > such that λF(x, t) for all t ∈ ‫ ޒ‬and a.e x ∈ λ (u) k1 S p |t| + Cλ 2p (3.4) Hence, = A(x, ∇u) dx −λ F(x, u) dx k1 h1 (x)|∇u|p dx − p k1 p u H − Cλ | |, 2p k1 S p |u| + Cλ dx 2p (3.5) where | | denotes the Lebesgue measure of in ‫ޒ‬N Thus, the functional coercive and hence bounded from below on H Let {um } be a Palais-Smale sequence in H, i.e | λ (um )| c for all m, λ (um ) → in H λ is (3.6) Since λ is coercive on H, {um } is bounded in H By Remark (ii), {um } is bounded in 1,p 1,p W0 ( ) It follows that there exists u ∈ W0 ( ) such that, passing to a subsequence, 1,p still denoted by {um }, it converges weakly to u in W0 ( ) We shall prove that {um } converges strongly to u in H Indeed, we observe by Remark 1(i), Proposition 3(i) and (3.6) that u ∈ H Hence, { um − u H } is bounded This and (3.6) imply that D λ (um )(um − u) converges to as m ă inequality we deduce that Using condition (F1 ) combined with Holder’s |f (x, um )||um − u| dx C C |um |p−1 |um − u| dx p−1 um Lp ( ) um − u Lp ( ) (3.7) 1,p Since the embedding W0 ( ) → Lp ( ) is compact, {um } converges strongly to u in Lp ( ) Therefore, relation (3.7) implies that lim DI(um )(um − u) = m→∞ Combining relations (3.6) and (3.8) with the fact that D (um )(um − u) = D λ (um )(um − u) + DI(um )(um − u), (3.8) MULTIPLICITY RESULT FOR A CLASS OF p-LAPLACIAN-TYPE EQUATIONS 521 we conclude that lim D (um )(um − u) = (3.9) m→∞ On the other hand, the convex property of functional that (u) − lim m→∞ (um ) = lim ( (u) − m→∞ (um )) (see Proposition 2(ii)) implies lim D (um )(u − um ) = m→∞ (3.10) Combining this with Proposition 3(i), we have (um ) = lim m→∞ (u) (3.11) We now assume by contradiction that {um } does not converge strongly to u in H, and then there exist a constant > and a subsequence {umk } of {um } such that Using Proposition 2(ii) we get umk − u H umk + u 1 (u) + (umk ) − 2 k0 umk − u p H k0 p (3.12) Letting k → ∞, relation (3.12) gives lim sup k→∞ umk +u } We remark that sequence { Proposition 3(i) again we get (u) umk + u (u) − k0 p (3.13) 1,p also converges weakly to u in W0 ( ) So, using lim inf k→∞ umk + u , (3.14) which contradicts (3.13) Therefore, {um } converges strongly to u in H LEMMA There exists a positive constant λ such that for all λ and hence u1 ≡ 0, i.e the solution u1 is not trivial λ, inf H λ < 0, Proof Let ⊂ be a compact subset large enough and a function ϕ0 ∈ C0∞ ( ) such that ϕ0 (x) = t1 in and ϕ0 (x) t1 in \ , where t1 is chosen as in assumption (F2 ): then we have p F(x, ϕ0 ) dx F(x, ϕ0 ) dx −Ct1 | \ 0| > (3.15) Thus, λ (ϕ0 ) < for λ λ with λ large enough This implies that inf H then λ (u1 ) < for λ λ, i.e u1 ≡ λ < and In the next part of this paper, we shall show the existence of the second solution u2 = u1 by using the Mountain pass theorem introduced in [3] To this purpose, we ´ˆ C ANH NGO ˆ NGUYEN THANH CHUNG AND QUO 522 first fix λ λ and set ⎧ ⎪ ⎨0, f (x, t) = f (x, t) ⎪ ⎩ f (x, u1 (x)) and F(x, t) = t for t < 0, for t u1 (x), for t > u1 (x), f (x, s) ds Define the functional λ (u) = λ A(x, ∇u) dx −λ : H → ‫ ޒ‬by F(x, u) dx With the same arguments as those used for the functional weakly continuously differentiable on H and D λ (u)(ϕ) = (3.16) a(x, ∇u)∇ϕ dx −λ λ, (3.17) we can show that λ is f (x, u)ϕ dx for all u, ϕ ∈ H LEMMA If u ∈ H is a critical point of (1.1)–(1.2) in the order interval [0, u1 ] Proof By the definitions of λ and before and by condition (A2 ) we have = (D = λ (u) λ, λ (u1 ))((u then u u1 So, u is a solution of problem if u is a critical point of λ then u as − u1 )+ ) (a(x, ∇u) − a(x, ∇u1 )) · ∇(u − u1 )+ dx −λ = −D λ {u>u1 } (3.18) (f (x, u) − f (x, u1 ))(u − u1 )+ dx (a(x, ∇u) − a(x, ∇u1 )) · (∇u − ∇u1 ) dx According to (3.18) and condition (A2 ), the equality holds if and only if ∇u = ∇u1 It follows that ∇u(x) = ∇u1 (x) for all x ∈ := {y ∈ : u(y) > u1 (y)} Hence, |∇(u − u1 )+ |p dx = |∇(u − u1 )|p dx = and thus Combining this with Remark 1(ii), we conclude that (u − u1 )+ (u − u1 )+ = in , that is, u u1 in LEMMA There exist a constant ρ ∈ (0, u1 α for all u ∈ H with u H = ρ H) Lp ( ) = and then and a constant α > such that λ (u) Proof We set u = {x ∈ : u(x) > {u1 (x), t0 }}, where t0 is given as in (F2 ) Then, by (3.16) and condition (F1 ), we have F(x, u(x)) on \ u Hence, λ (u) k1 u p H −λ F(x, u) dx u (3.19) MULTIPLICITY RESULT FOR A CLASS OF p-LAPLACIAN-TYPE EQUATIONS 523 ă Using (F1 ), Holders inequality and Remark 1(ii), we get F(x, u) dx |u|p dx C u where q = Np N−p C| u| u| 1− pq 1− pq u p H, (3.20) u Therefore, λ (u) u p H k1 − λC| (3.21) In order to prove Lemma 6, it is enough to show that | u | → as u H → Indeed, ⊂ a compact subset, large enough such that let > be arbitrary; we choose | \ | < , and denote by u, := u ∩ Then it is clear that u p H u |u|p dx p up dx lp| u, |, (3.22) u, where l = {min u1 ( ), t0 } Letting u H → we deduce that | u, | → Since we have | u | | u, | + with > as arbitrary Thus, | u | → u ⊂ u, ∪ \ as u H → LEMMA Functional λ satisfies the (PS) condition on H Proof We observe by (3.21) that λ is coercive Therefore, every Palais-Smale sequence of λ is bounded in H We follow the method as those used in the proof of Lemma It can be shown that the functional λ satisfies the (PS) condition on H Proof of Theorem By Lemmas 2–4, using Theorem 3, problem (1.1)–(1.2) admits a non-negative, non-trivial weak solution u1 Setting c := inf max γ ∈ u∈γ ([0,1]) λ (u), (3.23) where := {γ ∈ C([0, 1], H) : γ (0) = 0, γ (1) = u1 }, Lemmas 6–7 show that all of the assumptions of Proposition are fulfilled, λ (u1 ) = λ (u1 ) < and u1 H > ρ Then, c > is a critical value of λ , i.e there exists u2 ∈ H such that D λ (u2 )(ϕ) = for all ϕ ∈ H and λ (u2 ) = c > By Lemma 5, u2 u1 in Therefore, using (3.16) some simple 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