Nonlinear Analysis 70 (2009) 1536–1546 www.elsevier.com/locate/na Multiplicity of weak solutions for a class of nonuniformly elliptic equations of p-Laplacian type Hoang Quoc Toan, Quˆo´c-Anh Ngˆo ∗ Department of Mathematics, College of Science, Viˆet Nam National University, H`a Nˆoi, Viˆet Nam Received October 2007; accepted 14 February 2008 Abstract This paper deals with the multiplicity of weak solutions in W01 (Ω ) to a class of nonuniformly elliptic equations of the form −div(a(x, ∇u)) = h(x)|u|r −1 u + g(x)|u|s−1 u p in a bounded domain Ω of R N Here a satisfies |a(x, ξ )| c0 (h (x) + h (x)|ξ | p−1 ) for all ξ ∈ R N , a.e x ∈ Ω , h ∈ L p−1 (Ω ), h ∈ L 1loc (Ω ), h (x) for a.e x in Ω , < r < p − < s < (N p − N + p)/(N − p) c 2008 Elsevier Ltd All rights reserved Keywords: p-Laplacian; Nonuniform elliptic equations; Multiplicity Introduction Let Ω be a bounded domain in R N In the present paper we study the multiplicity of nontrivial solutions of the following Dirichlet elliptic problem: − div(a(x, ∇u)) = h(x)|u|r −1 u + g(x)|u|s−1 u (x)|ξ | p−1 ) (1) where |a(x, ξ )| c0 (h (x) + h for any ξ in and a.e x ∈ Ω , h (x) and h (x) for any x in Ω For h and h belonging to L ∞ , the problem has been studied Here we study the case in which h and h p belong to L p−1 (Ω ) and L 1loc (Ω ) respectively The equation now may be nonuniformly elliptic To our knowledge, such equations were first studied by Duc et al and Vu [3,6] In both papers, the authors studied the following problem: − div(a(x, ∇u)) = f (x, u) RN (2) where the nonlinear term f verifies the Ambrosetti–Rabinowitz type condition, see [1] They then obtained the existence of a weak solution by using a variation of the Mountain-Pass Theorem introduced in [2] We also point out the fact that for the case when h ≡ 1, our problem (1) was studied in [4] Our goal is to extend the results of [4] (for the “nonuniform case”) and of [3,6] (the existence of at least two weak solutions) ∗ Corresponding author Tel.: +84 8581135 E-mail address: bookworm vn@yahoo.com (Q.-A Ngˆo) 0362-546X/$ - see front matter c 2008 Elsevier Ltd All rights reserved doi:10.1016/j.na.2008.02.033 H.Q Toan, Q.-A Ngˆo / Nonlinear Analysis 70 (2009) 1536–1546 1537 In order to state our main theorem, let us introduce our hypotheses on the structure of problem (1) Assume that N and p < N Let Ω be a bounded domain in R N having C boundary ∂Ω Consider N N N a : R × R → R , a = a(x, ξ ), as the continuous derivative with respect to ξ of the continuous function ) A : R N × R N → R, A = A(x, ξ ), that is, a(x, ξ ) = ∂ A(x,ξ ∂ξ Assume that there are a positive real number c0 and two p nonnegative measurable functions h , h on Ω such that h ∈ L 1loc (Ω ), h ∈ L p−1 (Ω ), h (x) Suppose that a and A satisfy the hypotheses below: for a.e x in Ω (A1 ) |a(x, ξ )| c0 (h (x) + h (x)|ξ | p−1 ) for all ξ ∈ R N , a.e x ∈ Ω (A2 ) There exists a constant k1 > such that 1 ξ +ψ A x, A(x, ξ ) + A(x, ψ) − k1 h (x)|ξ − ψ| p 2 for all x, ξ , ψ, that is, A is p-uniformly convex (A3 ) A is p-subhomogeneous, that is, a(x, ξ )ξ p A(x, ξ ) for all ξ ∈ a.e x ∈ Ω (A4 ) There exists a constant k0 > such that RN , A(x, ξ ) k0 h (x)|ξ | p for all ξ ∈ R N , a.e x ∈ Ω (A5 ) A(x, 0) = for all x ∈ Ω (A6 ) A(x, −ξ ) = A(x, ξ ) for all ξ ∈ R N , a.e x ∈ Ω Example (i) A(x, ξ ) = (ii) A(x, ξ ) = p p |ξ | , p p |ξ | + θ (x)( a(x, ξ ) = |ξ | p−2 ξ with p + |ξ |2 − 1), a(x, ξ ) = |ξ | p−2 ξ We get the p-Laplacian operator + θ (x) √ ξ with p and θ a suitable function 1+|ξ | We get the operator div(|∇u| p−2 ∇u) + div θ (x) ∇u + |∇u|2 which can be regarded as the sum of the p-Laplacian operator and a degenerate form of the mean curvature operator p 2 p p−2 (iii) A(x, ξ ) = 1p ((θ p−1 (x) + |ξ |2 ) − θ p−1 (x)), a(x, ξ ) = (θ p−1 (x) + |ξ |2 ) ξ with p and θ a suitable function Now we get the operator div θ p−1 (x) + |∇u| p−2 ∇u which is a variant of the generalized mean curvature operator div (1 + |∇u|2 ) p−2 ∇u Regarding the functions h and g, we assume that (H) h(x) for all x ∈ Ω and h ∈ L r0 (Ω ) ∩ L ∞ (Ω ), where (r + 1) + = 1, r0 p∗ Np that is r0 = N p−(r +1)(N − p) (G) g(x) > a.e x ∈ Ω and g ∈ L s0 (Ω ) ∩ L ∞ (Ω ), where (s + 1) + = 1, s0 p∗ that is s0 = Np N p−(s+1)(N − p) 1538 H.Q Toan, Q.-A Ngˆo / Nonlinear Analysis 70 (2009) 1536–1546 1, p Let W 1, p (Ω ) be the usual Sobolev space Next, we define X := W0 (Ω ) as the closure of C0∞ (Ω ) under the norm p p |∇u| dx u = Ω (3) 1, p We now consider the following subspace of W0 (Ω ): 1, p E = u ∈ W0 (Ω ) : h (x)|∇u| p dx < +∞ Ω The space E can be endowed with the norm u E h (x)|∇u| dx p = Ω p As in [3], it is known that E is an infinite dimensional Banach space We say that u ∈ E is a weak solution for problem (1) if h(x)|u|r −1 uϕdx − a(x, ∇u)∇ϕdx − Ω Ω g(x)|u|s−1 uϕdx = Ω for all ϕ ∈ E Let J (u) = 1 h(x)|u|r +1 dx + g(x)|u|s+1 dx, r +1 Ω s+1 Ω Λ(u) = Ω A (x, ∇u) dx, and I (u) = Λ(u) − J (u) for all u ∈ E The following remark plays an important role in our arguments Remark (i) u u E for all u ∈ E since h (x) E → X → L i (Ω ), p i Thus the continuous embeddings p hold true (ii) By (A4 ) and (i) in Lemma 5, it is easy to see that 1, p 1, p E = {u ∈ W0 (Ω ) : Λ(u) < +∞} = {u ∈ W0 (Ω ) : I (u) < +∞} (iii) C0∞ (Ω ) ⊂ E since |∇u| is in Cc (Ω ) for any u ∈ C0∞ (Ω ) and h ∈ L 1loc (Ω ) Our main results are included in a couple of theorems below Theorem Assume < r < p − < s < (N p − N + p)/(N − p) and conditions (A1 )– (A5 ), (H), and (G) are fulfilled Then problem (1) has at least two nontrivial weak solutions in E provided that the product h s+1− p s−r L r0 (Ω ) g p−r −1 s−r L s0 (Ω ) is small enough Theorem Assume < r < p − < s < (N p − N + p)/(N − p) and conditions (A1 )– (A6 ), (H), and (G) are fulfilled Then problem (1) has infinitely many nontrivial generalized solutions in E To see the power of the theorems, we compare our assumptions to those considered in [5,3,4,6] Our problem (1) covers the following cases which have been considered in the literature: 1539 H.Q Toan, Q.-A Ngˆo / Nonlinear Analysis 70 (2009) 1536–1546 (i) A(x, ξ ) = (ii) A(x, ξ ) = p p |ξ | with p p 2 p ((1 + |ξ | ) − 1) with p Moreover, our assumption includes the following situations which could not be handled in [5,4] (i) A(x, ξ ) = (ii) A(x, ξ ) = h(x) p p |ξ | with p p h(x) 2 p ((1 + |ξ | ) and h ∈ L 1loc (Ω ) − 1) with p p and h ∈ L p−1 (Ω ) Auxiliary results In this section we recall certain properties of functionals Λ and J But firstly, we list here some properties of A Lemma (See [3]) (i) A verifies the growth condition |A(x, ξ )| c0 (h (x)|ξ | + h (x)|ξ | p ) for all ξ ∈ R N , a.e x ∈ Ω (ii) A(x, zξ ) A(x, ξ )z p for all z 1, x, ξ ∈ R N Due to the presence of h , the functional Λ does not belong to C (E, R) This means that we cannot apply directly the Mountain-Pass Lemma of Ambrosetti and Rabinowitz In this situation, we recall the following concept of weakly continuous differentiability Our approach is based on a weak version of the Mountain-Pass Lemma introduced by Duc [2] Definition Let F be a map from a Banach space Y to R We say that F is weakly continuous differentiable on Y if and only if the following two conditions are satisfied: (i) For any u ∈ Y there exists a linear map DF(u) from Y to R such that F(u + tv) − J (u) = DF(u)(v) t→0 t for every v ∈ Y (ii) For any v ∈ Y , the map u → DF(u)(v) is continuous on Y lim Denote by Cw1 (Y ) the set of weakly continuously differentiable functionals on Y It is clear that C (Y ) ⊂ Cw1 (Y ) where we denote by C (Y ) the set of all continuously Frechet differentiable functionals on Y For simplicity of notation, we shall denote DF(u) by F (u) The following lemma concerns the smoothness of the functional Λ Lemma (See [3]) (i) If {u n } is a sequence weakly converging to u in X , denoted by u n (ii) For all u, z ∈ E, Λ u+z 1 Λ(u) + Λ(z) − k1 u − z 2 p E u, then Λ(u) lim infn→∞ Λ(u n ) (iii) Λ is continuous on E (iv) Λ is weakly continuously differentiable on E and Λ (u), v = a(x, ∇u)∇vdx Ω for all u, v ∈ E (v) Λ(u) − Λ(v) Λ (v), u − v for all u, v ∈ E The following lemma concerns the smoothness of the functional J The proof is standard and simple, so we omit it 1540 H.Q Toan, Q.-A Ngˆo / Nonlinear Analysis 70 (2009) 1536–1546 Lemma (i) If u n u in X , then limn→∞ J (u n ) = J (u) (ii) J is continuous on E (iii) J is weakly continuously differentiable on E and J (u), v = h(x)|u|r −1 uvdx + Ω g(x)|u|s−1 uvdx Ω for all u, v ∈ E Our main tool is a variation of the Mountain-Pass Theorem introduced in [2] and the Z version of it introduced in [6] Lemma (Mountain-Pass Lemma) Let F be a continuous function from a Banach space E into R Let F be weakly continuously differentiable on E and satisfy the Palais–Smale condition Assume that F(0) = and there exist a positive real number ρ and z ∈ E such that (i) z E > ρ, F(z ) F (0) (ii) α = inf{F(u) : u ∈ E, u E = ρ} > Put G = {φ ∈ C([0, 1], E) : φ(0) = 0, φ(1) = z } Assume that G = ∅ Set β = inf{max F(φ([0, 1])) : φ ∈ G} Then β α and β is a critical value of F Lemma 10 (Symmetric Mountain-Pass Lemma) Let E be an infinite dimensional Banach space Let F be weakly continuously differentiable on E and satisfy the Palais–Smale condition Assume that F(0) = and: (i) There exist a positive real number α and ρ such that inf F(u) u∈∂ Bρ α>0 where Bρ is an open ball in E of radius ρ centered at the origin and ∂ Bρ is its boundary (ii) For each finite dimensional linear subspace Y in E, the set u ∈ Y : F(u) is bounded Then F possesses an unbounded sequence of critical values Proofs We remark that the critical points of the functional I correspond to the weak solutions of (1) In order to apply Lemma we need to verify the following facts Lemma 11 (i) I is a continuous function from E to R (ii) I be weakly continuously differentiable on E and I (u), v = Ω a (x, ∇u) ∇vdx − h(x)|u|r −1 uvdx − Ω for all u, v ∈ E (iii) I (0) = (iv) There exist two positive real numbers ρ and α such that inf{I (u) : u ∈ E, u E = ρ} > α (v) There exists ψ ∈ E such that limt→∞ I (tψ) = −∞ (vi) I satisfies the Palais–Smale condition on E (vii) There exists z ∈ E such that z E > ρ, I (z ) g(x)|u|s−1 uvdx Ω H.Q Toan, Q.-A Ngˆo / Nonlinear Analysis 70 (2009) 1536–1546 1541 (viii) The set G = {ϕ ∈ C ([0, 1] , E) : ϕ (0) = 0, ϕ (1) = z } is not empty Proof (i) This comes from (iii) in Lemma and (ii) in Lemma (ii) This comes from (iv) in Lemma and (iii) in Lemma (iii) This comes from the definition of I (iv) First, let S be the best Sobolev constant of the embedding W 1, p (Ω ) → L p (Ω ), that is, S= |∇u| p dx Ω inf u∈W 1, p (Ω )\{0} p Ω |u| dx p p Thus, we obtain Sp v L p (Ω ) v for all v ∈ E Since r0 = Np N p − (r + 1) (N − p) then 1 N p − (r + 1) (N − p) N−p + p = + (r + 1) = r0 Np Np r +1 By Hăolders inequality and the above relation we deduce h(x)|u|r +1 dx h L r0 (Ω ) h L r0 (Ω ) Ω h L r0 (Ω ) r +1 L p (Ω ) u S r +1 Sp h r +1 p S (4) r +1 p r +1 E u (5) L p (Ω ) =: (r + 1) µ u r +1 E (6) With similar arguments, we have (s + 1) ν u g(x)|u|s+1 dx Ω s+1 E Thus, we obtain k0 , I (u) p =: λ − µ u p E u r +1− p E −µ u −ν u r +1 E −ν u s+1− p E u s+1 E p E (7) (8) We show that there exists t0 > such that r +1− p λ − µt0 s+1− p − νt0 > (9) To that, we define the function Q (t) = µt r +1− p + νt s+1− p , t > Since limt→∞ Q (t) = limt→0 Q (t) = ∞, it follows that Q possesses a positive minimum, say t0 > In order to find t0 , we have to solve equation Q (t0 ) = 0, where Q (t) = (r + − p) µt r − p + (s + − p) νt s− p A simple computation yields t0 = p−r −1µ s+1− p ν s−r 1542 H.Q Toan, Q.-A Ngˆo / Nonlinear Analysis 70 (2009) 1536–1546 Thus, relation (9) holds provided that r +1− p s−r p−r −1µ s+1− p ν or equivalently µ µ s+1− p s−r ν r +1− p s−r or h r +1− p s−r p−r −1 s+1− p s+1− p s−r L r0 (Ω ) g Ω + µ s+1− p s−r ν r +1− p s−r s+1− p s−r ρ, I (z ) is followed from the fact that limt→∞ I (tψ) = −∞ (viii) We consider a function ϕ ∈ C ([0, 1] , E) defined by ϕ (t) = t z , for every t ∈ [0, 1] It is clear that ϕ ∈ G Λ(u) lim inf Λ m→∞ Proof of Theorem Using Lemmas and 11 we deduce the existence of u ∈ E as a nontrivial generalized solution of (1) We prove now that there exists a second weak solution u ∈ E such that u = u By Lemma 11, it follows that there exists a ball centered at the origin B ⊂ E, such that inf I > ∂B On the other hand, by the lemma there exists φ ∈ E such that I (tφ) < 0, for all t > small enough Recalling that relation (7) holds for all u ∈ E, that is, I (u) λ u p E −µ u r +1 E −ν u s+1 E we get that −∞ < c := inf I < B We let now < ε < inf I − inf I ∂B B Applying Ekeland’s Variational Principle for the functional I : B → R, there exists u ε ∈ B such that I (u ε ) < inf I + ε (25) B I (u ε ) < I (u) + ε u − u ε E , u = uε (26) Since I (u ε ) < inf I + ε < inf I + ε < inf I B ∂B B it follows that u ε ∈ B Now we define M : B → R by M(u) = I (u) + ε u − u ε point of M and thus M (u ε + tν) − M (u ε ) t E It is clear that u ε is a minimum H.Q Toan, Q.-A Ngˆo / Nonlinear Analysis 70 (2009) 1536–1546 1545 for a small t > and ν in the unit sphere of E The above relation yields I (u ε + tν) − I (u ε ) +ε ν t E Letting t → it follows that I (u ε ) , ν + ε ν >0 E and we infer that I (u ε ) E ε We deduce that there exists {u n } ⊂ B such that I (u n ) → c and I (u n ) → Using the fact that J satisfies the Palais–Smale condition on E we deduce that {u n } converges strongly to u in E Thus, u is a weak solution for (1) and since > c = I (u ) it follows that u is nontrivial Finally, we point out the fact that u = u since I (u ) = c > > c = I (u ) The proof is complete Proof of Theorem In view of (A6 ), I is even In order to apply Lemma 10, it is enough to verify condition (ii) in Lemma 10 It is known that   p−1 A (x, ∇u n ) dx Ω c0  Ω p |h (x)|dx un p E E + un =: c1 u n E + c0 u n p E − g(x)|u|s+1 dx s+1 Ω p E (27) (28) This gives I (u) c1 u n E + c0 u n Suppose that E is a finite dimensional subspace of E Setting u E s+1 g(x)|u| = E dx Ω for all u ∈ E, we see that positive constant K such that u s+1 K u E is a norm in E We also note that in E the norms are equivalent Thus, there exists a E This implies that I (u) c1 u n E + c0 u n p E − K u s+1 s+1 E Since p < s + then u ∈ E : I (u) is bounded Hence, I possesses an unbounded sequence of critical values Therefore, I possesses infinitely many critical points in E This completes the proof References [1] [2] [3] [4] A Ambrosetti, P Rabinowitz, Dual variational methods in critical point theory and applications, J Funct Anal 14 (1973) 349–381 D.M Duc, Nonlinear singular elliptic equations, J London Math Soc (2) 40 (1989) 420–440 D.M Duc, N.T Vu, Nonuniformly elliptic equations of p-Laplacian type, Nonlinear Anal 61 (2005) 1483–1495 M Mihailescu, Existence and multiplicity of weak solutions for a class of degenerate nonlinear elliptic equations, Bound Value Probl (2006) 1–17 Article ID 41295 [5] P de N´apoli, M.C Mariani, Mountain pass solutions to equations of p-Laplacian type, Nonlinear Anal 54 (2003) 1205–1219 [6] N.T Vu, Mountain pass theorem and nonuniformly elliptic equations, Vietnam J Math 33 (4) (2005) 391–408 1546 H.Q Toan, Q.-A Ngˆo / Nonlinear Analysis 70 (2009) 1536–1546 Further reading [1] R.A Adams, Sobolev Spaces, Academic Press, London, 1975 [2] G Dinca, P Jebelean, J Mawhin, Variational and topological methods for Dirichlet problems with p-Laplacian, Protugaliae Math 58 (2001) 340–377 [3] M Struwe, Variational Methods, Springer, New York, 1996  ... equations of p-Laplacian type, Nonlinear Anal 61 (2005) 1483–1495 M Mihailescu, Existence and multiplicity of weak solutions for a class of degenerate nonlinear elliptic equations, Bound Value Probl... 1–17 Article ID 41295 [5] P de N´apoli, M.C Mariani, Mountain pass solutions to equations of p-Laplacian type, Nonlinear Anal 54 (2003) 1205–1219 [6] N.T Vu, Mountain pass theorem and nonuniformly. .. |∇u|2 which can be regarded as the sum of the p-Laplacian operator and a degenerate form of the mean curvature operator p 2 p p−2 (iii) A( x, ξ ) = 1p ((θ p−1 (x) + |ξ |2 ) − θ p−1 (x)), a( x, ξ )