ON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM. Paweł Goncerz

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1 Opuscula Mathematica Vol. 32 No ON THE EXISTENCE OF THREE SOLUTIONS FOR QUASILINEAR ELLIPTIC PROBLEM Paweł Goncerz Abstract. We consider a quasilinear elliptic problem of the type pu = λfu+µgu in, u = 0, where R N is an open and bounded set, f, g are continuous real functions on R and λ, µ R. We prove the existence of at least three solutions for this problem using the so called three critical points theorem due to Ricceri. Keywords: critical point, elliptic problem, minimax inequality, p-laplacian, three critical points theorem, weak solution. Mathematics Subject Classification: 35J20, 35J25, 35J92, 58E INTRODUCTION In this paper we prove that the problem { p u = λfu + µgu in, u = 0, 1.1 has at least three weak solutions in the Sobolev space W 1,p 0. Here p stands for the p-laplacian defined by p := div p 2 with p 1, +. The main result of the paper is the following. Theorem 1.1. Let R N be an open and bounded set with smooth boundary and let f, g : R R be continuous functions. Put F ξ := ξ 0 ft dt, Gξ := ξ 0 gt dt. 473

2 474 Paweł Goncerz Assume that ξ R F ξ > 0 and that there exists four positive constants a, q, s, γ with s [1, p, γ p, p and q 0, p 1, where p := { Np N p if p < N, + if p N, such that max { fξ, gξ } a 1 + ξ q, ξ R, 1.2 and max{f ξ, Gξ } a 1 + ξ s, ξ R, 1.3 lim ξ 0 F ξ < ξ γ Then, there exists δ > 0 such that, for each µ [ δ, δ], there exist ρ > 0 and an open nonempty interval Λ [0, + such that, for each λ Λ, problem 1.1 has at least three weak solutions in W 1,p 0 whose norms are less than ρ. Existence and multiplicity results for problems involving the p-laplacian or px-laplacian have been investigated in recent years by many mathematicians and three critical points theorem has been often used as a main tool in their proofs. Problem 1.1 has been studied by Ricceri in [15,16] when p = 2 while in [13,14] he formulated modified versions of the three critical points theorem and the existence theorems for both Dirichlet and Neumann problems with the p-laplacian. Bonanno and Livrea obtained in [3] the existence of at least three weak solutions for the Dirichlet problem with the p-laplacian when p > N and Carathéodory function satisfying appropriate growth conditions. By the similar assumptions Bonanno and Candito established in [1] the same result for the Neumann problem with the p-laplacian. Existence of three solutions for the Dirichlet problem involving the p-laplacian was also proved by Bonanno and Giovanelli in [2] by using the non-smooth version of the three critical points theorem established by Marano-Motreanu in [10]. On the other hand there are similar results for the problems driven by the px-laplace operator, for example Liu obtained in [9] the existence of at least three solutions for both Dirichlet and Neumann problems in which functions occurred satisfy the subcritical growth condition. In [11] Mihǎilescu proved the existence of at least three weak solutions for Neumann problem under the assumptions that inf x px > N 3 and the right hand side nonlinearity has the form fx, t = t qx 2 t t, where q { h C : hx > 1 x } and 2 < qx < inf x px for any x. A few years later Wang, Fan and Ge established in [18] analogous result as above for more general f. Another multiplicity result for anisotropic variable exponent problems can be found in Stancu-Dumitru [17]. Finally, let us mention the papers of Gasiński and Papageorgiou [4 7] where the multiplicity results existence of at least three or five nontrivial solutions for Dirichlet quasilinear boundary value problems are obtained using different methods than presented here critical point theory based on the minimax theorems due to Chang.

3 On the existence of three solutions for quasilinear elliptic problem 475 This paper generalizes the result of Ricceri [16, Theorem 4] to a general p-laplacian with p > 1. Here we do not require the relation between N and p but we assume that the functions which occur in the right hand side of the studied equation satisfy subcritical growth conditions together with their antiderivatives. 2. PRELIMINARIES In this section we recall some notions and facts needed in the proof of the main theorem. Theorem 2.1 [16, Theorem 1]. Let X be a separable and reflexive real Banach space, Φ: X R continuously Gâteaux differentiable and a sequentially weakly lower semicontinuous functional whose Gâteaux derivative admits a continuous inverse on X, Ψ: X R continuously Gâteaux differentiable functional whose Gâteaux derivative is compact and I R an interval. Assume that lim Φx + λψx = + for all λ I, x + and that there exists a continuous concave function h: I R such that λ I inf Φx + λψx + hλ < inf x X x X λ I Φx + λψx + hλ. Then there exist ρ > 0 and an open interval Λ I such that, for each λ Λ, the equation Φ x + λψ x = 0 has at least three solutions in X whose norms are less than ρ. Proposition 2.2 [15, Proposition 3.1]. Let X be a nonempty set, and Φ, Ψ two real functions on X. Assume that there are r > 0 and x 0, x 1 X such that Φ x 0 = Ψ x0 = 0, Φ x1 > r, Ψx < r Ψ x 1 x Φ 1,r] Φ. x 1 Then for each ρ satisfying one has Ψx < ρ < r Ψ x 1 x Φ 1,r] Φ, x 1 inf Φx + λρ Ψx < inf λ 0 x X x X λ 0 Φx + λρ Ψx.

4 476 Paweł Goncerz Proposition 2.3 [12, Proposition B.1]. Let R N be a bounded domain and let z satisfy: i z C R; R, ii there are constants b 1, b 2 1 and a 1, a 2 0 such that zx, ξ a 1 + a 2 ξ b 1 b2, x, ξ R. Then the map v z, v belongs to C L b1 ; L b2. Proposition 2.4 [8, Proposition 1.4.1]. If X is a reflexive Banach space, Y is a Banach space, D X is nonempty, closed and convex and J : D Y is completely continuous, then J is compact. Now we formulate and prove the generalization of Proposition B.10 of Rabinowitz [12]. Proposition 2.5. Let be a bounded domain in R N whose boundary is a smooth manifold and p 1, +. Let z satisfy: z 1 z C R; R, z 2 there are constants a 1, a 2 > 0 and ν 0, p 1 such that If Zx, ξ := ξ then T C 1 W 1,p 0 ; R T u, v = 0 zx, ξ a 1 + a 2 ξ ν, x, ξ R. zx, t dt and T u := and 1 p ux p Zx, ux dx, ux p 2 ux. vx zx, uxvx dx, v W 1,p 0, where. denotes the scalar product in R N. Moreover, Ju := Zx, ux dx is weakly continuous and J u is compact. Proof. Let X = W 1,p 0 and denote by the norm in X, by i the norm in L i and by c i the constant of the embedding W 1,p L i. We will show that T is defined on X.

5 On the existence of three solutions for quasilinear elliptic problem 477 Let u X. We have ux Ju zx, t dt dx so then 0 ux 0 a1 + a 2 t ν dt dx = = a 1 u 1 + a 2 ν + 1 u a 1c 1 u + a 2c ν + 1 u < +, T u 1 p u p + Ju < +. In order to show that T u is defined on X for u fixed, let v X. Using the Hölder inequality, we obtain T u, v ux p 1 vx dx + vx a 1 + a 2 ux ν dx ux p dx p 1 p vx p dx 1 + a 2 vx dx ux dx u p 1 v + a 1 c 1 v + a 2 c v u ν < +. 1 p + a 1 v 1 + The first term in T is C 1 and its Fréchet derivative is the first term in T u so we need to show that the second one belongs to C 1 X; R. First we will show that J is Fréchet differentiable on X and then that J u is continuous. Put Υ := Z, u + v Z, u z, u v. We want to show that for all u, v X and ε > 0 there exists δ = δε, u, p > 0 such that Ju + v Ju zx, uxvx dx ε v 2.1 provided v δ. We have Ju + v Ju zx, uxvx dx Υx dx. For any β, η > 0, we define 1 := { x : ux β }, 2 := { x : vx η }, 3 := { x : ux β vx η }. ν

6 478 Paweł Goncerz Then Υx dx 3 By the Mean Value Theorem, there exists θ 0, 1 such that i=1 i Υx dx. 2.2 Zx, ux + vx Zx, ux = zx, ux + θvxvx. Hence, using the Hölder inequality, we obtain Zx, ux + vx Zx, ux dx vx a1 + a 2 ux + vx ν dx 1 1 p 1 a 1 1 p 1 v p + a 1 2 σ v p a 1 c p a 3 v ux + vx dx σ v u, p 1 1 p v + 2 ν2 a2 c p 1 p 1 p σ u + v ν ν + ν v where σ 1, + is such that 1 σ = 1 ν p } and a 3 := c p max {a 1, 2 ν2 a2 c ν. Using the Hölder inequality again, we have p zx, uxvx dx a p 1 v p + a 2 1 σ v p u ν 1 1 p a 4 v 1 p σ u ν, where a 4 := c p max { a 1, a 2 c ν }. Due to the definition of 1, we have the following estimates u c p u p c p 1 β p dx 1 p = c p β 1 1 p, and 1 1 σ u c p β p σ =: M1, 1 p 1 p u c p β pp 1 p =: M 2,

7 On the existence of three solutions for quasilinear elliptic problem 479 where M 1, M 2 0 as β +. Then Υx dx a 5 v M 2 + M 1 u + v ν, 1 where a 5 := max { a 3, a 4 }. Assume that δ 1 and choose β so large that Hence a 5 M2 + M 1 u + 1 ε 3. 1 Υx dx ε v If we put a 6 := 2 max { a 1, a 2 } and use the same tools as above, we obtain Υx dx a vx a 6 vx η 2 p 1 a 6 η p v p vx 1 + ux + vx ν dx p 1 dx 1 + u + v ν. If we put a 7 := a 6 max { } 1, c ν p c p, then Υx dx a 7 η 1 p 2 2 v p 1 + ux + vx ν ν ν dx 1 + u + v ν. 2.4 Since Z C 1 R; R, given any ˆε, ˆβ > 0, there exists ˆη = ˆη ˆε, ˆβ, p such that Zx, ux + vx Zx, ux zx, uxvx ˆε vx, x 3. In particular, if ˆβ := β and η ˆη, then 3 Υx dx ˆε v 1 c 1ˆε v. 2.5 Choose ˆε so that c 1ˆε ε 3. This determines ˆη, so take η := ˆη. Combining 2.2, 2.3, 2.4 and 2.5 yields Υx dx 2ε 3 v + a 7η 1 p p v 1 + u + v ν.

8 480 Paweł Goncerz Eventually choose δ so small that p 1 a 7 η 2 + u ν δ p 1 ε 3, thereby obtaining 2.1. In order to prove continuity of J u, let u n u in X. Then u n u in L by the Sobolev Embedding Theorem. By the Hölder inequality, J u n J u X = From hypothesis z 2, we have v 1 z x, un x zx, ux vx dx c z, un z, u. ν zx, ξ a 1 + a 2 ξ αν α, x ξ R α 1. Choosing α := ν and using Proposition 2.3, we obtain that the map u z, u belongs to C L ; L ν, so J u n J u as n +. We proved that J C 1 X; R. To prove that J is weakly continuous, let u n u in X. By the Sobolev Embedding Theorem, u n u in L. Then using the Mean Value Theorem, hypothesis z 2, the Hölder inequality and the Sobolev Embedding Theorem, we have J u n Ju u n x ux a 1 + a 2 ux + θ u n x ux ν dx a un 1 u 1 + a un 2 u u + θ un u ν un u a 1 ν + a2 c u + θ un u ν and hence J u n Ju as n +. The last step is to show that J u is compact. Let v n v in X. Then v n v in L. Using the same tools as above, we obtain J u, v n J u, v vn x vx a 1 + a 2 ux ν dx v n v a 1 ν + a2 c ν u ν and hence J u, v n J u, v as n +. This means that J u is completely continuous so it is compact by Proposition 2.4.

9 On the existence of three solutions for quasilinear elliptic problem PROOF OF THE MAIN RESULT In this section we prove Theorem 1.1. Let I = [0, + and X = W 1,p 0. The norm in X is defined by u := 1 ux p p dx. We put J 1 u := F ux dx, J 2 u := Gux dx for all u X. From 1.4, there exist η 0, 1] and c > 0 such that F ξ c ξ γ, ξ [ η, η]. By assumption 1.3, putting we obtain c 1 := max { c, ξ >η a 1 + ξ s } ξ γ, F ξ c 1 ξ γ, ξ R. By Sobolev the Embedding Theorem, there exists c 2 > 0 such that v γ c 2 v 1,p, v W 1,p as γ < p and then If r > 0 and u p pr, then where c 3 := c 1 c γ 2 p γ p. Hence, we have J 1 u c 1 c γ 2 u γ, u X. J 1 u c 3 r γ p, lim u p pr J 1 u = 0. r 0 + r By assumption ξ R F ξ > 0, so we can choose w X \ {0} in such a way that J 1 w > 0. We fix r, ε > 0 with r < 1 p w p, so that and fix δ > 0 satisfying δ J 1 u pr J 1w u p pr w p ε u p pr J2 u J2 w + pr w p < ε.

10 482 Paweł Goncerz If we put σ := ε δ then simple calculations show that u p pr J 2 u J2 w + pr w p, J1 u + µj 2 u pr J 1w + µj 2 w u p pr w p σ, µ [ δ, δ]. Further we fix µ [ δ, δ] and define functionals Φ, Ψ: X R and function h: I R by Φu := 1 p u p, Ψu := J 1 u + µj 2 u, hλ := ρλ, where ρ is a fixed number satisfying J1 u + µj 2 u < ρ < pr J 1w + µj 2 w u p pr w p. The functional Φ is convex and continuous so it is sequentially weakly lower semicontinuous see [19, Proposition 41.8]. Due to 1.2 and Proposition 2.5, the functional Ψ is continuously Gâteaux differentiable with compact Gâteaux derivative. For each u, v X one has Φ u, v = ux p 2 ux. vx dx, Ψ u, v = fux + µguxvx dx, so the weak solutions of the problem { p u = λfu + µgu in, u = 0 are the critical points in X of the functional Φ + λψ. We will show that lim Φu + λψu = +, λ I. 3.1 u + Let u X. Since s [1, p and 1.3 holds, we have the following estimate: Ψu F ux + µ Gux dx a1 + µ 1 + ux s dx s a1 + µ + 1 s 1 p u p a1 + µ + p s p u s.

11 On the existence of three solutions for quasilinear elliptic problem 483 Hence Φu + λψu 1 p u p λa1 + µ + p s p u s, which yields 3.1 while u +. Using Proposition 2.2 with x 0 = 0 and x 1 = w, we obtain inf Φu + λρ + Ψu < inf λ 0 u X which is equivalent to λ I inf Φu + λψu + hλ < inf u X u X λ 0 u X λ I Φu + λρ + Ψu, Φu + λψu + hλ. All the assumptions of Theorem 2.1 are satisfied so there exists an open interval Λ I such that, for each λ Λ, problem 1.1 has at least three weak solutions in X whose norms are less than ρ. 4. REMARKS In this section we formulate some remarks which show that the assumptions of Theorem 1.1 cannot be omitted. Remark 4.1. The condition ξ R F ξ > 0 is essential. Suppose that this condition does not hold. Then taking f = g = 0, problem 1.1 has only a trivial solution. Remark 4.2. We cannot drop assumption 1.4. Suppose that f = 1 and g = 0. Then we can take a = 1. Since γ > p > 1, then lim ξ 0 F ξ ξ γ = +. Let λ 0 otherwise we have the same situation as in Remark 4.1. Taking N = 1, p = 4, = A, B, where A < B, we obtain that the function ux = 3 3 λ 8 2 A + B 2x 4 3 B A 4 3 is the unique solution of the problem { 3 u 2 u = λ in A, B, ua = ub = 0., x A, B, Remark 4.3. Theorem 1.1 does not hold, in general, for any µ R. If f satisfies all the assumptions of Theorem 1.1, then taking µ = 1 and g = f, we have the situation from Remark 4.1.

12 484 Paweł Goncerz 5. EXAMPLES In this section we give some examples of problems to which one can use Theorem 1.1. Example 5.1. Let p = 3, N = 1 and let be an open and bounded interval. Consider functions ft = 8t 3 cos t 2 and gt = 3 t + 1. Easy calculations show that these functions satisfy the assumptions of Theorem 1.1 with constants a = 8, q = 3, s = 2 and γ = 4 so there exists δ > 0 such that, for each µ [ δ, δ], there exist ρ > 0 and an open interval Λ [0, + such that, for each λ Λ, the problem { div u u = λ 8u 3 cos u 2 + µ 3 u + 1 in, u = 0 has at least three weak solutions in W 1,3 0 whose norms are less than ρ. Example 5.2. Let p = 12 7, N = 3 and R3 be open and bounded. Consider functions ft = 4t3 t 4 + 1, gt = t2 cos t sin t t2 + 1 Then f and g satisfy the assumptions of Theorem 1.1 with constants a = 4, q = s = 1 and γ = 3 so there exists δ > 0 such that, for each µ [ δ, δ], there exist ρ > 0 and an open interval Λ [0, + such that, for each λ Λ, the problem div u 2 4u 3 7 u = λ u µ u 2 cos u sin u in, u2 + 1 u = 0 has at least three weak solutions in W0 whose norms are less than ρ. 1, 12 7 Acknowledgements The author would like to thank Professor Leszek Gasiński for helpful suggestions and discussing the problem, and the reviewer for his/her comments on this work. REFERENCES [1] G. Bonanno, P. Candito, Three solutions to a Neumann problem for elliptic equations involving the p-laplacian, Arch. Math. Basel , [2] G. Bonanno, N. Giovanelli, An eigenvalue Dirichet problem involving the p-laplacian with discontinuous nonlinearities, J. Math. Anal. Appl , [3] G. Bonanno, R. Livrea, Multiplicity theorems for the Dirichlet problem involving the p-laplacian, Nonlinear Anal , 1 7. [4] L. Gasiński, N.S. Papageorgiou, Solutions and multiple solutions for quasilinear hemivariational inequalities at resonance, Proc. Roy. Soc. Edinburgh Sect. A ,

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14 486 Paweł Goncerz Paweł Goncerz Jagiellonian University Faculty of Mathematics and Computer Science ul. Łojasiewicza 6, Kraków, Poland Received: June 27, Revised: September 17, Accepted: September 19, 2011.