Existence of solutions to a superlinear p-laplacian equation

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1 Electronic Journal of Differential Equations, Vol. 2001(2001), No. 66, ISSN: URL: htt://ejde.math.swt.edu or htt://ejde.math.unt.edu ft ejde.math.swt.edu (login: ft) Existence of solutions to a suerlinear -Lalacian equation Shibo Liu Abstract Using Morse theory, we establish the existence of solutions to the equation u = f(x, u) with Dirichlet boundary conditions. We assume that s f(x, t) dt lies between the first two eigenvalues of the -Lalacian. 0 1 Introduction Consider the Dirichlet roblem for the -Lalacian ( > 1), u = f(x, u), in Ω, u = 0, on Ω. (1.1) Here Ω is a bounded domain in R N with smooth boundary Ω, and u is the -Lalacian: u := div( u 2 u). We assume that f : Ω R R is a Carathéodory function with subcritical growth; that is, F1) The inequality f(x, u) C(1 + u q 1 ) holds for all u R, x Ω, and for some ositive constant C, where 1 q < N N if N + 1, and 1 q < if 1 N <. It is well known that weak solutions u W 1, 0 (Ω) of (1.1) are the critical oints of the C 1 functional Φ(u) = 1 u dx F (x, u) dx, where F (x, s) = s 0 f(x, t) dt. Let λ 1 and λ 2 be the first and the second eigenvalues of on W 1 0 (Ω). It is known that λ 1 > 0 is a simle eigenvalue, and that σ( ) (λ 1, λ 2 ) =, where σ( ) is the sectrum of, (cf. [2]). We shall assume the following conditions: Mathematics Subject Classifications: 49J35, 35J65, 35B34. Key words: -Lalacian, critical grou. c 2001 Southwest Texas State University. Submitted August 21, Published October 11,

2 2 Existence of solutions EJDE 2001/66 F2) There exist r > 0, λ (λ 1, λ 2 ) such that u r imlies λ 1 u F (x, u) λ u, F3) There exist θ >, M > 0 such that u M imlies 0 < θf (x, u) uf(x, u). Now, we are ready to state our main result. Theorem 1.1 Assume (F1), (F2), and (F3). Then (1.1) has a nontrivial weak solution in W 1, 0 (Ω). There are many aers devoted to the existence of solutions of (1.1); see for examle [1, 4, 5]. In these aers, the main tool is the minmax argument. However, it seems difficult to use the minmax argument in our situation. Thus we will use a different aroach: Morse theory [3]. To the best of our knowledge, [7] is the only work using Morse theory to obtain the solvability of -Lalacian equations. Our work is motivated by [7]. 2 Proof of main theorem In this section we give the roof of Theorem 1.1. Let E denote the Sobolev sace W 1, 0 (Ω), and. denote the norm in E. For Φ a continuously Fréchet differentiable ma from E to R, let Φ (u) denote its Fréchet derivative. As stated in Section 1, weak solutions u W 1, 0 (Ω) of (1.1) are the critical oints of the C 1 functional Φ(u) = 1 u dx F (x, u) dx. We will try to find a nontrivial critical oint of the functional Φ. First we state the following lemmas. Lemma 2.1 Under conditions (F1) and (F3), the functional Φ satisfies the Palais-Smale condition. Proof Assume (u n ) E, Φ(u n ) B for some B R, and Φ (u n ) 0. Let d := su n Φ(u n ). Then by (F3) we have θd + u n θφ(u n ) + Φ (u n ), u n = ( θ 1) u n u n M [θf (x, u n ) f(x, u n )u n ] [θf (x, u n ) f(x, u n )u n ] u n M ( θ 1) u n [θf (x, u n ) f(x, u n )u n ] u n M ( θ 1) u n D, for some D R.

3 EJDE 2001/66 Shibo Liu 3 Thus (u n ) is bounded in E. U to a subsequence, we may assume that u n u in E. Now because of condition (F1), a standard argument shows that u n u in E and the roof is comlete. Let V = san φ 1 be the one-dimensional eigensace associated to λ 1, where φ 1 > 0 in Ω and φ 1 = 1. Taking a subsace W E comlementing V, that is E = V W. Obviously the genus of W \0 satisfies γ(w \0) 2. Therefore, by the variational characterization of λ 2, for u W, u λ 2 u. Lemma 2.2 Under Assumtion (F2), the functional Φ has a local linking at the origin with resect to E = V W. That is, there exists ρ > 0, such that Φ(u) 0, u V, u ρ, Φ(u) > 0, u W, 0 < u ρ. The roof of this lemma can be found in [7, Lemma 3.3]. For a C 1 -functional Φ : E R and u an isolate critical oint of Φ, Φ(u) = c, we define the critical grou of Φ at u as C q (Φ, u) := H q (Φ c, Φ c \{u). Where H q (X, Y ) is the q-th homology grou of the toological air (X, Y ) over the ring Z. Since dim V = 1 < +, from Lemma 2.2 and Theorem 2.1 in [6], we have Lemma 2.3 Under assumtion (F2), 0 is a critical oint of Φ and C 1 (Φ, 0) 0. To find a nontrivial critical oint of Φ, we investigate the behavior of Φ near infinity. Lemma 2.4 Under Assumtion (F3), there exists a constant A > 0 such that Φ a S, for a < A, where S is the unit shere in E. Proof that Integrating on the inequality of (F2), we obtain a constant C 1 > 0 such F (x, t) C 1 t θ, for t M. Thus, for u S, we have Φ(tu), as t +. Set ( A := ) M Ω max f(x, u) + 1. Ω [ M,M]

4 4 Existence of solutions EJDE 2001/66 Using (F3) we obtain F (x, v) 1 vf (x, v) = F (x, v) + F (x, v) 1 vf (x, v) 1 vf (x, v) v M v M v M v M ( 1 θ 1 ) vf (x, v) + F (x, v) 1 vf (x, v) v M v M v M ( 1 θ 1 ) ( vf (x, v) ) M Ω max f (x, u) v M Ω [ M,M] ( 1 θ 1 ) vf (x, v) + A 1. For a < A and we have v M Φ(tu) = t F (x, tu) a, (u S ), d dt Φ(tu) = Φ (tu), u = t 2 t uf(x, tu) { F (x, tu) 1 tuf(x, tu) + a t {( 1 t θ 1 ) tuf(x, tu) + A 1 + a tu M {( 1 t θ 1 ) tuf(x, tu) 1 tu M { ( 1 t θ 1 )C 1θ tu θ 1 < 0. tu M By the Imlicit Function Theorem, there is a unique T C(S, R) such that Φ(T (u)u) = a, u S. For u 0, set T (u) = 1 u T ( u u ). Then T C(E\0, R) and for all u E\0, Φ( T (u)u) = a. Moreover, if Φ(u) = a, then T (u) = 1. We define a function ˆT : E\0 R as { T (u), if Φ(u) a, ˆT (u) := 1, if Φ(u) a. Since Φ(u) = a imlies T (u) = 1, we conclude that ˆT C(E\{0, R). Finally we set η : [0, 1] (E\0) E\0 as η(s, u) = (1 s)u + s ˆT (u)u.

5 EJDE 2001/66 Shibo Liu 5 It is easy to see that η is a strong deformation retract from E\0 to Φ a. Thus Φ a E\0 S and resent roof is comlete. We also use the following toological result,which was roved by Perera [8]. Lemma 2.5 Let Y B A X be toological saces and q Z. If H q (A, B) 0 and H q (X, Y ) = 0 then H q+1 (X, A) 0 or H q 1 (B, Y ) 0. Now we can rove the main theorem. Proof of Theorem 1.1 By Lemma 2.1, Φ satisfies the Palais-Smale condition. Note that Φ(0) = 0, from [3] Chater I, Theorem 4.2, there is a ε > 0, such that H 1 (Φ ε, Φ ε ) = C 1 (Φ, 0) 0. By Lemma 2.4, for a < A (A is as in the lemma) we have Φ a S. Since dim E = +, H 1 (E, Φ a ) = H 1 (E, S ) = 0. So that Lemma 2.5 yields H 2 (E, Φ ε ) 0 or H 0 (Φ ε, Φ a ) 0. It follows that Φ has a critical oint u for which Φ(u) > ε or ε > Φ(u) > a. Therefore, u is a nonzero critical oint of Φ, and (1.1) has a nontrivial solution. Remark Result similar to Lemma 2.4 has been roved (for = 2) in [9] and [3], under the additional conditions f C 1 (Ω R, R), f(x, 0) = f(x, t) t = 0. t=0 From these two references, we have obtained the motivation for this aer. References [1] A. R. El Amrouss & M. Moussaoui, Minimax rinciles for critical-oint theory in alications to quasilinear boundary-value roblems, Electron. J. Diff. Eqns.,, 2000(2000), No. 18, 1 9. [2] A. Anane & N. Tsouli, On the second eigenvalue of the -Lalacian, Nonlinear Partial Differential Equations, Pitman Research Notes 343(1996), 1 9.

6 6 Existence of solutions EJDE 2001/66 [3] K. C. Chang, Infinite dimensional Morse theory and multile solution roblems, Birkhäuser, Boston, [4] D. G. Costa & C. A. Magalhães, Existence results for erturbations of the -Lalacian, Nonlinear Analysis, 24(1995), [5] X. L. Fan & Z. C. Li, Linking and existence results for erturbations of the -Lalacian, Nonlinear Analysis, 42(2000), [6] J. Q. Liu, The Morse index of a saddle oint, Syst. Sc. & Math. Sc., 2(1989), [7] J. Q. Liu & J. B. Su, Remarks on multile nontrivial solutions for quasilinear resonant roblems, J. Math. Anal. Al., 258(2001), [8] K. Perera, Critical grous of critical oints roduced by local linking with alications, Abstract and Alied Analysis, 3(1998), [9] Z. Q. Wang, On a suerlinear ellitic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 8(1991), Shibo Liu Institute of Mathematics, Academy of Mathematics and Systems Sciences, Academia Sinica, Beijing, , P. R. China address: liusb@math08.math.ac.cn