Multiplicity results for some quasilinear elliptic problems

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1 Multilicity results for some uasilinear ellitic roblems Francisco Odair de Paiva, Deartamento de Matemática, IMECC, Caixa Postal 6065 Universidade Estadual de Caminas - UNICAMP , Caminas - SP, Brasil João Marcos do Ó and Everaldo Souto de Medeiros Deartamento de Matemática Universidade Federal da Paraíba , João Pessoa - PB, Brazil Abstract In this aer, we study multilicity of weak solutions for the following class of uasilinear ellitic roblems of the form u u = g(u) λ u u in with u = 0 on, where is a bounded domain in R n with smooth boundary, 1 < < < n, λ is a real arameter, u = div( u u) is the -Lalacian and the nonlinearity g(u) has subcritical growth. The roofs of our results rely on some linking theorems and critical grous estimates. Keywords and hrases: Quasilinear ellitic roblems, -Lalace oerator, multilicity of solutions, critical grous, linking theorems. AMS Subject Classification: 35J65, 35J0. 1 Introduction In this aer we look for multile solutions of a class of uasilinear ellitic euations of the form { u u = g(u) λ u u in, u = 0 on, (P λ ) where is a bounded domain in R n with smooth boundary, 1 < < < n, λ is a real arameter, u = div( u u) is the -Lalacian and the nonlinearity g(t) enjoys the following conditions: (g 0 ) g C 1 (R), g(0) = 0; (g 1 ) there are constants C 1 > 0 and α with < α < such that g(s) C 1 (1 + s α 1 ) for all s R, where = n/(n ) when < n and = when = n is the critical Sobolev exonent, and Work artially suorted by CNP/Brazil Grants CNP/Brazil Grants # 00648/006 3, 47399/006 6 and /006-6 and Millennium Institute for the Global Advancement of Brazilian Mathematics-IM-AGIMB. addressess: odair@ime.unicam.br (Francisco Odair de Paiva), jmbo@mat.ufb.br (João Marcos do Ó), everaldo@mat.ufb.br (Everaldo Medeiros) 1

2 Multilicity for Quasilinear Problems (g ) there are constants C > 0 and β > such that g (s) C (1 + s β ) for all s R. In what follows we will denote by λ 1 () the first eigenvalue of the following nonlinear eigenvalue roblem { u = λ() u u in, (1) u = 0 on, and λ k (), k = 1,,..., the k-th eigenvalue of the lalacian with homogeneous Dirichlet boundary condition, which corresonds to roblem (1) with =. On roblem (P λ ), our main results concern the multilicity of weak solutions when the nonlinearity g(t) satisfies some additional hyotheses. Our first and second theorems treat the case when the g(t) has -sublinear growth at infinity, more recisely, we assume that (g 3 ) lim su s G(s) s < λ 1 (), where G(t) = t g(s) ds. They are formulated as follow. 0 Theorem 1. Assume that g satisfies (g 0 ) (g 3 ) and suose that g (0) > λ 1 (). Then there exists λ > 0 such that roblem (P λ ) has at least four nontrivial weak solutions for λ (0, λ ). Next we consider the case when the associated functional of roblem (P λ ) has a local linking at origin. This geometry structure imlies the existence of another nontrivial weak solution. Theorem. Assume that g satisfies (g 0 ) (g 3 ). Moreover, we assume that g (0) (λ k (), λ k+1 ()], k, and G(s) 1 λ k+1() s + 1 λ 1() s for all s R. () Then there exists λ > 0 such that roblem (P λ ) has at least five nontrivial weak solutions for λ (0, λ ). In our next results we consider the case when G(u) has -sueruadratic growth at infinity, that is, we assume the following version of the Ambrosetti-Rabinowitz condition: (g 4 ) there are constants θ > and s 0 > 0 such that for s s 0, 0 θg(s) sg(s). In the -sueruadratic case our main results are formulated as follow. Theorem 3. Assume that g satisfies (g 0 ) (g ), (g 4 ) and g (0) > λ 1 (). Then there exists λ > 0 such that roblem (P λ ) has at least two nontrivial solutions for λ (0, λ ). Finally, we consider the case λ < 0. This case is similar to the concave-convex roblems studied in []. Theorem 4. Assume that g satisfies (g 0 ) (g ), (g 4 ) and, in addition suose that, g (0) < λ 1 (). Then there exists λ < 0 such that roblem (P λ ) has at least two ositive solutions for λ (λ, 0). There has been recently a good amount of work on uasilinear ellitic roblems. Some of these roblems come from a variety of different areas of alied mathematics and hysics. For examle, they can be found in the study of non-newtonian fluids, nonlinear elasticity and reaction-diffusions, for discussions about roblems modelled by these boundary value roblems see for examle [15].

3 Multilicity for Quasilinear Problems 3 The study of multile solutions for ellitic roblems has received considerable attention in recent years. First, we would like to mention the rogress involving the following class of semilinear ellitic roblems u = λ u u + g(u) in and u = 0 in, where 1 < <. Ambrosetti at al. in [], studied the case g(u) = λ u r u, < r <. Among others results, they roved the existence of two ositive solutions for small ositive λ. Perera in [18] roved the existence of multile solutions when g(u) is sublinear at infinity and λ is small and negative(see also [14] for assymtotically linear and suerlinear cases). Multilicity results involving the -Lalacian of uasilinear roblems of the form u = λ u s u + g(u) in and u = 0 in, where 1 < s <, has been studied in [3] and [17], when g(u) = u r u, < r <. Recently, critical grous comutations via Morse theory for a functional like I(u) = 1 u dx + 1 u dx G(u) dx, u W 1, 0 () (3) has been studied in [10], where the authors obtained a version of Shifting Theorem in the case g (u) C(1 + u r ), with 0 r <. Cingolani and Degiovanni [11], has roved a existence result for the functional (3) when g(u) has -linear growth at infinity, that is, lim t g(u)/ u u = µ. In fact, they roved a version of the classical existence theorem by Amann and Zehnder [1] for the semilinear roblems. Benci at al. [6], [7] have studied a roblem that involves oerator like in left-hand side of (P λ ), which are motivated by roblems from hysics, in fact arising in the mathematical descrition of roagation henomena of solitary waves. Finally, we refer to [16] and [5] where the authors roved multile solutions for a roblems involving more general class of oerator than in left hand side of (P λ ). The rest of this aer is organized as follows. Section contains reliminary results, including a result of Sobolev versus Holder local minimizers. Section 3 is devoted to roving ours main results. Preliminary results In this aer we make use of the following notation: C, C 0, C 1, C,... denote ositive (ossibly different) constants. For 1 <, L. () denotes the usual Lebesgue sace with norm u = [ u dx ] 1/ and W 1, 0 () denotes the Sobolev sace endowed with the usual norm u 1, = u. Here we search for weak solutions of roblem (P λ ), that is, functions u W 1, 0 () such that u u ϕ dx + u ϕ dx + λ u uϕ dx g(u)ϕ dx = 0, for all ϕ W 1, 0 (). It is well known that under conditions (g 0 ) (g 1 ) the associated functional of (P λ ), I λ : W 1, 0 () R, given by I λ (u) = 1 u dx + 1 u dx + λ u dx G(u) dx, (4) is well defined, continuously differentiable on W 1, 0 (), and its critical oints corresond to weak solutions of (P λ ) and conversely (see [1, 4]). Remark 5. Notice that condition (g 3 ) imlies that the functional I λ is coercive and therefore satisfies the Palais-Smale condition (see Lemma 9 in Section 3). On the other hand, under the hyothesis (g 4 ), the Palais-Smale condition for the functional I λ can be roved by standard arguments. Also, it is well known that there exists a smallest ositive eigenvalue λ 1 (), and an associated function ϕ 1 > 0 in that solves (1), and that λ 1 () is a simle eigenvalue (see [5]). We recall that we have the

4 Multilicity for Quasilinear Problems 4 following variational characterization { λ 1 () = inf u dx : u W 1, 0 (), } u dx = 1. (5) Next we show that the local minimum of the associated functional I λ in C 1 toology is still a local minimum in W 1, 0 (). This result was roved by Brezis and Nirenberg for = (see [8]) and for the uasilinear case we referee to [1, 17], we will include here a roof for the sake of comleteness. Lemma 6. Assume that g satisfies (g 1 ) and u 0 W 1, 0 () C 1 0() is a local minimizer of I λ in the C 1 - toology, that is, there exits r > 0 such that I λ (u 0 ) I λ (u 0 + v), v C0() 1 with v C 1 0 () r. (6) Then u 0 is a local minimizer of I λ in W 1, 0 (), that is, there exists α > 0 such that I λ (u 0 ) I λ (u 0 + v), v W 1, 0 () with v 1, α. Proof. If u 0 is a local minimizer of I λ in the C 1 -toology, we see that it is a weak solution of (P λ ). By regularity results in Tolksdorf [0], u 0 C 1,α () (0 < α < 1). Now, suose that the conclusion does not holds. Then ɛ > 0, v ɛ B ɛ such that I λ (u 0 + v ɛ ) < I λ (u 0 ), (7) where B ɛ := {v W 1, 0 () : v 1, ɛ}. It is easy to see that I λ is lower semicontinuous on the convex set B ɛ. Notice that B ɛ is weakly seuentially comact and weakly closed in W 1, 0 (). By standard lower semicontinuous argument, we know that I λ is bounded from below on B ɛ and there exists v ɛ B ɛ such that I λ (u 0 + v ɛ ) = inf v B ɛ I λ (u 0 + v). We shall rove that v ɛ 0 in C 1 as ɛ 0, which is a contradiction with (6) and (7). The corresonding Euler euation for v ɛ involves a Lagrange multilier µ ɛ 0, namely, v ɛ satisfies I λ(u 0 + v ɛ )(h) = µ ɛ v ɛ v ɛ h, h W 1, 0 (), that is, Thus, (u 0 + v ɛ ) (u 0 + v ɛ ) g(u 0 + v ɛ ) u 0 + v ɛ (u 0 + v ɛ ) = µ ɛ v ɛ. u 0 u 0 [ (u 0 + v ɛ ) u 0 + v ɛ ] + µ ɛ v ɛ = g(u 0 + v ɛ ) + u 0 + v ɛ (u 0 + v ɛ ). This imlies that [ (u 0 + v ɛ ) u 0 ] + µ ɛ v ɛ = g(u 0 + v ɛ ) g(u 0 )+ u 0 + v ɛ (u 0 + v ɛ ) u 0 u 0. (8) Notice that we can write (8) as div(a(v ɛ )) := div ( (u 0 + v ɛ ) (u 0 + v ɛ ) u 0 u 0 + v ɛ µ ɛ v ɛ ) v ɛ =g(u 0 + v ɛ ) g(u 0 ) + u 0 + v ɛ (u 0 + v ɛ ) u 0 u 0 =g (ξ) + u 0 + v ɛ (u 0 + v ɛ ) u 0 u 0

5 Multilicity for Quasilinear Problems 5 where ξ (min{u 0, u 0 + v ɛ }, max{u 0, u 0 + v ɛ }). We know (see Tolksdorf [0]) that for > there exits ρ > 0 indeendent of u 0 and v ɛ such that [ (u0 + v ɛ ) (u 0 + v ɛ ) u 0 u 0 ] ρ vɛ. Thus, A(v ɛ ). v ɛ (ρ µ ɛ ) v ɛ + v ɛ C v ɛ, since µ ɛ 0. Using the growth condition (g ) we have g (ξ) C 1 + C ( u 0 β 1 + v ɛ β 1 + u v ɛ 1 ). Since β 1 > 0 and 1 > 0, by regularity results obtained in [], we have that for some 0 < α < 1, there exists C > 0 indeendent of ɛ such that By the regularity results in [3], we also have that v ɛ C α () v ɛ 1, C. v ɛ C 1,α 0 () C 1. This imlies that v ɛ 0 in C 1 as ɛ 0. Since v ɛ 1, 0, we have v 0 0. This comletes the roof. Now, for u W 1, 0 () we define I ± λ (u) = 1 u dx + 1 u dx + λ u ± dx G(u ± ) dx, (9) where u + = max{u, 0} and u = min{u, 0}. Since g(0) = 0, I ± λ C1 and the critical oints u ± of I ± λ satisfy ±u ± 0, we conclude that u ± are also critical oints of I λ. In fact, (I ± λ ) (u ± )[(u ± ) ] = (u ±) dx + (u ±) dx = 0. Lemma 7. The origin u 0 is a local minimizer for I λ and I ± λ, for any λ > 0. Proof. By Lemma 6, is sufficient to show that u = 0 is a local minimizer of I λ in the C 1 toology. First we observe that from (g 1 ) and the regularity of g, we have G(s) C s + C s α, for s R, for some ositive constant C. Then, for u C0() 1 we have I λ (u) = 1 u dx + 1 u dx + λ λ u dx G(u) dx λ u dx C u dx C u α dx ( ) λ C u C C u α 0 C 0 if C u C C u α C 1 0 λ. The same argument works for I± λ. u dx 0, u dx G(u) dx We do not include here the roof of next lemma because it follows using the same ideas of [18, Lemma.1]. Lemma 8. If u ± is a local minimizer for I ± λ, then it is a local minimizer of I λ, for any λ > 0.

6 Multilicity for Quasilinear Problems 6 3 Proofs of Main Theorems Proof of Theorem 1 We already know that the origin is a local minimum of the functional I ± λ. In the next two lemmas we rove that I ± λ is coercive and min u W0 1() I ± λ (u) < 0. Thus, I± λ has a global minimum u± 0 with negative energy. Finally, by alying the Mountain Pass Theorem, we get critical oints u ± 1 with ositive energy. Lemma 9. The functional I ± λ is coercive, lower bounded and satisfies the (PS) condition. Proof. By (g 3 ) there exists ɛ > 0, small enough, and a constant C such that Then Thus I ± λ (u) as u 1,. G(s) (λ 1 () ɛ) s + C, for all s R. I ± λ (u) 1 u dx G(u) dx 1 u dx (λ 1() ɛ) u dx C 1 ( 1 (λ ) 1() ɛ) u 1, λ 1 () C. Lemma 10. Let ϕ 1 be the first eigenfunction associated to λ 1 (). Then there exists t 0 > 0 such that I ± λ (±t 0ϕ 1 ) < 0, for all λ in a limited set. Proof. Since g(0) = 0, g (0) > λ 1 () and (g 1 ) holds, for each ɛ > 0, there exists C > 0 such that Then, for t > 0 we obtain G(s) (λ 1() + ɛ) s C s α, for all s R. I ± λ (±tϕ 1) t ϕ 1 dx + t ( = t 1 (λ 1() + ɛ) λ 1 () ϕ 1 dx + t λ Since α > > we can conclude the lemma. ϕ 1 dx (λ 1() + ɛ)t ) ϕ 1 1, + t ϕ 1 1, + t λ ϕ 1 + Ct α ϕ 1 α α. ϕ 1 dx + Ct α ϕ 1 α dx Finishing the roof of Theorem 1: By the Mountain Pass Theorem, I ± λ has a nontrivial critical oint u ± 1 with I± λ (u± 1 ) > 0. Since I± λ is bounded bellow, it also has a global minimizer u± 0 with I± λ (u± 0 ) < 0. Proof of Theorem In what follows we assume that the reader is somewhat familiar with Morse theory (see [9] for necessary rereuisites). In articular, we recall that the critical grous of a real C 1 functional Φ at an isolated critical oint u 0 with Φ(u 0 ) = c, are defined by C (Φ, u 0 ) = H (Φ c U, (Φ c U) \ {u 0 }) for N.

7 Multilicity for Quasilinear Problems 7 Here Φ c := {u : Φ(u) c}, U is a neighborhood of u 0 such that u 0 is the only critical oint of Φ in Φ c U, and H (, ) denote the singular relative homology grous with coefficients in Z. By the roof of Theorem 1, I λ has four critical oints u ± 1, with I λ(u ± 1 ) > 0, and u± 0, with I λ(u ± 0 ) < 0. By Lemma 8, we have that u ± 0 are local minimum of I λ since they are global minimum of I ± λ. Then C j (I λ, u ± 0 ) = δ j0z. Now, in order to rove the existence of another nontrivial solution we will aly the following abstract theorem due to Perera (see [19, Theorem 3.1]): Theorem 11. Let X = X 1 X be a Banach sace with 0 < k = dimx 1 <. Suose that Φ C 1 (X, R), has a finite number of critical oints and satisfies the Palais-Smale comactness condition (P. S). Moreover, assume the following conditions: (H 1 ) there exists ρ > 0 such that su S 1 ρ Φ < 0, where S 1 ρ = {u X 1 ; u = ρ}, (H ) Φ 0 in X, (H 3 ) there is e X 1 \ {0} such that Φ is bounded from below on {se + x ; s 0, x X }. Then Φ has a critical oint u 0 with Φ(u 0 ) < 0 and C k 1 (Φ, u 0 ) 0. First we consider H k := k j=1 ker( λ j()i) and W k = W 1, 0 () Hk, where H k orthogonal subsace of H k in H0 1 (). Thus we have denotes the W 1, 0 () = H k W k, u λ k () u, u H k, (10) u λ k+1 () u, u W k. The next lemma is a verification of the hyotheses of Theorem 11. Lemma 1. The functional I λ enjoys the following roerties: (I 1 ) there exist λ > 0 and ρ > 0 such that su S k ρ I λ < 0, for 0 < λ < λ, where S k ρ = {u H k ; u 1, = ρ}; (I ) I λ 0 in W k ; (I 3 ) I λ is bounded from below. Proof. Verification of (I 1 ): Since g(0) = 0, by (g 1 ) given ɛ > 0 there exists C > 0 such that G(s) (g (0) ɛ) s C s α, for all s R. Since the norm are euivalent in H k, for each u H k we have I λ (u) 1 u dx + 1 u dx + λ u dx (g (0) ɛ) 1 ( ) 1 (g (0) ɛ) u + C(λ u + u + u α ) λ k () c ( ) 1 (g (0) ɛ) u + C(λ u + u + u α ), λ k () u dx + C u α dx

8 Multilicity for Quasilinear Problems 8 where, in the last ineuality, we use that the norms are euivalent in a finite dimensional sace. Since, we can choose ɛ such that g (0) ɛ > λ k (), and using that α > > then we can take ρ and λ, small enough, such that (I 1 ) holds. Verification of (I ): If u W k, by (), we have I λ (u) 1 u dx + 1 u dx + λ We notice that (I 3 ) already was roved. u dx λ k+1() u dx λ 1() u dx 0. Finishing the roof of Theorem : Thus I λ has a critical oint u with I λ (u ) < 0 and C k 1 (I λ, u ) 0. Then we can conclude, using k, that u is different of u ± 0 and u± 1. Proof of Theorem 3 This theorem is a direct alication of the Mountain Pass Theorem (see [4, 4]). Lemma 13. Let ϕ 1 be the first eigenfunction associated to λ 1 (). Under the above conditions, there exist t 0 > 0 and λ > 0 such that I ± λ (±t 0ϕ 1 ) < 0, for all λ (0, λ ). Proof. By g(0) = 0, g (0) > λ 1 () and (g 1 ) we have that, given ɛ > 0 there exists C > 0 such that Then, for t > 0, G(s) (λ 1() + ɛ) s C s α, for all s R. I ± λ (±tϕ 1) = t ϕ 1 dx + t ( = t 1 (λ 1() + ɛ) λ 1 () t {( 1 (λ 1() + ɛ) λ 1 () ϕ 1 dx + t λ ) ϕ 1 1, + t ϕ 1 1, + t λ ϕ 1 dx (λ 1() + ɛ)t ϕ 1 + Ct α ϕ 1 α α. ) ϕ 1 1, + t ϕ 1 1, + t λ ϕ 1 + C 1 t α ϕ 1 α α ϕ 1 dx + Ct α Since α > > > > 1, we can choose λ > 0 such that I ± λ (±t 0ϕ 1 ) < 0, for all λ (0, λ ). }. ϕ 1 α dx Now, the roof of Theorem 3 follows from standard argument using the Mountain Pass Theorem. Proof of Theorem 4 We will look for critical oints of the C 1 functional I + λ (u) = 1 u dx + 1 u dx + λ u + dx G(u + ) dx, u W 1, 0 (). The roof of this theorem is similar in sirit to that of Theorem.1 in [13]. We will shown that the functional has the mountain-ass structure which together with the comactness condition will give us a ositive I + λ

9 Multilicity for Quasilinear Problems 9 solution in a ositive level. After that, we will roof that I + λ the origin with negative energy. We made it in three stes. has a ositive local minimum in a ball around Ste 1: There exist r > 0 and a > 0 such that I + λ (u) a if u 1, = r. Indeed, using condition (g 1 ) and g (0) < λ 1 (), for ɛ small and s R we have G(s) 1 (λ 1() ɛ) s + C s α, s R. Thus, (remember that λ < 0) I + λ (u) 1 u dx + 1 u dx + λ u dx (λ 1() ɛ) u dx C u α dx 1 u dx + 1 ( 1 (λ ) 1() ɛ) u dx + λ u dx C u α dx λ 1 () A u 1, + λb u 1, C u α 1,. By [13, Lemma 3.] there exists λ < 0 such that if λ (λ, 0), I + λ satisfies the roerty above. Ste : There exists t M > r such that I + λ (t M ϕ 1 ) 0. Indeed, let ϕ 1 the first eigenfunction associated to λ 1 (). For t > 0 we have I + λ (tϕ 1) = t ϕ 1 dx + t ϕ 1 dx + t λ ϕ 1 dx G(tϕ 1 ) dx { } 1 =t ϕ 1 dx + t ϕ 1 dx + t λ ϕ 1 G(tϕ 1 ) dx t dx. Follows from condition (g 4 ) that there exists a ositive constant C such that G(s) Cs θ, for all s s 0. Thus { } 1 I + λ (tϕ 1) t ϕ 1 dx + t ϕ 1 dx + t λ ϕ 1 dx Ct θ ϕ 1 θ dx. Now, observing that t lim ϕ 1 t λ dx = lim ϕ t + t + 1 dx = 0, for all λ (λ, 0) and θ >, we obtain I + λ (t M ϕ 1 ) 0 for some t M > 0. Thus, we can aly the Mountain Pass Theorem to obtain a critical oint u 1 such that I + λ (u 1) > 0. Ste 3: There exists 0 < t m < r such that I + λ (t mϕ 1 ) < 0. Indeed, let ϕ 1 the first eigenfunction associated to λ 1 (). We have that { t I + λ (tϕ 1) =t ϕ 1 dx + 1 } ϕ 1 dx + t λ ϕ 1 G(tϕ 1 ) dx t dx. Since <, λ < 0 and G(t)/t is bounded next to t = 0, our claim follows. Now, the minimum of the functional I + 1, λ in a closed ball of W0 () with center in zero and radius r such that I + λ (u) 0 u with u 1, = r, is achieved in the corresondent oen ball and thus yields a nontrivial critical oint u of I + λ, with I+ λ (u ) < 0 and u 1, < r.

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