A Variational Approach for a Nonlocal and Nonvariational Elliptic Problem Francisco Julio S.A Corrˆ ea,
,
UFCG - Unidade Acadˆemica de Matem´atica e Estat´ıstica, 58.109-970 - Campina Grande - PB - Brazil E-mail:
[email protected],
Giovany M. Figueiredo UFPA - Faculdade de Matem´atica - PPGME 66075-110 Bel´em - PA - Brazil. E-mail:
[email protected].
¯ The Abstract: We prove results concerning the for all s1 , s2 ∈ [0, R1 ] and for all x ∈ Ω. existence of solutions for the problem nonlinearity f : Ω × R → R is a continuous function satisfying Z −a x, u ∆u = f (x, u) in Ω, f (x, s) = 0 for all s < 0 and x ∈ Ω, (f1 ) Ω where Ω is bounded regular domain and f : R → R is a function having subcritical growth. |f (x, s)| lim = 0, uniformly in x ∈ Ω. (f2 ) Although we are facing a problem with lack of s |s|→0 variational structure we will be able to apply variational technique (the Mountain Pass Theorem) by suitably using a device introduced in There exists 2 < q < 2∗ = 2N such that N −2 De Figueiredo-Giradi-Matzeu [6]. |f (x, s)| = 0, uniformly in x ∈ Ω, |s|q−1 1 Introduction (f3 ) 2N ∗ ∗ In this paper we investigate questions of exis- where 2 < q < 2 and 2 = . N −2 tence of solutions for the following problem From assumptions (f2 )−(f3 ), given > 0, there Z exist C such that, −a x, u ∆u = f (x, u) in Ω, Ω f (x, s) ≤ |s| + C |s|q−1 , (1) (P1 ) u = 0 on ∂Ω u > 0 for all x ∈ Ω, for s ∈ R and x ∈ Ω. lim
s→+∞
where Ω ⊂ RN is a bounded smooth domain, N ≥ 3 and the functions a and f enjoy the following assumptions: The function a : Ω × R → R is continuous and there are constants a0 , a∞ , R1 and L1 such that
In this article, the classical Palais-Smale condition will play a key role. Related to this condition, we have the well known AmbrosettiRabinowitz superlinear condition, that is, there exists θ ∈ R with 2 < θ < q such that Z s 0 < θF (x, s) = θ f (x, t)dt ≤ sf (x, s), (f4 ) 0
¯ × R, 0 < a0 ≤ a(x, t) ≤ a∞ for all (x, t) ∈ Ω ¯ (a1 ). for all s > 0 and for all x ∈ Ω. The function and f (x, s) is increasing in (0, +∞), s→ |a(x, s1 ) − a(x, s2 )| ≤ L1 |s1 − s2 |, (a2 ) s —8—
(f5 )
for all x ∈ Ω. We also suppose that there exists Kirchhoff-type like a constant L2 such that Z 2 ∆u = f (x, u) in Ω, −M x, |∇u| |f (x, t1 ) − f (x, t2 )| ≤ L2 |t1 − t2 | (f6 ) Ω u = 0 on ∂Ω ¯ for all t1 , t2 ∈ [0, R1 ] and for all x ∈ Ω. u > 0 for all x ∈ Ω, 1 We say that u ∈ H0 (Ω) is a weak solution of (P4 ) the problem (P1 ) if where Ω is as before and M : Ω × R+ → R is a given function. Z Z f (x, u)φ In this work, we denote by Sr is the best Z ∇u∇φ = Ω Ω constant of the embedding of H01 (Ω) into a x, u kuk Ω Lr (Ω), that is, Sr = inf ,where kuk = u6≡0 |u|r 1 Z 1/2 Z 1/r for all φ ∈ H0 (Ω). 2 r Problem (P1 ) is a generalization of the equa|∇u| and |u|r = |u| are, Ω Ω tion respectively, the usual norms in H01 (Ω) and Z Lr (Ω). −a u ∆u = f in Ω, (P2 ) Note that if it exists a constant K2 such that Ω u = 0 on ∂Ω, |s| ≤ K2 , then, from (1), there exists Z a constant C1 , depending on K2 , such that |f (x, s)|2 ≤ f ∈ H −1 (Ω), which is the steady-state counterΩ part of the parabolic problem C1 for all x ∈ Ω. Our main result is as follows: Z u ∆u = f in Ω × (0, T ), ut − a Theorem 1.1 Assume conditions (a1 ) − (a2 ) Ω u(x, t) = 0 on ∂Ω × (0, T ) and (f1 ) − (f6 ) hold. If u(x, 0) = u0 (x). 1/2 (P3 ) S2 L2 C1 < 1, Such an equation arises in various situations. S1 (a0 S22 − L1 a∞ ) For instance, u could describe the density of a population (bacteria, for instance) subject to then problem (P1 ) has a positive solution. spreading. The diffusion coefficient a is then supposed to depend on the entire population in the domain Ω, rather than on the local density, 2 The variational framework that is, moves are guided by considering the As in [6], the technique used in this paper conglobal state of the medium. Furthermore, with the respect to the statio- sists of associating with problem (P1 ) a family elliptic problems. Namely, nary problem (P2 ), it has the special feature of of local semilinear 1 (Ω) we consider the problem for each w ∈ H 0 not being variational. It has been studied by several authors as [2], [3], [4] and [5] by using f (x, u) in Ω, some techniques as Fixed Point Theory, Sub −∆u = Z and Supersolution, Quasi-variational inequali(P5 ) a x, w Ω ties, Galerkin Method an so on. u = 0 on ∂Ω and u > 0 in Ω. In problem (P1 ), besides the lack of variational structure, the function a also depends on Now, problem (P5 ) is variational and we can the variable x ∈ Ω situation that, al least to our treat it by Variational Methods. knowledge, has not been studied in the existing As usual, a weak solution of a problem as literature. in (P5 ) is obtained as a critical point of the However, inspired by ideas developed in [6], associated functional we use the Mountain Pass Theorem to find a Z Z solution of (P1 ). F (x, u) 1 2 Z . Iw (u) = |∇u| − We point out that the techniques we will use 2 Ω Ω a x, w are valid, mutatis mutandis, for equation of the Ω —9—
The proof of Theorem 1.1 is broken in several lemmas. We prove that the functional Iw has the geometry of the mountain pass theorem, that it satisfies the Palais-Smale condition and finally that the obtained solutions have the uniform bounds stated in the theorem.
Lemma 2.4 Let w ∈ H01 (Ω). There exists a positive constant K1 independent of w, such that kuw k ≥ K1 , for all solutions uw obtained in Lemma 2.3.
Proof Using uw as a test function in (P2 ), we obtain Z Lemma 2.1 Let w ∈ H01 (Ω). Then there exist f (x, uw )uw 2 . Z kuw k = positive numbers ρ and α, which are indepenΩ a x, w dent of w, such that Ω
Iw (u) ≥ α > 0, ∀u ∈ H01 (Ω) : kuk = ρ
From (a1 ), (1) and using Sobolev embedding Proof. From (a1 ), (1) and using Sobolev em- theorem, we conclude bedding theorem, we conclude C (1 − 2 )kuk2 ≤ q kukq . Sq a0 S2 a0 1 C q Iw (u) ≥ kuk2 − − q kuk . 2 2 2a0 S2 a0 Sq So, the result follows. 2 Since 2 < q, the result follows. 2 Lemma 2.5 Let w ∈ H01 (Ω). There exists a 1 1 Lemma 2.2 Let w ∈ H0 (Ω). Fix v0 ∈ H0 (Ω), positive constant K2 independent of w, such with v0 > 0 and kv0 k = 1. Then there is a that kuw k ≤ K2 , for all solutions uw obtained in Lemma 2.3. T > 0, independent of w, such that Iw (tv0 ) ≤ 0,
for all t ≥ T.
Proof. It follows from (f4 ) that there exist constants C3 and C4 such that t2
Proof Using (f5 ), we obtain the inf max characterization of uw in Lemma 2.3. So,
tθ
cw ≤ max Iw (tv0 ) t≥0
C3 C4 Iw (tv0 ) ≤ − − |Ω|. θ 2 a∞ a∞ Sθ
with v0 choosen in Lemma 2.3. We estimate cw using (f4 ): Since θ > 2, we obtain T independent of v0 and 2 θ e also of w, such that the result holds. 2 cw ≤ max Iw (tv0 ) ≤ max t − C3 t −C4 |Ω| = K. t≥0 t≥0 2 Sθθ Lemma 2.3 Assume (f1 ) − (f4 ). Then problem (P2 ) has at least one positive solution uw Also from (f4 ), we obtain for any w ∈ H01 (Ω). 1 1 1 e − kuw k ≤ Iw (uw )− Iw0 (uw )uw = cw ≤ K. 2 θ θ Proof. Lemmas 2.1 and 2.2 show that the functional Iw has the mountain pass geometry. The result follows by considering K2 = −1 1/2 From (1) we conclude that Iw satisfies the (PS) 1 1 e . 2 condition. So, by the mountain pass theorem, K 2 − θ a weak solution uw of (P2 ) is obtained as a criRemark 2.6 (On the regularity of the solutical point of Iw at an inf max level. Namely tion of (P2 )). In Lemma 2.3 we have obtaiIw0 (uw ) = 0 ned a weak solution uw of (P2 ) for each given w ∈ H01 (Ω). Since q < 2∗ , a standard bootsand trap argument, using the Lp -regularity theory, Iw (uw ) = cw = inf max Iw (γ(t)), (2) shows that uw is, in fact, in C 1,β (Ω). As a conγ∈Γw t∈[0,1] sequence of the Sobolev embedding theorems where Γw = {γ ∈ C([0, 1], H01 (Ω)) : γ(0) = and Lemma 2.5 we conclude with the following: 0, γ(1) = T v0 }, for some v0 and T as in Lemma T 1,β 1 2.2. From now on we fix such a v0 and such a T . Lemma 2.7 Let w ∈ H0 (Ω) C (Ω).Then Multiplying both sides of the equation in (P2 ) there exists a positive constant R1 , independent of w, such that the solution uw obtained by u− w , using (f1 ) and integrating by parts, we − in Lemma 2.3 satisfies kuw kC 0,β ≤ R1 . conclude that uw ≡ 0. So uw is positive. 2 — 10 —
3
Proof of Theorem 1.1
We construct a sequence (un ), n ∈ IN , of solutions as f (x, un ) in Ω, −∆un = Z a x, un−1 Ω un = 0 on ∂Ω and un > 0 for all x ∈ Ω. (Pn ) obtained by the mountain pass theorem, starT ting with an arbitrary u0 ∈ H01 (Ω) C 1,β (Ω). By Remark 4, we see that kun kC 0,β (Ω) ≤ R1 . On the other hand, using (Pn ) and (Pn+1 ) we obtain Z ∇un+1 (∇un+1 − ∇un ) Ω
Z = Ω
So, a20 S22 − L1 a∞ kun+1 − un k ≤ S2 1/2 L2 C1 kun − un−1 k. S1 Hence, 1/2 L2 C1 S2 kun − un−1 k kun+1 − un k ≤ 2 2 S1 a0 S2 − L1 a∞ or kun+1 − un k ≤: kkun − un−1 k. Since the coefficient k is less than 1, it follows, by a straightforward argument, that the sequence (un ) converges strongly in H01 (Ω) to some function u ∈ H01 (Ω). Since K1 ≤ kun k for all n, we have u > 0 in Ω. From (Pn ), we obtain
f (x, un+1 ) Z (un+1 − un ) a x, un
Z
Z ∇un ∇φ = Ω
Ω
Ω
f (x, u )φ Z n . a x, un−1 Ω
and
Z ∇un (∇un+1 − ∇un ) Ω
Z = Ω
f (x, u ) Z n (un+1 − un ). a x, un−1
Since un → u in H01 (Ω), we conclude that Z
Z ∇u∇φ =
Ω
Ω
Ω
f (x, u)φ Z , for all φ ∈ H01 (Ω), a x, u Ω
Note that from (1) and Lemma 2.5, we have and the proof of the theorem is over. 2 Z 1/2 1/2 2 that |f (x, un )| ≤ C1 . Ω Z 1 encias 2 Thus, kun+1 − un k ≤ 2 (f (x, un+1 ) − Referˆ a Ω 0 Z [1] Ambrosetti A., and Rabinowitz P. H., f (x, un ))a x, un−1 |un+1 − un | Dual variational methods in critical point Z Z Z 1 theory and applications, J. Funct. Anal., + 2 [ f (x, un ) a x, un−1 −a x, un 14(1973)349-381. a0 Ω |un+1 − un |]. Using Z(a2 ) and (f6 ), kun+1 − un k2 ≤ [2] Bueno H., Ercole G., Ferreira W. and L1 a∞ Zumpano A., Existence and multiplicity of |un+1 − un |2 2 positive solutions for the p-Laplacian with a0 Z Ω Z L2 nonlocal coefficient, J. Math. Anal. Appl. + 2 |f (x, un )| |un − un−1 | |un+1 − un |. (2008), doi:10.1016/j.jmaa.2008.01.001. a0 Ω Ω Using Sobolev embedding theorem Article in press. L1 a∞ 2 2 kun+1 − un k ≤ 2 2 kun+1 − un k a0 SZ2 [3] Chipot M. and Lovat B., Some remarks Z L2 on nonlocal elliptic and parabolic problems, + 2 |un − un−1 | |f (x, un )||un+1 − un |. a0 Ω Nonlinear Anal., Vol. 30(1997)4619-4627. Ω From H¨ o lder inequality 2 2 a0 S2 − L1 a∞ [4] Chipot M. and Rodrigues J. F., On a class kun+1 − un k ≤ a20 S22 of nonlocal nonlinear problems, RAIRO 1/2 Mod´elisation Math. Anal. Num´er., Vol. L2 C1 kun+1 − un kkun − un−1 k. 2 26(1992)447-467. a S2 S 1 0
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[5] Corrˆea F.J.S.A., On positive solutions of nonlocal and nonvariational elliptic problems, Nonlinear Anal., 59(2004)11471155. [6] De Figueiredo D., Girardi M. and Matzeu M., Semilinear elliptic equations with dependence on the gradient via mountain pass techniques, Differential and Integral Equations, 17(2004)119-126.
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