NONLOCAL ELLIPTIC PROBLEMS

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1 EVOLUTION EQUATIONS: EXISTENCE, REGULARITY AND SINGULARITIES BANACH CENTER PUBLICATIONS, VOLUME 52 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES WARSZAWA 2 NONLOCAL ELLIPTIC PROBLEMS ANDRZE J KR ZYWICKI Mathematical Institute, University of Wroc law Pl. Grunwaldzki 2/4, Wroc law, Poland krzywick@math.uni.wroc.pl TADEUS Z NA DZIEJA Institute of Mathematics, Technical University of Zielona Góra Podgórna 5, Zielona Góra, Poland T.Nadzieja@im.pz.zgora.pl Abstract. Some conditions for the existence and uniqueness of solutions of the nonlocal elliptic problem ϕ = M f(ϕ) (, ϕ = are given. f(ϕ))p In this paper we study the following nonlocal elliptic problem: (1) ϕ = M f(ϕ) ( f(ϕ))p, (2) ϕ =. Here ϕ : IR is an unknown function from a bounded subdomain of IR n into IR, f : IR n IR + is a given function and M >, p > are real parameters. The physical motivations for the study of nonlocal elliptic problems come from statistical mechanics [A], [B], [BKN], [BN1], [BN2], [W], [S], theory of electrolytes [BN2], [BHN], and theory of thermistors [C], [L]. If the nonlinearity f(ϕ) has the form f(ϕ) = e ±ϕ and p = 1 then (1) is identical with the Poisson equation ϕ = ρ with density ρ of the Boltzmann form. In this case (1) is called the Poisson Boltzmann equation. If f(ϕ) = e ϕ (f(ϕ) = e ϕ ) ϕ can be interpreted as the gravitational (electric) potential of systems of particles in thermodynamical equilibrium interacting via gravitational (Coulomb) potential. In this interpretation, the parameter M, is the total mass (charge) of the particles of the system. 2 Mathematics Subject Classification: Primary 35J6. Key words and phrases: nonlinear nonlocal elliptic equations. Grant support from KBN (324/P3/97/12) is gratefully acknowledged. The paper is in final form and no version of it will be published elsewhere. [147]

2 148 A. KRZYWICKI AND T. NADZIEJA The Poisson Boltzmann equation arises also in investigations of phenomena associated with the occurrence of shear bands in metals being deformed under high strain rates [BT], and in modelling of turbulent behavior of flows [CLMP]. The general problem (1), (2) with given f(ϕ) and positive p appears in modelling of stationary temperature ϕ when an electric current flows through a material with temperature dependent electrical resistivity f(ϕ), subject to a fixed potential difference [L]. The main tool in the proof of the existence of a solution of (1), (2) is the technique of sub- (super-)solutions [BL], [BT], [T], variational methods [CLMP], [GL], and topological methods (Leray Schauder theory) [BHN], [KN1], [KN2]. The nonexistence results are a consequence of the Pokhožaev identity [BT], [KN2], or construction of some special subsolutions [BL]. The existence and uniqueness of solutions for the Poisson Boltzmann problem with f(ϕ) = e ϕ and arbitrary M > was proved in [GL] and [KN1]. In the first paper the variational and in the second topological methods were applied. When f has the form f(ϕ) = e ϕ, the solutions do not exist for large M and generally are not unique. Moreover, the existence and uniqueness depend on the geometry of the domain [KN2], [KN3]. When f(ϕ) = e ϕ and p > the problem (1), (2) was considered in [C] by using the technique of sub- and supersolutions, maximum principle and rearrangement method. Under the assumption + f(ϕ) d ϕ < +, the problem (1), (2) was investigated in [BL], [T]. We consider our problem in a bounded domain in IR n with the boundary of class C 1+ɛ. This assumption guarantees the existence of the Green function G(x, y) corresponding to and zero boundary data, satisfying the estimate (3) G(x, y) K x y 2 n, K xg(x, y) x y 1 n, with some constant K depending on the domain [GW]. To prove the existence of a solution of (1), (2) we shall use the Leray Schauder theorem. To do this we consider the family of integral equations G(x, y)f(ϕ(y)) dy (4) ϕ(x) = λ ( f(ϕ(y)), dy)p where λ is a parameter from [, M]. When λ = M the problem (4) is equivalent to (1), (2). It follows from the estimates (3) that the right hand side of (4) is a continuous, compact operator on the space of continuous functions C () with the uniform norm ϕ = sup ϕ. To apply the Leray Schauder theorem, a uniform for λ [, M] a priori estimate of solutions ϕ of (4) is needed. First of all, replacing M by M(f()) 1 p we may put f() = 1. Assume that f is decreasing. Then denoting z := ϕ we have z λγ() p (f(z)) p, where Γ() = sup x G(x, y) dy and is the volume of.

3 NONLOCAL ELLIPTIC PROBLEMS 149 Therefore z will be uniformly bounded for all λ [, M], if (5) lim z + z(f(z))p > MΓ() p. Hence Theorem 1. If f is continuous, positive and decreasing, then (1), (2) has a solution for M, f and p satisfing (5). In a similar way, for f positive, increasing we have from (4) Hence we get z λγ()f(z) p. Theorem 2. If f is continuous, positive and increasing, then (1), (2) has a solution for M, f and p satisfing z lim z + f(z) > MΓ() p. To get a stronger results, a more subtle estimate of f(ϕ) is needed. We start with a simple Lemma 1. Let f, F : IR IR be continuous functions, f positive, F nondecreasing. Then for any continuous function ϕ : IR f(ϕ)f (f(ϕ)) (6) f(ϕ) 1 F (f(ϕ)). Proof. The inequality (6) is equivalent to f(ϕ(x))f (f(ϕ(x))) dx dy f(ϕ(y))f (f(ϕ(x))) dx dy. The left hand side may be transformed to 1 (F (f(ϕ(x))) F (f(ϕ(y)))(f(ϕ(x)) f(ϕ(y))) dx dy, 2 which is nonnegative. It is clear that the assumed continuity of all functions may be relaxed, however only in this weak form the Lemma 1 will be used. Theorem 3. If f is a positive, decreasing differentiable function such that sup f /f < + and < p 1, then the problem (1), (2) has a unique solution for all M >. Proof. Applying the Jensen inequality we have ( ) 1 (7) exp ln f(ϕ) 1 f(ϕ). Using the Cauchy and Poincaré inequality, we have with some positive constant C ( 2 ln f(ϕ)) C (f (ϕ)/f(ϕ)) 2 ϕ 2 C sup f /f f (ϕ)/f(ϕ) ϕ 2.

4 15 A. KRZYWICKI AND T. NADZIEJA Now we multiply (1) by ln f(ϕ) and integrate over, what gives (f /f) ϕ 2 λ = ( λ f(ϕ) ln f(ϕ) f(ϕ))p ( f(ϕ))p 1 The last inequality follows from Lemma 1. Two inequalities above imply ( 1 p (8) ln f(ϕ) λc sup f /f f(ϕ)). From (8) ln f(ϕ) MC sup f /f 1 p, hence, using (4) and (7), we get ϕ MΓ() p exp(mcp sup f /f p ), which is the desired estimate. To prove the uniqueness, let ϕ i, i = 1, 2 satisfy (9) ϕ i = Mµ p i f(ϕ i), ϕ i =, ln f(ϕ). with µ i = ( f(ϕ i(x)) dx) 1. We distinguish two cases: µ = µ 1 = µ 2 and µ 1 µ 2. In the first case, we take the difference of two equations (9), multiply it by ϕ 1 ϕ 2 and integrate over, which gives (ϕ 1 ϕ 2 ) 2 = Mµ p (f(ϕ 1) f(ϕ 2 ))(ϕ 1 ϕ 2 ). The right hand side of the last equation is nonpositive, hence (ϕ 1 ϕ 2 ) =, what implies ϕ 1 = ϕ 2 in due to ϕ 1 = ϕ 2 on. Now let µ 1 > µ 2. First we show that ϕ 1 > ϕ 2 in. If not, there exists x such that ϕ 1 (x ) ϕ 2 (x ) and (ϕ 1 ϕ 2 )(x ), whereas the difference Mµ p 1 f(ϕ 1(x )) Mµ p 2 f(ϕ 2(x )) of the right hand side members of (9) at x is positive, a contradiction. We ϕ2 now have ϕ1 (9) over we get Mµ p 1 ϕ 1 = ϕ2 on. Thus along, which is a consequence of ϕ 1 ϕ 2. Moreover, integrating 1 = ϕ 1 ϕ 2 = Mµp 1 2. These two facts give us (1) (ϕ 1 ϕ 2 ) = at any boundary point. From Theorem 1. 2, Ch. IV in [LU] applied to (9) which guarantees that ϕ i C 2 () and from µ 1 > µ 2 it follows that near the boundary (ϕ 1 ϕ 2 ) <. Therefore zero is the minimal value of ϕ 1 ϕ 2 there. Due to the Hopf lemma (ϕ 1 ϕ 2 ) < along, contrary to (1). Consider now the special case when is a ball of radius 1 centered at the origin and ϕ is radially symmetric. Then the problem (1), (2) reads (11) (r n 1 ϕ ) = Mr n 1 µ p f(ϕ), ϕ () = ϕ(1) =, d dr =, where µ = µ ϕ = (σ n 1 rn 1 f(ϕ(r)) dr) 1 and σ n is the area of the unit sphere in IR n. Theorem 4. If f is decreasing, p 1, and M is such that (12) lim z + z(f(z))p 1 > Mσ p n n p 1, then the problem (11) has a solution.

5 NONLOCAL ELLIPTIC PROBLEMS 151 Proof. Integrating (11) twice and introducing the operator A λ 1 t ) A λ ϕ(r) = λµ (t p n 1 s n 1 f(ϕ(s)) ds dt we may replace (11) by (13) ϕ = A λ ϕ, λ [, M], r if we put λ = M. Consider the family of problems (13). The operator A λ is continuous on the space C [, 1]. Now r (15) (A λ ϕ(r)) = λr 1 n µ p s n 1 f(ϕ(s)) ds is uniformly bounded if ϕ belongs to a bounded subset of C [, 1]. Hence A λ is a compact operator. To apply the Leray-Schauder theorem, it is enough to prove a uniform a priori estimate of solutions ϕ of (13) for λ [, M]. First we show that ϕ <. In fact, coming back to the form (11) of (13) we see that r r n 1 ϕ (r) = λr n 1 µ p f(ϕ(r)) + (n 1)r 1 λµ p s n 1 f(ϕ(s)) ds < λr n 1 µ p f(ϕ(r)) + (n 1)r 1 λµ p f(ϕ(r)) r s n 1 ds <. Hence the minimum ϕ is attained at r = 1. Putting r = 1 in (15) we get ϕ (1) = λσn 1 µ p 1, so z = ϕ < Mσn 1 µ p 1. We have assumed p 1, therefore µ p 1 < (f(z)) 1 p n p 1 σn 1 p which implies z(f(z)) p 1 Mσn p n p 1. From (12) and the last inequality the desired a priori estimate follows. References [A] [B] [BL] [BT] [BHN] [BKN] [BN1] [BN2] J. J. Aly, Thermodynamics of a two-dimensional self-gravitating system, Physical Review E 49 (1994), F. Bavaud, Equilibrium properties of the Vlasov functional: the generalized Poisson Boltzmann Emden equation, Rev. Mod. Phys. 63 (1991), J. W. Bebernes and A. A. Lacey, Global existence and finite-time blow-up for a class of nonlocal parabolic problems, Adv. Diff. Equations 2 (1997), J. W. Bebernes and P. Talaga, Nonlocal problems modelling shear banding, Comm. Appl. Nonlinear Analysis 3 (1996), P. Biler, W. Hebisch and T. Nadzieja, The Debye system: existence and large time behavior of solutions, Nonlinear Analysis T.M.A. 23 (1994), P. Biler, A. Krzywicki and T. Nadzieja, Self-interaction of Brownian particles coupled with thermodynamic processes, Reports Math. Physics 42 (1998), P. Biler and T. Nadzieja, A class of nonlocal parabolic problems occurring in statistical mechanics, Colloq. Math. 66 (1993), P. Biler and T. Nadzieja, Nonlocal parabolic problems in statistical mechanics, Nonlinear Analysis T. M. A. 3 (1997),

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