On the existence and uniqueness of positive solutions of some mildly nonlinear elliptic boundary value problems

Călin-Ioan Gheorghiu
(original), Fixed point theory, Numerical Analysis, paper, PDEs (numerical)

Abstract

For a homogeneous Dirichlet problem attached to a semilinear elliptic equation we study the existence and uniqueness of non-negative solutions. Our analysis is based on  straightforward use of Schauder’s fixed point principle for existence and comparatively, based on the Banach’s contraction principle and on a generalized maximum principle for the uniqueness. We have obtained two independent conditions for uniqueness of solution. Both depend on the geometry of the domain as well as on the parameters of the problem.

Authors

C.I. Gheorghiu
Tiberiu Popoviciu Institute of Numerical Analysis

Al. Tămăşan
Babeş-Bolyai University, Cluj-Napoca, Romania

Keywords

boundary value problem; elliptic; quadratic nonlinearity; non-negative solution; existence; uniqueness; Schauder fixed point; Banach contraction; generalized maximum ;

References

See the expanding block below.

Cite this paper as

C.I. Gheorghiu, Al. Tămăşan, On the existence and uniqueness of positive solutions of some mildly nonlinear elliptic boundary value problems, Rev. Anal. Numér. Théor. Approx. 24 (1995), pp. 125-129.

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Rev. Anal. Numér. Théor. Approx.

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Editions de l’Academie Roumaine

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https://ictp.acad.ro/jnaat/journal/article/view/1995-vol24-nos1-2-art13

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1222-9024

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2457-8126

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[1] Shampine, L.F., Wing, G.M., Existence and Uniqueness of Solutions of a Class olf Nonlinear Elliptic Boundary Value Problems, J. Math. Mech., 19 (1970), pp. 971-979.

[2] Rus, I.A., Fixed Points Principles, Ed. Dacia, 1979 (in Romanian).

[3] Petrila T., Gheorghiu, C.I., Finite Elements Method and Applications, Ed. Academiei, Bucureşti 1987 (in Romanian).

[4] Protter, H.M., Weinberger, H.F., Maximum Principles in Differential Equations, Prentice Hall, Inc. 1967.

[5] Kantorovici, L.V., Akilov, G.P., Functional analysis (Romanian translation), Nauka ed. 1977.

[6] Berger, M.S., Fraenkel, L.E., On the Asymptotic Solution of a Nonlinear Dirichlet Problem, J. Math. Mech., 19 (1970), pp. 553-585.

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ON THE EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS OF SOME MILDLY NONLINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

C.I. GHEORGHIU, AL. TĂMĂŞAN(Cluj-Napoca)

1. INTRODUCTION

The aim of this paper is to find some simple criteria of the existence and uniqueness of the non-negative solutions of the following problem
(1) { − Δ u + c u 2 − B 2 u = S ( x ) , x ∈ Ω u = 0 , x ∈ ∂ Ω (1) − Δ u + c u 2 − B 2 u = S ( x ) , x ∈ Ω u = 0 , x ∈ ∂ Ω {:(1){[-Delta u+cu^(2)-B^(2)u=S(x)","x in Omega],[u=0","x in del Omega]:}:}\left\{\begin{array}{r} -\Delta u+c u^{2}-B^{2} u=S(x), x \in \Omega \tag{1}\\ u=0, x \in \partial \Omega \end{array}\right.(1){−Δu+cu2−B2u=S(x),x∈Ωu=0,x∈∂Ω
where Ω Ω Omega\OmegaΩ is a bounded domain in R n R n R^(n)R^{n}Rn with boundary ∂ Ω , c ∂ Ω , c del Omega,c\partial \Omega, c∂Ω,c and B 2 B 2 B^(2)B^{2}B2 are positive (physical) constants and S ( x ) > 0 , x ∈ Ω S ( x ) > 0 , x ∈ Ω S(x) > 0,x in OmegaS(x)>0, x \in \OmegaS(x)>0,x∈Ω (the source term) is a given function.
We shall use a straightforward analysis based on Schauder's fixed point principle for existence and comparatively, based on the Banach's contraction principle and on a generalized maximum principle for the uniqueness.
This problem is also considered in [1]. Our analysis is simpler that that and consists in finding bounded solutions as well as positive solutions. Similar problems are also available in [2]. The integral formulation we will carry out opens the possibility of a numerical approximation [3]. Problems like (1) comes from nonlinear diffusion theory and some physical insight into these problems could be found, for example, in [1].

2. EXISTENCE

For the sake of simplicity, let us denote the nonlinear (quadratic) term in the left hand side of (1) by N() .
With that [ N ( u ) = c u 2 − B 2 u ] , ( 1 ) N ( u ) = c u 2 − B 2 u , ( 1 ) [N(u)=cu^(2)-B^(2)u],(1)\left[N(u)=c u^{2}-B^{2} u\right],(1)[N(u)=cu2−B2u],(1) becomes
{ (2) Δ Ψ = N ( Ψ ) − S ( x ) , x ∈ Ω Ψ = 0 , x ∈ ∂ Ω . (2) Δ Ψ = N ( Ψ ) − S ( x ) , x ∈ Ω Ψ = 0 , x ∈ ∂ Ω . {[(2)Delta Psi=N(Psi)-S(x)","quad x in Omega],[Psi=0","quad x in del Omega.]:}\left\{\begin{align*} \Delta \Psi & =N(\Psi)-S(x), \quad x \in \Omega \tag{2}\\ \Psi & =0, \quad x \in \partial \Omega . \end{align*}\right.{(2)ΔΨ=N(Ψ)−S(x),x∈ΩΨ=0,x∈∂Ω.
If Ψ Ψ Psi\PsiΨ is a solution of ( 1 ) , Ψ ∈ C 2 ( Ω ) ∩ C ( Ω ¯ ) ( 1 ) , Ψ ∈ C 2 ( Ω ) ∩ C ( Ω ¯ ) (1),Psi inC^(2)(Omega)nn C( bar(Omega))(1), \Psi \in C^{2}(\Omega) \cap C(\bar{\Omega})(1),Ψ∈C2(Ω)∩C(Ω¯), then Ψ Ψ Psi\PsiΨ is a solution for the following nonlinear integral equation
(3) Ψ ( x ) = − ∫ Ω G ( x , s ) [ N ( Ψ ( s ) ) − S ( s ) ] d s (3) Ψ ( x ) = − ∫ Ω   G ( x , s ) [ N ( Ψ ( s ) ) − S ( s ) ] d s {:(3)Psi(x)=-int_(Omega)G(x","s)[N(Psi(s))-S(s)]ds:}\begin{equation*} \Psi(x)=-\int_{\Omega} G(x, s)[N(\Psi(s))-S(s)] d s \tag{3} \end{equation*}(3)Ψ(x)=−∫ΩG(x,s)[N(Ψ(s))−S(s)]ds
Conversely, if Ψ Ψ Psi\PsiΨ is a solution of (3), ψ ∈ C ( Ω ¯ ) ψ ∈ C ( Ω ¯ ) psi in C( bar(Omega))\psi \in C(\bar{\Omega})ψ∈C(Ω¯), and is sufficiently regular, then Ψ Ψ Psi\PsiΨ is a solution of (1). Here G ( x , s ) G ( x , s ) G(x,s)G(x, s)G(x,s) is Green's function for Laplace's operator. The integral formulation (3) comes usually by applying the boundary condition in the integral representation of the solution of Poisson's equation. We are looking for a solution of (3) in a subset of C ( Ω ¯ ) C ( Ω ¯ ) C( bar(Omega))C(\bar{\Omega})C(Ω¯), namely
B R 1 , R 2 = { u ∈ C ( Ω ¯ ) : R 1 ≤ u ( x ) ≤ R 2 , x ∈ Ω } B R 1 , R 2 = u ∈ C ( Ω ¯ ) : R 1 ≤ u ( x ) ≤ R 2 , x ∈ Ω B_(R_(1),R_(2))={u in C(( bar(Omega))):R_(1) <= u(x) <= R_(2),x in Omega}B_{R_{1}, R_{2}}=\left\{u \in C(\bar{\Omega}): R_{1} \leq u(x) \leq R_{2}, x \in \Omega\right\}BR1,R2={u∈C(Ω¯):R1≤u(x)≤R2,x∈Ω}
where 0 ≤ R 1 < R 2 0 ≤ R 1 < R 2 0 <= R_(1) < R_(2)0 \leq R_{1}<R_{2}0≤R1<R2. Our solutions are bounded by R 2 R 2 R_(2)R_{2}R2 (i.e. belong to the sphere B 0 , R 2 ⊂ C ( Ω ¯ ) B 0 , R 2 ⊂ C ( Ω ¯ ) B_(0,R_(2))sub C( bar(Omega))B_{0, R_{2}} \subset C(\bar{\Omega})B0,R2⊂C(Ω¯) when R 1 = 0 R 1 = 0 R_(1)=0R_{1}=0R1=0 ) and they are positive when R 1 > 0 R 1 > 0 R_(1) > 0R_{1}>0R1>0.
In order to apply Schauder's fixed point principle we consider the operator
T : C ( Ω ¯ ) → C ( Ω ¯ ) T : C ( Ω ¯ ) → C ( Ω ¯ ) T:C( bar(Omega))rarr C( bar(Omega))T: C(\bar{\Omega}) \rightarrow C(\bar{\Omega})T:C(Ω¯)→C(Ω¯)
defined by
(4) T ( u ) ( x ) = − ∫ Ω G ( x , s ) [ N ( u ( s ) ) − S ( s ) ] d s (4) T ( u ) ( x ) = − ∫ Ω   G ( x , s ) [ N ( u ( s ) ) − S ( s ) ] d s {:(4)T(u)(x)=-int_(Omega)G(x","s)[N(u(s))-S(s)]ds:}\begin{equation*} T(u)(x)=-\int_{\Omega} G(x, s)[N(u(s))-S(s)] \mathrm{d} s \tag{4} \end{equation*}(4)T(u)(x)=−∫ΩG(x,s)[N(u(s))−S(s)]ds
It is well known that T T TTT is compact and continuous when C ( Ω ¯ ) C ( Ω ¯ ) C( bar(Omega))C(\bar{\Omega})C(Ω¯), is the Banach space of continuous functions on Ω ¯ Ω ¯ bar(Omega)\bar{\Omega}Ω¯ with the topology of uniform convergence. The only thing which concerns us is the invariance of B R 1 , R 2 B R 1 , R 2 B_(R_(1),R_(2))B_{R_{1}, R_{2}}BR1,R2 This means
(5) T ( u ) ∈ B R 1 , R 2 for all u ∈ B R 1 , R 2 . (5) T ( u ) ∈ B R 1 , R 2  for all  u ∈ B R 1 , R 2 . {:(5)T(u)inB_(R_(1),R_(2))" for all "u inB_(R_(1),R_(2)).:}\begin{equation*} T(u) \in B_{R_{1}, R_{2}} \text { for all } u \in B_{R_{1}, R_{2}} . \tag{5} \end{equation*}(5)T(u)∈BR1,R2 for all u∈BR1,R2.
Let
m ≤ inf x ∈ Ω S ( x ) and M ≥ sup x ∈ Ω S ( x ) , 0 < m ≤ M m ≤ inf x ∈ Ω   S ( x )  and  M ≥ sup x ∈ Ω   S ( x ) , 0 < m ≤ M m <= i n f_(x in Omega)S(x)quad" and "quad M >= s u p_(x in Omega)S(x),0 < m <= Mm \leq \inf _{x \in \Omega} S(x) \quad \text { and } \quad M \geq \sup _{x \in \Omega} S(x), 0<m \leq Mm≤infx∈ΩS(x) and M≥supx∈ΩS(x),0<m≤M
be some bounds of S S SSS on Ω ¯ Ω ¯ bar(Omega)\bar{\Omega}Ω¯. The following inequality holds
(6) ∫ Ω G ( x , s ) [ N ( u ( s ) ) − M ] d s ≤ ∫ Ω G ( x , s ) [ N ( u ( s ) ) − S ( s ) ] d s ≤ ≤ ∫ Ω G ( x , s ) [ N ( u ( s ) ) − m ] d s , ∀ u ∈ C ( Ω ¯ ) , ∀ x ∈ Ω . (6) ∫ Ω   G ( x , s ) [ N ( u ( s ) ) − M ] d s ≤ ∫ Ω   G ( x , s ) [ N ( u ( s ) ) − S ( s ) ] d s ≤ ≤ ∫ Ω   G ( x , s ) [ N ( u ( s ) ) − m ] d s , ∀ u ∈ C ( Ω ¯ ) , ∀ x ∈ Ω . {:[(6)int_(Omega)G(x","s)[N(u(s))-M]ds <= int_(Omega)G(x","s)[N(u(s))-S(s)]ds <= ],[ <= int_(Omega)G(x","s)[N(u(s))-m]ds","quad AA u in C( bar(Omega))","quad AA x in Omega.]:}\begin{align*} & \int_{\Omega} G(x, s)[N(u(s))-M] \mathrm{d} s \leq \int_{\Omega} G(x, s)[N(u(s))-S(s)] \mathrm{d} s \leq \tag{6}\\ & \leq \int_{\Omega} G(x, s)[N(u(s))-m] \mathrm{d} s, \quad \forall u \in C(\bar{\Omega}), \quad \forall x \in \Omega . \end{align*}(6)∫ΩG(x,s)[N(u(s))−M]ds≤∫ΩG(x,s)[N(u(s))−S(s)]ds≤≤∫ΩG(x,s)[N(u(s))−m]ds,∀u∈C(Ω¯),∀x∈Ω.
From a physical point of view, the most important case appears when
(7) R 2 ≤ B 2 2 c (7) R 2 ≤ B 2 2 c {:(7)R_(2) <= (B^(2))/(2c):}\begin{equation*} R_{2} \leq \frac{B^{2}}{2 c} \tag{7} \end{equation*}(7)R2≤B22c
This will be the case we are dealing with.
By simple manipulations, and taking into account the sign of G ( x , s ) G ( x , s ) G(x,s)G(x, s)G(x,s) and N ( u ( s ) ) − S ( s ) N ( u ( s ) ) − S ( s ) N(u(s))-S(s)N(u(s))-S(s)N(u(s))−S(s), we have
∫ Ω G ( x , s ) [ N ( u ( s ) ) − m ] d s ≤ [ N ( R 1 ) − m ] min x ∈ Ω ∫ Ω G ( x , s ) d s ∫ Ω   G ( x , s ) [ N ( u ( s ) ) − m ] d s ≤ N R 1 − m min x ∈ Ω   ∫ Ω   G ( x , s ) d s int_(Omega)G(x,s)[N(u(s))-m]ds <= [N(R_(1))-m]min_(x in Omega)int_(Omega)G(x,s)ds\int_{\Omega} G(x, s)[N(u(s))-m] \mathrm{d} s \leq\left[N\left(R_{1}\right)-m\right] \min _{x \in \Omega} \int_{\Omega} G(x, s) \mathrm{d} s∫ΩG(x,s)[N(u(s))−m]ds≤[N(R1)−m]minx∈Ω∫ΩG(x,s)ds
and
∫ Ω G ( x , s ) [ N ( u ( s ) ) − M ] d s ≥ [ N ( R 2 ) − M ] max x ∈ Ω ∫ Ω G ( x , s ) d s ∫ Ω   G ( x , s ) [ N ( u ( s ) ) − M ] d s ≥ N R 2 − M max x ∈ Ω   ∫ Ω   G ( x , s ) d s int_(Omega)G(x,s)[N(u(s))-M]ds >= [N(R_(2))-M]max_(x in Omega)int_(Omega)G(x,s)ds\int_{\Omega} G(x, s)[N(u(s))-M] \mathrm{d} s \geq\left[N\left(R_{2}\right)-M\right] \max _{x \in \Omega} \int_{\Omega} G(x, s) \mathrm{d} s∫ΩG(x,s)[N(u(s))−M]ds≥[N(R2)−M]maxx∈Ω∫ΩG(x,s)ds
Let us denote by α = max x ∈ Ω ∫ Ω G ( x , s ) d s and by β = min x ∈ Ω Ω ∫ Ω G ( x , s ) d s and ob- serve that α > 0 and β ≥ 0 .  Let us denote by  α = max x ∈ Ω   ∫ Ω   G ( x , s ) d s  and by  β = min x ∈ Ω Ω   ∫ Ω   G ( x , s ) d s  and ob-   serve that  α > 0  and  β ≥ 0 .  {:[" Let us denote by "alpha=max_(x in Omega)int_(Omega)G(x","s)ds" and by "beta=min_(x inOmega_(Omega))int_(Omega)G(x","s)ds" and ob- "],[" serve that "alpha > 0" and "beta >= 0". "]:}\begin{aligned} & \text { Let us denote by } \alpha=\max _{x \in \Omega} \int_{\Omega} G(x, s) \mathrm{d} s \text { and by } \beta=\min _{x \in \Omega_{\Omega}} \int_{\Omega} G(x, s) \mathrm{d} s \text { and ob- } \\ & \text { serve that } \alpha>0 \text { and } \beta \geq 0 \text {. }\end{aligned} Let us denote by α=maxx∈Ω∫ΩG(x,s)ds and by β=minx∈ΩΩ∫ΩG(x,s)ds and ob-  serve that α>0 and β≥0. 
In order to fulfill (5), with inequality (6) coupled with the last two inequalities, we are led to the following system for R 1 R 1 R_(1)R_{1}R1 and R 2 R 2 R_(2)R_{2}R2.
(8) { β [ N ( R 1 ) − m ] ≤ − R 1 α [ N ( R 2 ) − M ] ≥ − R 2 (8) β N R 1 − m ≤ − R 1 α N R 2 − M ≥ − R 2 {:(8){[beta[N(R_(1))-m] <= -R_(1)],[alpha[N(R_(2))-M] >= -R_(2)]:}:}\left\{\begin{array}{l} \beta\left[N\left(R_{1}\right)-m\right] \leq-R_{1} \tag{8}\\ \alpha\left[N\left(R_{2}\right)-M\right] \geq-R_{2} \end{array}\right.(8){β[N(R1)−m]≤−R1α[N(R2)−M]≥−R2
Thus, we get the following result:
THEOREM 1. Consider problem (1). If the parameters B 2 B 2 B^(2)B^{2}B2 and c c ccc, the domain Ω Ω Omega\OmegaΩ and the source term S S SSS are such that the system of inequalities (7) and (8) has a solution R 1 , R 2 R 1 , R 2 R_(1),R_(2)R_{1}, R_{2}R1,R2 with 0 ≤ R 1 < R 2 0 ≤ R 1 < R 2 0 <= R_(1) < R_(2)0 \leq R_{1}<R_{2}0≤R1<R2 then problem (1) has at least one continuous solution u : Ω → [ R 1 , R 2 ] u : Ω → R 1 , R 2 u:Omega rarr[R_(1),R_(2)]u: \Omega \rightarrow\left[R_{1}, R_{2}\right]u:Ω→[R1,R2].
Remark 1. If Ω Ω Omega\OmegaΩ is a sphere of radius r r rrr in R 2 R 2 R^(2)\mathbb{R}^{2}R2 centred at the origin, a simple computation gives us
α ≤ r ( 1 4 + ln ⁡ 2 ) . α ≤ r 1 4 + ln ⁡ 2 . alpha <= r((1)/(4)+ln 2).\alpha \leq r\left(\frac{1}{4}+\ln 2\right) .α≤r(14+ln⁡2).

3. UNIQUENESS

We have already got T : B R 1 , R 2 → B R 1 , R 2 T : B R 1 , R 2 → B R 1 , R 2 T:B_(R_(1),R_(2))rarrB_(R_(1),R_(2))T: B_{R_{1}, R_{2}} \rightarrow B_{R_{1}, R_{2}}T:BR1,R2→BR1,R2 where R 1 R 1 R_(1)R_{1}R1 and R 2 R 2 R_(2)R_{2}R2 are from the theorem above. All we need is T T TTT to be a contraction.
We simply have
or
| T ( u ) ( x ) − T ( v ) ( x ) | ≤ ∫ Ω G ( x , s ) | N ( u ( s ) ) − N ( v ( s ) ) | d s ‖ T ( u ) − T ( v ) ‖ ≤ 2 c α R 2 ‖ u − v ‖ | T ( u ) ( x ) − T ( v ) ( x ) | ≤ ∫ Ω   G ( x , s ) | N ( u ( s ) ) − N ( v ( s ) ) | d s ‖ T ( u ) − T ( v ) ‖ ≤ 2 c α R 2 ‖ u − v ‖ {:[|T(u)(x)-T(v)(x)| <= int_(Omega)G(x","s)|N(u(s))-N(v(s))|ds],[||T(u)-T(v)|| <= 2c alphaR_(2)||u-v||]:}\begin{gathered} |T(u)(x)-T(v)(x)| \leq \int_{\Omega} G(x, s)|N(u(s))-N(v(s))| \mathrm{d} s \\ \|T(u)-T(v)\| \leq 2 c \alpha R_{2}\|u-v\| \end{gathered}|T(u)(x)−T(v)(x)|≤∫ΩG(x,s)|N(u(s))−N(v(s))|ds‖T(u)−T(v)‖≤2cαR2‖u−v‖
where the norm is the Chebyshev norm corresponding to C ( Ω ¯ ) C ( Ω ¯ ) C( bar(Omega))C(\bar{\Omega})C(Ω¯).
Thus, if in addition to (7) and (8) we have
(9) 2 α c R 2 < 1 (9) 2 α c R 2 < 1 {:(9)2alpha cR_(2) < 1:}\begin{equation*} 2 \alpha c R_{2}<1 \tag{9} \end{equation*}(9)2αcR2<1
the solution u : Ω → [ R 1 , R 2 ] u : Ω → R 1 , R 2 u:Omega rarr[R_(1),R_(2)]u: \Omega \rightarrow\left[R_{1}, R_{2}\right]u:Ω→[R1,R2] of (1) is unique.
Remark 2. For R 1 = 0 R 1 = 0 R_(1)=0R_{1}=0R1=0, inequalities (7), (8) and (9) reduce to a system of simultaneous inequalities for R 2 R 2 R_(2)R_{2}R2.
In the remaining part of this paper we try to get a generalized maximum principle. The uniqueness problem by a generalized maximum principle (see [4], p. 73) needs the assumption that our domain Ω Ω Omega\OmegaΩ lies within a slab (i.e. there are two parallel hyperplans at distance b − a b − a b-ab-ab−a ). Suppose for all x = ( x 1 , … , x n ) ∈ Ω x = x 1 , … , x n ∈ Ω x=(x_(1),dots,x_(n))in Omegax=\left(x_{1}, \ldots, x_{n}\right) \in \Omegax=(x1,…,xn)∈Ω we have x 1 ∈ ( a , b ) x 1 ∈ ( a , b ) x_(1)in(a,b)x_{1} \in(a, b)x1∈(a,b). Let u 1 , u 2 ∈ B R 1 , R 2 u 1 , u 2 ∈ B R 1 , R 2 u_(1),u_(2)inB_(R_(1),R_(2))u_{1}, u_{2} \in B_{R_{1}, R_{2}}u1,u2∈BR1,R2 be two solutions of (1). Then the following identity holds
Δ ( u 1 − u 2 ) − [ c ( u 1 + u 2 ) − B 2 ] ( u 1 − u 2 ) = 0 Δ u 1 − u 2 − c u 1 + u 2 − B 2 u 1 − u 2 = 0 Delta(u_(1)-u_(2))-[c(u_(1)+u_(2))-B^(2)](u_(1)-u_(2))=0\Delta\left(u_{1}-u_{2}\right)-\left[c\left(u_{1}+u_{2}\right)-B^{2}\right]\left(u_{1}-u_{2}\right)=0Δ(u1−u2)−[c(u1+u2)−B2](u1−u2)=0
Consider now the problem
(10) { Δ u + h ( x ) u = 0 , x ∈ Ω u = 0 , x ∈ ∂ Ω (10) Δ u + h ( x ) u = 0 , x ∈ Ω u = 0 , x ∈ ∂ Ω {:(10){[Delta u+h(x)u=0","x in Omega],[u=0","x in del Omega]:}:}\left\{\begin{array}{r} \Delta u+h(x) u=0, x \in \Omega \tag{10}\\ u=0, x \in \partial \Omega \end{array}\right.(10){Δu+h(x)u=0,x∈Ωu=0,x∈∂Ω
where h ( x ) = B 2 − c ( u 1 ( x ) + u 2 ( x ) ) h ( x ) = B 2 − c u 1 ( x ) + u 2 ( x ) h(x)=B^(2)-c(u_(1)(x)+u_(2)(x))h(\mathrm{x})=B^{2}-c\left(u_{1}(x)+u_{2}(x)\right)h(x)=B2−c(u1(x)+u2(x)) and hence 0 ≤ h ( x ) ≤ B 2 − 2 c R 1 , x ∈ Ω ¯ 0 ≤ h ( x ) ≤ B 2 − 2 c R 1 , x ∈ Ω ¯ 0 <= h(x) <= B^(2)-2cR_(1),x in bar(Omega)0 \leq h(x) \leq B^{2}-2 c R_{1}, x \in \bar{\Omega}0≤h(x)≤B2−2cR1,x∈Ω¯. We shall prove that (10) has a unique solution. Since both u 1 − u 2 u 1 − u 2 u_(1)-u_(2)u_{1}-u_{2}u1−u2 and 0 are solutions we get u 1 = u 2 u 1 = u 2 u_(1)=u_(2)u_{1}=u_{2}u1=u2. In order to apply the generalized maximum principle we build up a function W W WWW such that
(11) W ( x ) > 0 , x ∈ Ω ¯ (11) W ( x ) > 0 , x ∈ Ω ¯ {:(11)W(x) > 0","x in bar(Omega):}\begin{equation*} W(x)>0, x \in \bar{\Omega} \tag{11} \end{equation*}(11)W(x)>0,x∈Ω¯
(12) Δ W + h ( x ) W ≤ 0 , x ∈ Ω . (12) Δ W + h ( x ) W ≤ 0 , x ∈ Ω . {:(12)Delta W+h(x)W <= 0","quad x in Omega.:}\begin{equation*} \Delta W+h(x) W \leq 0, \quad x \in \Omega . \tag{12} \end{equation*}(12)ΔW+h(x)W≤0,x∈Ω.
Let W ( x ) = 1 − μ e γ ( x 1 − a ) W ( x ) = 1 − μ e γ x 1 − a W(x)=1-mue^(gamma(x_(1)-a))W(x)=1-\mu e^{\gamma\left(x_{1}-a\right)}W(x)=1−μeγ(x1−a) where μ μ mu\muμ and γ γ gamma\gammaγ are to be specified later on. We estimate
Δ W + h ( x ) W ≤ − μ γ 2 e γ ( x 1 − a ) + ( B 2 − 2 c R 1 ) Δ W + h ( x ) W ≤ − μ γ 2 e γ x 1 − a + B 2 − 2 c R 1 Delta W+h(x)W <= -mugamma^(2)e^(gamma(x_(1)-a))+(B^(2)-2cR_(1))\Delta W+h(x) W \leq-\mu \gamma^{2} e^{\gamma\left(x_{1}-a\right)}+\left(B^{2}-2 c R_{1}\right)ΔW+h(x)W≤−μγ2eγ(x1−a)+(B2−2cR1)
and select μ = ( B 2 − 2 c R 1 ) / γ 2 μ = B 2 − 2 c R 1 / γ 2 mu=(B^(2)-2cR_(1))//gamma^(2)\mu=\left(B^{2}-2 c R_{1}\right) / \gamma^{2}μ=(B2−2cR1)/γ2. Also we need to ensure W > 0 W > 0 W > 0W>0W>0 and so we choose γ γ gamma\gammaγ such that
B 2 − 2 c R 1 < γ 2 e − γ ( b − a ) B 2 − 2 c R 1 < γ 2 e − γ ( b − a ) B^(2)-2cR_(1) < gamma^(2)e^(-gamma(b-a))B^{2}-2 c R_{1}<\gamma^{2} e^{-\gamma(b-a)}B2−2cR1<γ2e−γ(b−a)
The best value γ γ gamma\gammaγ is given by γ = ( b − a ) / 2 γ = ( b − a ) / 2 gamma=(b-a)//2\gamma=(b-a) / 2γ=(b−a)/2.
Academia Română
Institutul de Calcul „Tiberiu Popoviciu"
P.O. Box 68
3400 Cluj-Napoca I
România
,,Babes-Bolyai" University Faculty of Mathematics
1, M. Kogălniceanu str.
3400 Cluj-Napoca
Romania
We end up with the following:
THEOREM 2. If
B 2 − 2 c R 1 < ( b − a 2 ) 2 e − 2 ( b − a 2 ) 2 B 2 − 2 c R 1 < b − a 2 2 e − 2 b − a 2 2 B^(2)-2cR_(1) < ((b-a)/(2))^(2)e^(-2((b-a)/(2))^(2))B^{2}-2 c R_{1}<\left(\frac{b-a}{2}\right)^{2} e^{-2\left(\frac{b-a}{2}\right)^{2}}B2−2cR1<(b−a2)2e−2(b−a2)2
then (1) has at most a solution in B R 1 , R 2 B R 1 , R 2 B_(R_(1),R_(2))B_{R_{1}, R_{2}}BR1,R2.
Remark 3. The continuous dependence of the solution of (1) on the data follows immediately from [5] p. 366. There one can find some more general results on the continuous dependence.

CONCLUDING REMARKS

We have obtained two independent conditions for uniqueness of the solution of (1). The first one, (9), is in term of R 2 R 2 R_(2)R_{2}R2, the second one, (13), is in term of R 1 R 1 R_(1)R_{1}R1 and both depend on the geometry of Ω Ω Omega\OmegaΩ and the parameters of the problem. For specific values of physical parameters B B BBB and c c ccc and for a given domain Ω Ω Omega\OmegaΩ these could be compared. Another important topic on problem (1) is bifurcation. We have already obtained some results and it is hoped that this matter can be investigated in a deep manner. The treatment of the case when the nonlinear term is a third order polynomial does not differ in any essential respect from that for N N NNN (.) quadratic. In [6], for example, the authors consider a similar case with N ( u ) = u − g 2 ( x ) u 3 N ( u ) = u − g 2 ( x ) u 3 N(u)=u-g^(2)(x)u^(3)N(u)=u-g^{2}(x) u^{3}N(u)=u−g2(x)u3 where g ( x ) g ( x ) g(x)g(x)g(x) is a given function, but their analysis is much more complicated.
Aknowledgements. We wish to thank Prof. I.A. Rus for many valuable discussions during the preparation of this paper.

REFERENCES

  1. Shampine, L.F., Wing, G.M., Existence and Uniqueness of Solutions of a Class of Nonlinear Elliptic Boundary Value Problems, J. Math. Mech., 19 (1970), 971-979.
  2. Rus L.A., Fixed Points Principles, Ed. Dacia, 1979 (in Romanian).
  3. Petrila T., Gheorghiu, C.I., Finite Elements Method and Applications, Ed. Academiei, Bucureşti 1987 (in Romanian).
  4. Protter, H.M., Weinberger, H.F., Maximum Principles in Differential Equations, Prentice Hall, Inc. 1967.
  5. Kantorovici, L.V., Akilov, G.P., Functional Analysis (Romanian translation), Nauka ed. 1977.
  6. Berger, M.S., Fraenkel, L.E., On the Asymptotic Solution of a Nonlinear Dirichlet Problem, J. Math. Mech., 19 (1970), 553-585.
Received 20 XII 1994
1995

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