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
https://ictp.acad.ro/gheorghiu/
https://ictp.acad.ro/gheorghiu/
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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1222-9024
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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.
Paper (preprint) in HTML form
ON THE EXISTENCE AND UNIQUENESS OF POSITIVE SOLUTIONS OF SOME MILDLY NONLINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS
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){[-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.
where Omega\Omega is a bounded domain in R^(n)R^{n} with boundary del Omega,c\partial \Omega, c and B^(2)B^{2} are positive (physical) constants and S(x) > 0,x in OmegaS(x)>0, x \in \Omega (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)=cu^(2)-B^(2)u],(1)\left[N(u)=c u^{2}-B^{2} u\right],(1) becomes
{[(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.
If Psi\Psi is a solution of (1),Psi inC^(2)(Omega)nn C( bar(Omega))(1), \Psi \in C^{2}(\Omega) \cap C(\bar{\Omega}), then Psi\Psi is a solution for the following nonlinear integral equation
{:(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*}
Conversely, if Psi\Psi is a solution of (3), psi in C( bar(Omega))\psi \in C(\bar{\Omega}), and is sufficiently regular, then Psi\Psi is a solution of (1). Here 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( bar(Omega))C(\bar{\Omega}), namely
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\}
where 0 <= R_(1) < R_(2)0 \leq R_{1}<R_{2}. Our solutions are bounded by R_(2)R_{2} (i.e. belong to the sphere B_(0,R_(2))sub C( bar(Omega))B_{0, R_{2}} \subset C(\bar{\Omega}) when R_(1)=0R_{1}=0 ) and they are positive when R_(1) > 0R_{1}>0.
In order to apply Schauder's fixed point principle we consider the operator
T:C( bar(Omega))rarr C( bar(Omega))T: C(\bar{\Omega}) \rightarrow C(\bar{\Omega})
defined by
{:(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*}
It is well known that TT is compact and continuous when C( bar(Omega))C(\bar{\Omega}), is the Banach space of continuous functions on bar(Omega)\bar{\Omega} 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}} This means
{:(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*}
Let
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 M
be some bounds of SS on bar(Omega)\bar{\Omega}. The following inequality holds
{:[(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*}
From a physical point of view, the most important case appears when
{:(7)R_(2) <= (B^(2))/(2c):}\begin{equation*}
R_{2} \leq \frac{B^{2}}{2 c} \tag{7}
\end{equation*}
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) and N(u(s))-S(s)N(u(s))-S(s), we have
By simple manipulations, and taking into account the sign of G(x,s)G(x, s) and N(u(s))-S(s)N(u(s))-S(s), we have
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
and
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
{:[" 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}
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} and R_(2)R_{2}.
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} and R_(2)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.
Thus, we get the following result:
THEOREM 1. Consider problem (1). If the parameters B^(2)B^{2} and cc, the domain Omega\Omega and the source term SS are such that the system of inequalities (7) and (8) has a solution R_(1),R_(2)R_{1}, R_{2} with 0 <= R_(1) < R_(2)0 \leq R_{1}<R_{2} then problem (1) has at least one continuous solution u:Omega rarr[R_(1),R_(2)]u: \Omega \rightarrow\left[R_{1}, R_{2}\right].
THEOREM 1. Consider problem (1). If the parameters B^(2)B^{2} and cc, the domain Omega\Omega and the source term SS are such that the system of inequalities (7) and (8) has a solution R_(1),R_(2)R_{1}, R_{2} with 0 <= R_(1) < R_(2)0 \leq R_{1}<R_{2} then problem (1) has at least one continuous solution u:Omega rarr[R_(1),R_(2)]u: \Omega \rightarrow\left[R_{1}, R_{2}\right].
Remark 1. If Omega\Omega is a sphere of radius rr in R^(2)\mathbb{R}^{2} centred at the origin, a simple computation gives us
alpha <= r((1)/(4)+ln 2).\alpha \leq r\left(\frac{1}{4}+\ln 2\right) .
3. UNIQUENESS
We have already got T:B_(R_(1),R_(2))rarrB_(R_(1),R_(2))T: B_{R_{1}, R_{2}} \rightarrow B_{R_{1}, R_{2}} where R_(1)R_{1} and R_(2)R_{2} are from the theorem above. All we need is TT to be a contraction.
We simply have
or
or
{:[|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}
where the norm is the Chebyshev norm corresponding to C( bar(Omega))C(\bar{\Omega}).
Thus, if in addition to (7) and (8) we have
Thus, if in addition to (7) and (8) we have
{:(9)2alpha cR_(2) < 1:}\begin{equation*}
2 \alpha c R_{2}<1 \tag{9}
\end{equation*}
the solution u:Omega rarr[R_(1),R_(2)]u: \Omega \rightarrow\left[R_{1}, R_{2}\right] of (1) is unique.
Remark 2. For R_(1)=0R_{1}=0, inequalities (7), (8) and (9) reduce to a system of simultaneous inequalities for R_(2)R_{2}.
Remark 2. For R_(1)=0R_{1}=0, inequalities (7), (8) and (9) reduce to a system of simultaneous inequalities for R_(2)R_{2}.
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 Omega\Omega lies within a slab (i.e. there are two parallel hyperplans at distance b-ab-a ). Suppose for all x=(x_(1),dots,x_(n))in Omegax=\left(x_{1}, \ldots, x_{n}\right) \in \Omega we have x_(1)in(a,b)x_{1} \in(a, b). Let u_(1),u_(2)inB_(R_(1),R_(2))u_{1}, u_{2} \in B_{R_{1}, R_{2}} be two solutions of (1). Then the following identity holds
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)=0
Consider now the problem
{:(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.
where 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) and hence 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}. We shall prove that (10) has a unique solution. Since both u_(1)-u_(2)u_{1}-u_{2} and 0 are solutions we get u_(1)=u_(2)u_{1}=u_{2}. In order to apply the generalized maximum principle we build up a function WW such that
{:(11)W(x) > 0","x in bar(Omega):}\begin{equation*}
W(x)>0, x \in \bar{\Omega} \tag{11}
\end{equation*}
{:(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*}
Let W(x)=1-mue^(gamma(x_(1)-a))W(x)=1-\mu e^{\gamma\left(x_{1}-a\right)} where mu\mu and gamma\gamma are to be specified later on. We estimate
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)
and select mu=(B^(2)-2cR_(1))//gamma^(2)\mu=\left(B^{2}-2 c R_{1}\right) / \gamma^{2}. Also we need to ensure W > 0W>0 and so we choose gamma\gamma such that
B^(2)-2cR_(1) < gamma^(2)e^(-gamma(b-a))B^{2}-2 c R_{1}<\gamma^{2} e^{-\gamma(b-a)}
The best value gamma\gamma is given by gamma=(b-a)//2\gamma=(b-a) / 2.
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We end up with the following:
THEOREM 2. If
THEOREM 2. If
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}}
then (1) has at most a solution in B_(R_(1),R_(2))B_{R_{1}, R_{2}}.
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.
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}, the second one, (13), is in term of R_(1)R_{1} and both depend on the geometry of Omega\Omega and the parameters of the problem. For specific values of physical parameters BB and cc and for a given domain Omega\Omega 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 NN (.) 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} where 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
- 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.
- Rus L.A., Fixed Points Principles, Ed. Dacia, 1979 (in Romanian).
- Petrila T., Gheorghiu, C.I., Finite Elements Method and Applications, Ed. Academiei, Bucureşti 1987 (in Romanian).
- Protter, H.M., Weinberger, H.F., Maximum Principles in Differential Equations, Prentice Hall, Inc. 1967.
- Kantorovici, L.V., Akilov, G.P., Functional Analysis (Romanian translation), Nauka ed. 1977.
- 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