Locating the peaks of least-energy solutions to a semilinear Neumann problem. (English) Zbl 0796.35056

Ni, Wei-Ming; Takagi, Izumi

Duke Math. J. 70, No. 2, 247-281 (1993).

We continue our study initiated in [C.-S. Lin, W.-M. Ni and I. Takagi, J. Differ. Equations 72, No. 1, 1-27 (1988; Zbl 0676.35030)] and [(*) W.-M. Ni and I. Takagi, Commun. Pure Appl. Math. 44, No. 7, 819-851 (1991; Zbl 0754.35042)] on the shape of certain solutions to a semilinear Neumann problem arising in mathematical models of biological pattern formation. Let \(\Omega\) be a bounded domain in \(\mathbb{R}^ N\) with smooth boundary \(\partial \Omega\) and let \(\nu\) be the unit outer normal to \(\partial \Omega\). We consider the Neumann problem for certain semilinear elliptic equations including \[ d\Delta u-u + u^ p=0\quad \text{ and } u>0 \text{ in } \Omega,\;\partial u/ \partial \nu=0 \text{ on } \partial \Omega, (BVP)_ d \] where \(d>0\) and \(p>1\) are constants. This problem is encountered in the study of steady-state solutions to some reaction-diffusion systems in chemotaxis as well as in morphogenesis.
Assume that \(p\) is subcritical, i.e., \(1<p<(N+2)/(N-2)\) when \(N \geq 3\) and \(1<p< + \infty\) when \(N=2\). Then we can apply the mountain-pass lemma to obtain a least-energy solution \(u_ d\) to \((BVP)_ d\), by which it is meant that \(u_ d\) has the smallest energy \(J_ d(u) = {1 \over 2} \int_ \Omega (d | \nabla u |^ 2 + u^ 2) dx-(p+1)^{-1} \int_ \Omega u_ +^{p+1} dx\), where \(u_ + = \max \{u,0\}\), among all the solutions to \((BVP)_ d\). It turns out that \(u_ d\equiv 1\) if \(d\) is sufficiently large, whereas \(u_ d\) exhibits a “point- condensation phenomenon” as \(d \downarrow 0\). More precisely, when \(d\) is sufficiently small, \(u_ d\) has only one local maximum over \(\overline{\Omega}\) (thus it is the global maximum), and the maximum is achieved at exactly one point \(P_ d\) on the boundary. Moreover, \(u_ d(x) \to 0\) as \(d \downarrow 0\) for all \(x \in \Omega\), while \(\max u_ d \geq 1\) for all \(d>0\). Hence, a natural question raised immediately is to ask where on the boundary the maximum point \(P_ d\) is situated, and it is the purpose of the present paper to answer this question. Indeed, we show that \(H(P_ d)\), the mean curvature of \(\partial \Omega\) at \(P_ d\), approaches the maximum of \(H(P)\) over \(\partial \Omega\) as \(d \downarrow 0\), as was announced in \((*)\).

MSC:

35J65	Nonlinear boundary value problems for linear elliptic equations
35J20	Variational methods for second-order elliptic equations

Keywords:

point-condensation phenomenon; semilinear Neumann problem; mountain-pass lemma; least-energy solution

Citations:

Zbl 0676.35030; Zbl 0754.35042

References:

[1]	C.-C. Chen and C. S. Lin, Uniqueness of the ground state solutions of \(\Delta u+f(u)=0\) in \(\mathbf R^ n,\;n\geq 3\) , Comm. Partial Differential Equations 16 (1991), no. 8-9, 1549-1572. · Zbl 0753.35034
[2]	B. Gidas, W.-M. Ni, and L. Nirenberg, Symmetry of positive solutions of nonlinear elliptic equations in \(\mathbf R\spn\) , Mathematical analysis and applications, Part A, Adv. in Math. Suppl. Stud., vol. 7, Academic Press, New York, 1981, pp. 369-402. · Zbl 0469.35052
[3]	D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of second order , Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. · Zbl 0562.35001
[4]	L. Hörmander, Estimates for translation invariant operators in \(L\spp\) spaces , Acta Math. 104 (1960), 93-140. · Zbl 0093.11402
[5]	A. I. Košelev, A priori estimates in \(L_p\) and generalized solutions of elliptic equations and systems , Six Papers on Partial Differential Equations, Amer. Math. Soc. Transl. (2), vol. 20, Amer. Math. Soc., Providence, 1962, pp. 105-171. · Zbl 0122.33702
[6]	M. K. Kwong and L. Q. Zhang, Uniqueness of the positive solution of \(\Delta u+f(u)=0\) in an annulus , Differential Integral Equations 4 (1991), no. 3, 583-599. · Zbl 0724.34023
[7]	C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system , J. Differential Equations 72 (1988), no. 1, 1-27. · Zbl 0676.35030
[8]	W.-M. Ni, X.-B. Pan, and I. Takagi, Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents , Duke Math. J. 67 (1992), no. 1, 1-20. · Zbl 0785.35041
[9]	W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem , Comm. Pure Appl. Math. 44 (1991), no. 7, 819-851. · Zbl 0754.35042
[10]	K. Yosida, Functional analysis , Grundlehren der Mathematischen Wissenschaften, vol. 123, Springer-Verlag, Berlin, 1978. · Zbl 0365.46001