Weak finite‐dimensional approximations of semi‐linear elliptic PDEs with near‐critical exponents

Article type: Research Article

Authors: Budd, C.J. | Humphries, A.R.

Affiliations: Department of Mathematics, University of Bath, Claverton Down, Bath, BA2 7AY, UK | School of Mathematical Sciences, University of Sussex, Brighton, BN1 9QH, UK

Abstract: We make a formal study of the differential equation \begin{equation} u_{rr} + \frac{2}{r} u_r + \lambda u + u^{5+\varepsilon} = 0,\quad u_r(0) = u(1) = 0,\quad u > 0\quad \hbox{if}\ 0\mathrel{\hbox{{\char"36}}}r<1, \end{equation} when posed as a variational problem over a finite‐dimensional subset S_h of H^1_0 comprising piecewise‐linear functions defined on a mesh of size h. We determine critical points U_h \in S_h of the variational form of (1). Such functions are perturbations of u when a solution of (1) exists, but we show that U_h can also exist when (1) has no solution and we determine an asymptotic expression for the solution branch (\lambda,U_h) when \Vert U_h\Vert _{\infty} is large and h \Vert U_h\Vert _{\infty}^2 is small. If \varepsilon = 0, then u exists if \lambda > \mbox{{\char"19}}^2/4, and we give a formula expressing U_h as a perturbation of u. If \lambda \mathrel{\hbox{{\char"36}}}\mbox{{\char"19}}^2/4, then a solution of the differential equation does not exist, and U_h grows as h \rightarrow 0. We show that the rate of growth is proportional to h^{-1/4} if \lambda = \mbox{{\char"19}}^2/4, and h^{-1/3} if \lambda = 0. We compare these results with estimates for the solutions of (1) when \varepsilon \rightarrow 0^-. Our results are obtained by using formal asymptotic methods – particulary the method of matched asymptotic expansions – and are supported by some numerical calculations.

Journal: Asymptotic Analysis, vol. 17, no. 3, pp. 185-220, 1998