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On characterization of solutions of some nonlinear differential equations and applications. (English) Zbl 0807.34050
The paper contains a nice description of the set of solutions on \(S^ 1= {\mathbf R}/2\pi\mathbb{Z}\) of the nonlinear eigenvalue problem \(\omega''- \mu\omega+ g(\omega)= 0\), where \(\mu\in {\mathbf R}\) and \(g\) satisfies the hypotheses: \(g\in C^ 1({\mathbf R})\cap C^ 3({\mathbf R}\backslash \{0\})\); \(g(0)= g'(0)= 0\); \(rg''(r)>0\), \(\forall r\neq 0\) and \(\lim_{r\to \pm\infty} {g(r)\over r}= +\infty\) (a typical function \(g\) satisfying the previous assumptions is \(g(\omega)= |\omega|^{q- 1}\omega\), \(q>1\)). Also, depending on the values of \(\mu\), the authors depict their levels of energy. After, they apply these results to study the large-time behavior of solutions of the parabolic equation \(u_ t= u_{xx}+ | u|^{q-1} u- \mu u\) with periodic, Neumann or Dirichlet conditions. Finally, they consider the behavior near the origin or near infinity of the solutions of the elliptic equation in \({\mathbf R}^ 2\backslash\{0\}\): \[ \Delta u- c{u\over | x|^ 2}+ | u|^{q-1} u= 0, \] where \(c,q\in {\mathbf R}\), \(c>0\) and \(q>1\).

34C23 Bifurcation theory for ordinary differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
35B40 Asymptotic behavior of solutions to PDEs
35J99 Elliptic equations and elliptic systems
35K99 Parabolic equations and parabolic systems
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