Stability and instability of solutions of semilinear problems

References

• Aris , R . 1975 . Mathematical Theory of Diffusion and Reaction in Permeable Catalysts , London : Oxford University Press .
   Google Scholar
• Brown , KJ , Ibrahim , MMA and Shivaji , R . 1981 . S-shaped bifurcation curves. . Nonlinear Analysis , 5 ( 5 ) : 475 – 486 .
   Web of Science ®Google Scholar
• Dancer , EN . 1980 . On the structure of solutions of an equation in catalysis theory when a parameter is large. . Journal of Differential Equations , 37 ( 3 ) : 404 – 437 .
   Web of Science ®Google Scholar
• Du , Y and Lou , Y . 2001 . Proof of a conjecture for the perturbed Gelfand equation from combustion theory. . Journal of Differential Equations , 173 ( 2 ) : 213 – 230 .
   Web of Science ®Google Scholar
• Hastings , SP and McLeod , JB . 1985 . The number of solutions to an equation from catalysis. . Proceedings of the Royal Society of Edinburgh. Section A. Mathematics , 101 ( 1–2 ) : 15 – 30 .
   Google Scholar
• Galstyan , A , Korman , P and Li , Y . 2006 . On the oscillations of the solution curve for a class of semilinear equations. . Journal of Mathematical Analysis and Applications , 321 ( 2 ) : 576 – 588 .
   Web of Science ®Google Scholar
• Gidas , B , Ni , W-M and Nirenberg , L . 1979 . Symmetry and related properties via the maximum principle. . Communications in Mathematical Physics , 68 : 209 – 243 .
   Web of Science ®Google Scholar
• Korman , P . 2006 . “ Global solution branches and exact multiplicity of solutions for two point boundary value problems ” . In Handbook of Differential Equations, Ordinary Differential Equations , Edited by: Canada , A , Drabek , P and Fonda , A . Vol. 3 , 547 – 606 . North Holland : Elsevier Science .
   Google Scholar
• Korman , P and Li , Y . 1999 . On the exactness of an . S-shaped bifurcation curve.Proceedings of the American Mathematical Society , 127 ( 4 ) : 1011 – 1020 .
   Web of Science ®Google Scholar
• Korman , P , Li , Y and Ouyang , T . 1996 . Exact multiplicity results for boundary-value problems with nonlinearities generalising cubic. . Proceedings of the Royal Society of Edinburgh. Section A. Mathematics , 126A : 599 – 616 .
   Google Scholar
• Korman , P and Shi , J . 1999 . “ Instability and exact multiplicity of solutions of semilinear equations. ” . In Proceedings of the Conference on Nonlinear Differential Equations 311 – 322 . Coral Gables, FL 2000(electronic), Electronic Journal of Differential Equations Conference, 5, Southwest Texas State Univ., San Marcos, TX
   Google Scholar
• Lin , C.S and Ni , W.-M . 1988 . A counterexample to the nodal domain conjecture and related semilinear equation. . Proceedings of the American Mathematical Society , 102 : 271 – 277 .
   Web of Science ®Google Scholar
• Ouyang , T and Shi , J . 1999 . Exact multiplicity of positive solutions for a class of semilinear problems, II. . Journal of Differential Equations , 158 ( 1 ) : 94 – 151 .
   Web of Science ®Google Scholar
• Parter , SV . 1974 . Solutions of a differential equation arising in chemical reactor processes. . SIAM Journal on Applied Mathematics , 26 : 687 – 716 .
   Google Scholar
• Schaaf , R . 1990 . “ Global Solution Branches of Two Point Boundary Value Problems ” . In Lecture Notes in Mathematics, No. 1458 , Berlin : Springer-Verlag .
   Google Scholar
• Schaaf , R and Schmitt , K . 1988 . A class of nonlinear Sturm-Liouville problems with infinitely many solutions. . Transactions of the American Mathematical Society , 306 ( 2 ) : 853 – 859 .
   Web of Science ®Google Scholar
• Wang , S-H . 1994 . On S-shaped bifurcation curves. . Nonlinear Analysis TMA , 22 ( 12 ) : 1475 – 1485 .
   Web of Science ®Google Scholar
• Wang , S-H . On the evolution and qualitative behaviors of bifurcation curves for a boundary value problem. . Nonlinear Analysis TMA , (To be appear)
   Google Scholar
• Wang , S-H and Lee , FP . 1994 . Bifurcation of an equation from catalysis theory. . Nonlinear Analysis , 23 ( 9 ) : 1167 – 1187 .
   Web of Science ®Google Scholar
• Wang , S-H . 1998 . Rigorous analysis and estimates of S-shaped bifurcation curves in a combustion problem with general Arrhenius reaction-rate laws. . The Royal Society of London. Proceedings. Series A. Mathematical, Physical and Engineering Sciences , 454 ( 1972 ) : 1031 – 1048 .
   Web of Science ®Google Scholar
• Wang , S-H . An S-shaped bifurcation curves theorem for a p-Laplacian Dirichlet problem and its application to the perturbed Gelfand problem (Preprint) ,
   Google Scholar