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Semilinear Biharmonic Problem with a Singular Term

Bayrami Aminlouee, Masoud | 2015

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  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 48127 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Hesaaraki, Mahmoud
  7. Abstract:
  8. The aim of this work is to study the optimal exponent p to have solvability of semilinear biharmonic problem with a singular term in a smooth and bounded domain such that contains origin in Euclidean space with dimension greater than 4. The singular term is related to the Hardy inequality. First of all, it is not difficult to show that any positive supersolution of problem is unbounded near the origin and then additional hypotheses on p are needed to ensure existence of solutions. We will say that problem blows up completely if the solutions to the truncated problems (with a bounded weight instead of the Hardy singularity) tend to infinity for every x in domain as n goes infinity. The main objective of this work is to explain the influence of the Hardy type term on the existence or nonexistence of solutions and to determine the threshold exponent to have a complete blow-up phenomenon if p is equal or greater than the threshold. In fourth-order problems, Navier boundary conditions play an important role to prove existence results. The problem can be rewritten as a second-order system with Dirichlet boundary conditions. By the classical elliptic theory, we easily prove a Maximum Principle. As a consequence, we deduce a Comparison Principle that allows us to prove the existence of solutions for p less than threshold as a limit of approximated problems. We will use Moreau type decomposition and Picone inequality in the nonexistence proofs
  9. Keywords:
  10. Hardy Inequality ; Singularity ; Semilinear Biharmonic Problems ; Navier Conditions ; Non Existence Results ; Existence Results

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