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Nonlinear Partial Differential Equations of Elliptic Type

Vicentiu Radulescu

TL;DR

This course notes address nonlinear elliptic boundary-value problems and the existence, multiplicity, and qualitative behavior of solutions. It develops a suite of methods, starting with the sub- and supersolution approach to trap solutions and establish minimal and maximal solutions via a monotone iteration, and complementing this with an implicit-function-theorem framework to obtain local and global bifurcation for problems like $- abla^2 u = \,\lambda\ f(u)$. It then extends the analysis to nonsmooth settings by introducing Clarke's generalized gradient, Ekeland's variational principle, and nonsmooth Mountain-Pass and Saddle-Point results for locally Lipschitz functionals, enabling critical-point theory without differentiability. Together, the notes identify threshold and regularity phenomena for problems with asymptotically linear nonlinearities, outline energy relations at critical parameter values, and provide a cohesive toolkit for existence and stability insights in nonlinear elliptic PDEs.

Abstract

Course Notes on Nonlinear Partial Differential Equations of Elliptic Type given to Master students at the University of Craiova, Romania.

Nonlinear Partial Differential Equations of Elliptic Type

TL;DR

This course notes address nonlinear elliptic boundary-value problems and the existence, multiplicity, and qualitative behavior of solutions. It develops a suite of methods, starting with the sub- and supersolution approach to trap solutions and establish minimal and maximal solutions via a monotone iteration, and complementing this with an implicit-function-theorem framework to obtain local and global bifurcation for problems like . It then extends the analysis to nonsmooth settings by introducing Clarke's generalized gradient, Ekeland's variational principle, and nonsmooth Mountain-Pass and Saddle-Point results for locally Lipschitz functionals, enabling critical-point theory without differentiability. Together, the notes identify threshold and regularity phenomena for problems with asymptotically linear nonlinearities, outline energy relations at critical parameter values, and provide a cohesive toolkit for existence and stability insights in nonlinear elliptic PDEs.

Abstract

Course Notes on Nonlinear Partial Differential Equations of Elliptic Type given to Master students at the University of Craiova, Romania.

Paper Structure

This paper contains 22 sections, 82 theorems, 871 equations.

Table of Contents

  1. Method of sub and super solutions
  2. Implicit Function Theorem and Applications to Boundary Value Problems
  3. Abstract theorems
  4. A basic bifurcation theorem
  5. Qualitative properties of the minimal solution in a neighbourhood of the bifurcation point
  6. Bifurcation problems associated to convex positive asymptotic linear functions
  7. Nonsmooth Mountain-Pass Theory
  8. Basic properties of locally Lipschitz functionals
  9. Ekeland's Variational Principle
  10. Mountain Pass and Saddle Point type theorems
  11. Applications of the Mountain-Pass Theorem
  12. The simplest model
  13. A bifurcation problem
  14. Critical points and coerciveness of locally Lipschitz functionals with the strong Palais-Smale property
  15. Critical point theorems of Lusternik-Schnirelmann type
  16. ...and 7 more sections

Key Result

Theorem 1

Let \hbox{$\underline{U}$} (resp., \hbox{$\overline{U}$}) be a subsolution (resp., a supersolution) to the problem (P1) such that \hbox{$\underline{U}$} \leq \hbox{$\overline{U}$} in \Omega. The following hold: (i) there exists a solution u of (P1) which, moreover, satisfies \hbox{$\underline{U}$} \

Theorems & Definitions (107)

  • Definition 1
  • Theorem 1
  • Remark 1
  • Remark 2
  • Definition 2
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Remark 3
  • Remark 4
  • ...and 97 more