Critical Point Theory and Nonlinear Differential Equations

Critical point theory provides a unifying variational framework for analysing nonlinear differential equations by identifying points at which an associated functional attains extremal values or stationary values. Through this approach, existence, multiplicity and qualitative properties of solutions to boundary value problems and dynamical systems can be established. Central tools include minimax theorems, Morse theory and deformation arguments, which allow one to locate critical points even in the absence of compactness or Palais–Smale conditions. Nonlinear differential equations of elliptic, parabolic and Hamiltonian type often give rise to functionals defined on Sobolev or Banach spaces, where the geometry of the functional landscape encodes bifurcation phenomena, symmetry breaking and homoclinic orbits. Recent work has extended classical mountain-pass and linking theorems to non-smooth settings, quasilinear operators and supercritical growth regimes, broadening both theoretical reach and practical applicability in fields ranging from material science to electrical engineering.

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Critical Point Theory and Nonlinear Differential Equations publication trend

The graph below shows the total number of articles in critical point theory and nonlinear differential equations across all publications each year (not limited to Nature Index journals).

Technical terms

Critical point: A point in the domain of a functional where its first variation vanishes, indicating a candidate for a minimum, maximum or saddle.

Variational method: An approach that reformulates differential equations as extremal problems for functionals on infinite-dimensional spaces.

Quasilinear elliptic equation: A second-order elliptic differential equation in which the highest-order term depends nonlinearly on the gradient of the unknown function.

Leray–Lions operator: A generalisation of the p-Laplace operator allowing nonstandard growth conditions and dependence on both solution and gradient.

Deflation technique: An algorithmic strategy to modify a system after finding each solution so as to render that solution invisible to subsequent searches, enabling discovery of multiple distinct solutions.

References

1. Nonhomogeneous quasilinear elliptic systems with small perturbations and lack of compactness. Bulletin of Mathematical Sciences (2025).
2. Entire radial bounded solutions for Leray-Lions equations of (p, q)-type. Advances in Nonlinear Analysis (2025).
3. A SPICE-Oriented Method for Finding Multiple DC Solutions in Nonlinear Circuits. Applied Sciences (2023).