Nonlinear analysis

Nonlinear analysis studies equations and variational problems in which superposition fails. Its tools include nonlinear functional inequalities, conserved energies, compactness, phase-plane methods, and concentration arguments.

The Emden-Fowler transformation replaces a radial variable $r$ by logarithmic time $s=\log r$ and rescales the dependent variable by a power of $r$. It converts many scale-invariant radial equations into autonomous ordinary differential equations.

A Lane-Emden equation is a semilinear elliptic equation of the formFor a radial function it becomes $u^{\prime \prime }+(d-1)u^{\prime }/r+u^p=0$.

When $a=2/(p-1)$ and $a<d-2$, the Lane-Emden equation has the singular scale-invariant solution

At the energy-critical exponent $p=(d+2)/(d-2)$, the normalized Aubin-Talenti bubblesolves $\Delta W+W^p=0$ and optimizes the critical Sobolev embedding theorem.

Nonlinear Schrödinger blowup analysis studies whether a solution of the Focusing nonlinear Schrodinger equation remains bounded in its natural Sobolev space or develops an unbounded gradient norm in finite time.

For sufficiently regular solutions of a gauge-invariant nonlinear Schrödinger equation,is independent of time.

For the focusing power equation $i{u}_t+\Delta u+|u{|}^{p-1}u=0$, the conserved energy is

An $H^1$ solution of the nonlinear Schrödinger equation either exists globally forward in time or its $H^1$ norm, equivalently its gradient norm when mass is conserved, becomes unbounded at the finite endpoint of its maximal lifespan.

A localized virial identity differentiates a weighted mass $\int \chi |u{|}^2$ and its associated momentum. Compactly supported or flattened weights retain the coercive interior contribution while replacing an infinite-variance assumption by controllable tail errors.

For a radial function $u\in H^1({\mathbb{R}}^d)$,This supplies spatial decay without requiring a weighted moment.

For the two-dimensional focusing cubic nonlinear Schrödinger equation, the NLS ground state is the positive radial solution ofIts symmetry orbit consists of the optimizers of the sharp Gagliardo-Nirenberg inequality.

In two dimensions,and equality holds exactly on the phase, translation, and scaling orbit of the NLS ground state $Q$.

Multiplication by $e^{ia\psi (x)}$ is a gauge transform that preserves pointwise modulus and mass while shifting the gradient by $ia(\nabla \psi )u$.

Young's inequality for products states that for conjugate exponents $q,q^{\prime }>1$ and nonnegative $a,b$,Its weighted form absorbs a lower power into a coercive higher power at the cost of a constant.