GENERALIZED DEGREE THEORY FOR SEMILINEAR OPERATOR EQUATIONS

Generalized Topological Degree and Semilinear Equations

The topological degree is a fundamental concept in algebraic topology and in analysis, and the number of its applications to nonlinear differential equations has increased at an impressive rate during the whole second half of this century. The concept can be rooted in the fundamental work of Kronecker [8] on systems of smooth real functions f0, f1, . . . , fn of n real variables such that 0 is ...

Analyticity of the Scattering Operator for Semilinear Dispersive Equations

We present a general algorithm to show that a scattering operator associated to a semilinear dispersive equation is real analytic, and to compute the coefficients of its Taylor series at any point. We illustrate this method in the case of the Schrödinger equation with powerlike nonlinearity or with Hartree type nonlinearity, and in the case of the wave and Klein–Gordon equations with power nonl...

Semilinear Elliptic Equations with Generalized Cubic Nonlinearities

A semilinear elliptic equation with generalized cubic nonlinearity is studied. Global bifurcation diagrams and the existence of multiple solutions are obtained and in certain cases, exact multiplicity is proved.

Oscillation Theory for Semilinear Elliptic Equations with Arbitrary Nonlinearities

Index theory for linear self-adjoint operator equations and nontrivial solutions for asymptotically linear operator equations

We will first establish an index theory for linear self-adjoint operator equations. And then with the help of this index theory we will discuss existence and multiplicity of solutions for asymptotically linear operator equations by making use of the dual variational methods and Morse theory. Finally, some interesting examples concerning second order Hamiltonian systems, first order Hamiltonian ...