Tobias Weth

Partial symmetry of least energy nodal solutions to some variational problems

We investigate the symmetry properties of several radially symmetric minimization problems. The minimizers which we obtain are nodal solutions of superlinear elliptic problems, or eigenfunctions of weighted asymmetric eigenvalue problems, or they lie on the first curve in the Fucik spectrum. In all instances, we prove that the minimizers are foliated Schwarz symmetric. We give examples showing that the minimizers are in general not radially symmetric. The basic tool which we use is polarization, a concept going back to Ahlfors. We develop this method of symmetrization for sign changing functions.

Sign Changing Solutions of Superlinear Schrödinger Equations

Abstract We prove the existence of sign changing solutions in H 1(ℝ N ) for a stationary Schrödinger equation −Δu + a(x)u = f(x, u) with superlinear and subcritical nonlinearity f, and control the number of nodal domains. If f is odd we obtain an unbounded sequence of sign changing solutions u k , k ≥ 1, so that u k has at most k + 1 nodal domains. The bound on the number of nodal domains follows from a nonlinear version of Courants nodal domain theorem which we also prove.

A note on additional properties of sign changing solutions to superlinear elliptic equations

We obtain upper bounds for the number of nodal domains of sign changing solutions of semilinear elliptic Dirichlet problems using suitable min-max descriptions. These are consequences of a generalization of Courants nodal domain theorem. The solutions need not to be isolated. We also obtain information on the Morse index of solutions and the location of sub- and supersolutions.

Nodal Solutions of a p-Laplacian Equation

We prove that the

Asymptotic behaviour of solutions of planar elliptic systems with strong competition

p

Symmetry of solutions to semilinear elliptic equations via Morse index

-Laplacian problem

Radial symmetry of positive solutions in nonlinear polyharmonic Dirichlet problems

-\Delta_p u = f(x, u)

On a fourth order Steklov eigenvalue problem

, with

Monotonicity and nonexistence results for some fractional elliptic problems in the half-space

u \in W^{1, p}_0 (\Omega)

Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay

on a bounded domain