**Title:** Nonsymmetric Positive Solutions for Symmetric Dirichlet Elliptic Problems

**Speaker:** Professor Daomin Cao

**Speaker Info:** Chinese Academy of Sciences

**Brief Description:**

**Special Note**: **This is a joint PDE and Analysis/Probability Seminar**

**Abstract:**

In this talk the speaker will present some results on the existence of non-radially symmetric positive solutions of the following radially symmetric problems \begin{equation}\label{1.2} \left\{ \begin{array}{lll} -\Delta u+(\lambda-h(x))u=(1-f(x))u^p, &\mbox{in} &\mathbb{R}^N, \\ \qquad u>0,\ \ \mbox{in}\ \ \mathbb{R}^N, &&\\ \qquad u\in H^1(\mathbb{R}^N), && \end{array} \right. \end{equation} where $h(x)$ and $f(x)$ are nonnegative radially symmetric functions in $L^\infty(\mathbb{R}^N)$, $h(x)$ and $f(x)$ have compact support in $\mathbb{R}^N$, $f(x)\leq1$ for all $x\in\mathbb{R}^N$, $1When $N=1,\sigma =1$, $h(x)=\chi_{_{[-d,d ]}}$, $f(x)=\chi_{_{[-d,d ]}}$, where $\chi_{_{[-d,d ]}}$ stands for the characteristic function of $[-d,d ]$, $d>0$ is a given number),problem (\ref{1.2}) is a model in nonlinear optics arising in the study of the asymmetric guided waves in a symmetric structure consisting of three layers of dielectric materials whose refractive indices depend on the density of the electric field.

When $N=2$, problem (\ref{1.2}) represents a convenient scalar approximation of a model arising from the study of the propagation of a monochromatic electric field in optical cylindrical waveguides. Such scalar approximation can be found in engineering and mathematical literature. A mathematical description of the propagation of electromagnetic fields in optical cylindrical waveguides having a defocusing or self-focusing dielectric response can be found in papers of O.John and C.Stuart and C.Stuart.

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