Nonlinear elliptic equations and systems with linear part at resonance

The famous result of Landesman and Lazer [10] dealt with resonance at a simple eigenvalue. Soon after publication of [10], Williams [14] gave an extension for repeated eigenvalues. The conditions in Williams [14] are rather restrictive, and no examples were ever given. We show that seemingly different classical result by Lazer and Leach [11], on forced harmonic oscillators at resonance, provides an example for this theorem. The article by Williams [14] also contained a shorter proof. We use a similar approach to study resonance for $2 \times 2$ systems. We derive conditions for existence of solutions, which turned out to depend on the spectral properties of the linear part.Comment: 17 page

On the localized wave patterns supported by convection-reaction-diffusion equation

A set of traveling wave solution to convection-reaction-diffusion equation is studied by means of methods of local nonlinear analysis and numerical simulation. It is shown the existence of compactly supported solutions as well as solitary waves within this family for wide range of parameter values

Elliptic equations with transmission and Wentzell boundary conditions and an application to steady water waves in the presence of wind

In this paper, we present results about the existence and uniqueness of solutions of elliptic equations with transmission and Wentzell boundary conditions. We provide Schauder estimates and existence results in H\"older spaces. As an application, we develop an existence theory for small-amplitude two-dimensional traveling waves in an air-water system with surface tension. The water region is assumed to be irrotational and of finite depth, and we permit a general distribution of vorticity in the atmosphere.Comment: 33 page

Capillary-gravity water waves with vorticity : steady wind-driven waves and waves with a submerged dipole

In this thesis, we study two mathematical problems on water waves in the setting of the incompressible Euler equations with vorticity, gravity, and surface tension. We investigate the existence of small-amplitude steady wind-driven water waves in finite depth, using the Crandall Rabinowitz theorem. As part of the result, elliptic equations with transmission and Wentzell boundary conditons are also examined, and Schauder type estimates on classical solutions are established. The second chapter considers the existence and instability of solitary water waves with a nite dipole in in nite depth. We construct waves of this type using an Implicit Function Theorem argument. Then we establish orbital instability. This is proved using a modi cation of the classical Grillaks Shatash Strauss method.Includes bibliographical reference

Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions

Assuming $B_{R}$ is a ball in $\mathbb R^{N}$, we analyze the positive solutions of the problem $$\begin{cases} -\Delta u+u= |u|^{p-2}u, &\text{ in } B_{R},\newline \partial_{\nu}u=0,&\text{ on } \partial B_{R}, \end{cases}$$ that branch out from the constant solution $u=1$ as $p$ grows from $2$ to $+\infty$. The non-zero constant positive solution is the unique positive solution for $p$ close to $2$. We show that there exist arbitrarily many positive solutions as $p\to\infty$ (in particular, for supercritical exponents) or as $R \to \infty$ for any fixed value of $p>2$, answering partially a conjecture in [Bonheure-Noris-Weth]. We give the explicit lower bounds for $p$ and $R$ so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.Comment: 37 pages, 24 figure