I am looking for names of methods, and examples of their use that can be used to find solutions for semilinear elliptic PDE equations of the below types:
$$\frac{\partial^ny}{\partial x^n}+\frac{\partial^ny}{\partial t^n}+\frac{\partial^{n-1}y}{\partial x^{n-1}}+\frac{\partial^{n-1}y}{\partial t^{n-1}}+...+\frac{\partial^2y}{\partial x^2}+\frac{\partial^2y}{\partial t^2}+\alpha\frac{\partial y}{\partial t}\frac{\partial y}{\partial x}+\beta y(x,t)+\gamma=g(x,t)$$
Alternative notation:
$$\nabla^ny+\nabla^{n-1}y+...+\nabla^2y+\alpha y_t y_x+\beta y(x,t)+\gamma=g(x,t)$$
where $y=y(x,t)$, $y(x)$ and $y(t)$ can be solutions. $g(x,t)$ is assumed to be continuous and differentiable, and $\alpha$, $\beta$, $\gamma \in \Re$ and <>0
Can one use eg Laplace Transforms to solve it? If no why not? If yes how?