American Institute of Mathematical Sciences

$ -\Delta_p u+V(|x|)|u|^{q-2}u=Q(|x|)|u|^{s-2}u, x\in R^N,$

$ u(x) \rightarrow 0,$ as $ |x|\rightarrow\infty. $

$\frac{\partial u}{\partial t}-\Delta_p u+$ div $(c(x,t)|u|^{\gamma-1}u) =f $ in $Q_T$

$u(x,t)=0$ on $\partial\Omega\times(0,T) $

$u(x,0)=u_0 (x)$ in $\Omega,$

where $Q_T=\Omega\times(0,T),$ $\Omega$ is an open and bounded subset of $ \mathcal{R} ^N$, $N\geq2,$ $T>0,$ $\Delta_p$ is the so called $p$-Laplace operator, $\gamma=\frac{(N+2)(p-1)}{N+p},$ $c(x,t)\in(L^{\tau }(Q_{T}))^N,$ $\tau=\frac{N+p}{p-1},$ $\ f\in L^1 (Q_T), $ $u_{0}\in L^1(\Omega).$

$ \Delta h = \frac{1}{\alpha } h^{ -\alpha }-p(r)$ in $R^N( N\geq3 ),$

where $ \alpha >1$ and the monotone decreasing function $p(r)$ satisfying $\lim_{r\rightarrow \infty}p(r)=c>0.$ In this paper we show that for any $ \eta >0 $ there is a unique radial solution $ h(r) $ with $ h(0)=\eta $ and $ h(r)$ is oscillatory in $ [0, \infty )$. We prove $ \lim_{r \rightarrow \infty } h(r)=( \alpha c)^{-\frac{1}{\alpha}}. $ We also obtain similar properties of singular solutions, of which the zero set is nonempty.

$i\frac{\partial \psi}{\partial t}=- \Delta \psi+\psi + \bar{\omega} (|\psi |^2)\psi- \lambda \rho(|\psi|^2)\psi-\kappa\Delta \rho(|\psi|^2)\rho'(|\psi|^2)\psi, x \in \Omega,$

where $\bar{\omega} (\tau^2) \tau \rightarrow +\infty$ as $\tau \rightarrow 0$ and, $\lambda>0$ is a parameter and $\Omega$ is a ball in $\mathcal{R}^N$. This problem is studied in connection with the following quasilinear eigenvalue problem

$-\Delta \Psi-\kappa\Delta \rho(|\Psi|^2)\rho'(|\Psi|^2)\Psi =\lambda_1 \rho(|\Psi|^2)\Psi, x \in \Omega,$

Indeed, we establish the existence of solutions for the above Schrödinger equation when $\lambda$ belongs to a certain neighborhood of the first eigenvalue $\lambda_1$ of the above eigenvalue problem. The main feature of this paper is that the nonlinearity $ \bar{\omega} ( |\psi |^2)\psi$ is unbounded around the origin and also the presence of the second order nonlinear term. Our analysis shows the importance of the role played by the parameter $\lambda$ combined with the nonlinear nonhomogeneous term $-\Delta \rho(|\psi|^2)\rho'(|\psi|^2)\psi$. The proofs are based on various techniques related to variational methods and implicit function theorem.

$ u_t + h(u)u_x = (d(u)u_x)_x + f(u)$

is here studied with respect to the diffusion, reaction and convection terms.