American Institute of Mathematical Sciences

$-\Delta u=\frac{\lambda}{(1-u)^2}$ in $\Omega$, $u=0$ on $\partial \Omega$, $0 < u < 1$ in $\Omega$

where $\Omega$ is a rather symmetric domain in $\mathbb R^2$. We prove that there exists a $\lambda_\star>0$ such that for $\lambda \in (0, \lambda_\star)$ the minimal solution is unique. Then we analyze the asymptotic behavior of touch-down solutions, i.e., solutions with max$_\Omega u_i (0) \to 1$. We show that after a rescaling, the solution will be asymptotically symmetric. As a consequence, we show that the branch of positive solutions must undergo infinitely many bifurcations as the maximums of the solutions on the branch go to 1 (possibly only changes of direction). This gives a positive answer to some open problems in [12]. Our result is new even in the radially symmetric case. Central to our analysis is the monotonicity formula, one-dimensional Sobloev inequality, and classification of solutions to a supercritical problem

$ \Delta U=\frac{1}{U^2}\quad$ in $\mathbb R^2, U(0)=1, U(z) \geq 1.$

$\Delta u = 0$ in $ \Omega$, $\qquad \frac{\partial u}{\partial \nu} =\lambda f(u)$ on $\Gamma_1, \qquad u = 0$ on $\Gamma_2$

where $\lambda>0$, $f(u) = e^u$ or $f(u) = (1+u)^p$, $\Gamma_1$, $\Gamma_2$ is a partition of $\partial \Omega$ and $\Omega\subset \mathbb R^N$. We determine sharp conditions on the dimension $N$ and $p>1$ such that the extremal solution is bounded, where the extremal solution refers to the one associated to the largest $\lambda$ for which a solution exists.

$-\epsilon^2 \Delta u+V(x)u=f(u), \quad u\in H^1(\mathbb R^N),$ $\qquad\qquad\qquad$ (*)$_\epsilon $

where $V$ satisfies $ V(x)\geq 0 $, liminf$_{ |x|\to \infty }V(x)>0$. $f(u)\in C^1(\mathbb R,\mathbb R)$ satisfies Ambrosetti--Rabinowitz condition and some properties and $\frac{f(u)}{u}$ is nondecreasing. We consider the case inf$_{x\in \mathbb R^N}V(x)>0$ and critical frequency case, that is, inf$_{x\in \mathbb R^N}V(x)=0$. We study the existence of sign-changing 2-peak solutions of (*)$_\epsilon$ whose one peak is positive, another peak is negative and both peaks concentrate to a same local minimum point of $V(x)$ as $\epsilon \to 0$.

$\varphi (u'(t))' + \lambda h(t) f (u(t)) = 0,\ \ $ a.e. $\ t \in (0,1), \qquad\qquad\qquad\qquad\qquad $ $(\Phi_\lambda)$

$u(0) = 0=u(1),$

where $\varphi : \mathbb R \to \mathbb R$ is an odd increasing homeomorphism, $\lambda$ a positive parameter and $h \in L^1(0,1)$ a nonnegative measurable function on $(0,1)$ which may be singular at $t = 0$ and/or $t = 1,$ and $f \in C(\mathbb R, \mathbb R)$ and is odd.