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CPAA

We investigate here the
elliptic equation -div$(a(x)\nabla u)+a(x)u=0$ posed on a
bounded smooth domain $\Omega$ in $\mathbb R^2$ with nonlinear Neumann
boundary condition $\frac{\partial u}{\partial \nu}=\varepsilon e^u$, where $\varepsilon$ is a
small parameter. We extend the work of Davila-del Pino-Musso
[5] and show that if a family of solutions $u_\varepsilon$ for
which $\varepsilon\int_{\partial Omega}e^{u_\varepsilon}$ is bounded, then it will develop up
to subsequences a finite number of bubbles $\xi_i\in\partial Omega$, in the
sense that $\varepsilon e^{u_\varepsilon}\to
2\pi\sum_{i=1}^k m_i\delta_{\xi_i}$
as $\varepsilon\rightarrow 0$ with $k, m_i \in \mathbb Z^+$. Location of
blow-up points is characterized in terms of function $a(x)$.

DCDS

Infinitely many positive and sign-changing solutions for nonlinear fractional scalar field equations

We consider the following nonlinear fractional scalar field equation
$$
(-\Delta)^s u + u = K(|x|)u^p,\ \ u > 0 \ \ \hbox{in}\ \ \mathbb{R}^N,
$$
where $K(|x|)$ is a positive radial function, $N\ge 2$, $0 < s < 1$, and
$1 < p < \frac{N+2s}{N-2s}$. Under various asymptotic assumptions on $K(x)$ at infinity, we
show that this problem has
infinitely many non-radial positive solutions and sign-changing solutions, whose energy can be made arbitrarily large.

CPAA

In this paper, we study the structure of solutions of a fourth order
elliptic equation with a singular nonlinearity. For different
boundary values $\kappa$, we establish the global bifurcation
branches of solutions to the equation. More precisely, we show that
$\kappa=1$ is a critical boundary value to change the structure of
solutions to this problem.

DCDS

In this paper, we study conserved quantities, blow-up criterions, and global existence of solutions for a generalized CH equation. We investigate the classification of self-adjointness, conserved quantities for this equation from the viewpoint of Lie symmetry analysis. Then, based on these conserved quantities, blow-up criterions and global existence of solutions are presented.

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