In this paper, we study positive solutions to problems involving the fractional Laplacian \begin{equation} \begin{cases} (-\Delta)^{\alpha} u(x)+|u|^{p-1}u(x)=0,\ \ \ \ & x\in\Omega\setminus\mathcal{C},\\ \ \quad \ u(x)=0,\ & x\in\Omega^c,\\ \displaystyle \lim_{x\in\Omega\setminus\mathcal{C}, \ x\to\mathcal{C}}u(x)=+\infty, \end{cases} \tag*{(0.1)} \end{equation} where $p>1$ and $\Omega$ is an open bounded $C^2$ domain in $\mathbb{R}^N$, $\mathcal{C}\subset \Omega$ is a compact $C^2$ manifold with $N-1$ multiples dimensions and without boundary, the operator $(-\Delta)^{\alpha}$ with $\alpha\in(0,1)$ is the fractional Laplacian. We consider the existence of positive solutions for problem (0.1). Moreover, we further analyze uniqueness, asymptotic behavior and nonexistence.

We derive a new variational model in the description of prestrained elastic thin films. The model consists of minimizing a biharmonic energy of the out-of plane displacements $v\in W^{2,2}(\Omega, \mathbb{R})$, satisfying the Monge-Ampèere constraint: $$ \det\nabla^2v = f . $$ Here, $f=-\mbox{curl}^T\mbox{curl} (S_g)_{2\times 2}$ is the linearized Gauss curvature of the incompatibility (prestrain) family of Riemannian metrics $G^h= \mbox{Id}_3 + 2 h^\gamma S_g+ h.o.t.$, imposed on the referential configurations of the thin films with midplate $\Omega$ and small thickness $h$. We further discuss multiplicity properties of the minimizers of this model in some special cases.

We give existence theorems for the Cauchy problem of a large class of semi-linear parabolic equations in $L^{\infty}$, $L^{\infty} \cap L^p$ or $L^{\infty} \cap \dot W^{1,p}$, using a contracting map argument. We then construct integral solutions to parabolic equations with data growing at infinity and defocusing nonlinearity, and give an example of instantaneous blow up when the nonlinearity is focusing and the initial data has tame growth.

This paper mainly concerns with the small data solution problem for the 3-D nonlinear wave equation: $\partial_t^2u-(1+u+\partial_t u)\Delta u=0$. This equation is prototypical of the more general equation $$ \sum_{i,j=0}^3g_{ij}(u, \nabla u)\partial_{ij}^2u =0 , $$ where $x_0=t$, $\nabla=(\partial_0, \partial_1, ..., \partial_3)$, and $$ g_{ij}(u, \nabla u)=c_{ij}+d_{ij}u+ \sum_{k=0}^3e_{ij}^k\partial_ku+O(|u|^2+|\nabla u|^2) $$ are smooth functions of their arguments with $c_{ij}, d_{ij}$ and $e_{ij}^k$ being constants, and $d_{ij}\not=0$ for some $(i,j)$; moreover, $ \sum_{i,j,k=0}^3e_{ij}^k\partial_ku\partial_{ij}^2u$ does not fulfill the null condition. For the 3-D nonlinear wave equations $\partial_t^2u-(1+u)\Delta u=0$ and $\partial_t^2u-(1+\partial_t u)\Delta u=0$, H. Lindblad, S. Alinhac, and F. John proved and disproved, respectively, the global existence of small data solutions. For radial initial data, we show that the small data solution of $\partial_t^2u-(1+u+\partial_t u)\Delta u=0$ blows up in finite time. The explicit asymptotic expression of the lifespan $T_{\varepsilon}$ as $\varepsilon\to 0+$ is also given.

We prove, using elementary methods of complex analysis, the following generalization of the isoperimetric inequality: if $p\in\mathbb{R}$, $\Omega\subset\mathbb{R}^2$, then the inequality $$ \big (\frac{|\Omega|}{\pi} \big )^{\frac{p+1}{2}} \leq\frac{1}{2\pi}\int_{\partial\Omega}|x|^pd\sigma(x), $$ holds true under appropriate assumptions on $\Omega$ and $p.$ This solves an open problem arising in the context of isoperimetric problems with density and poses some new ones (for instance generalizations to $\mathbb{R}^n$). We prove the equivalence with a Hardy-Sobolev inequality, giving the best constant, and generalize thereby the equivalence between the classical isoperimetric inequality and the Sobolev inequality. Furthermore, the inequality paves the way for solving another problem: the generalization of the harmonic transplantation method of Flucher to the singular Moser-Trudinger embedding.

The existence of at least three classical solutions for a parametric ordinary Dirichlet problem involving the mean curvature operator are established. In particular, a variational approach is proposed and the main results are obtained simply requiring the sublinearity at zero of the considered nonlinearity.

We study the first eigenvalue of the $p$-Laplacian under the Dirichlet boundary condition. For a convex domain, we give an a priori estimate for the first eigenvalue in terms of the radius $d$ of the maximum ball contained in the domain. As a consequence, we prove that the first eigenvalue diverges to infinity as $p\to\infty$ if the domain is convex and $d\leq 1$. Moreover, we show that in the annulus domain $a < |x| < b$, the first eigenvalue diverges to infinity if $b-a\leq 2$ and converges to zero if $b-a>2$.

This paper is concerned with non-selfadjoint elliptic problems having a principal part in divergence form and involving an indefinite weight. We study the asymptotic behavior of the principal eigenvalues when the zero order term becomes larger and larger. Use is made of a minimax formula for these principal eigenvalues.