Abstract:
We examine the regularity of the extremal solution of the nonlinear eigenvalue problem $\Delta^2 u = \lambda f(u)$ on a general bounded domain $\Omega$ in $ \IR^N$, with the Navier boundary condition $ u=\Delta u =0 $ on $ \pOm$. Here $ \lambda$ is a positive parameter and $f$ is a non-decreasing nonlinearity with $f(0)=1$. We give general pointwise bounds and energy estimates which show that for any convex and superlinear nonlinearity $f$, the extremal solution $ u^*$ is smooth provided $N\leq 5$.

Abstract:
Consider the problem {ll} \Delta^2 u= \lambda e^{u} &\text{in} B u=\frac{\partial u}{\partial n}=0 &\text{on}\partial B, where $B$ is the unit ball in $\R^N$ and $\lambda$ is a parameter. Unlike the Gelfand problem the natural candidate $u=-4\ln(|x|)$, for the extremal solution, does not satisfy the boundary conditions and hence showing the singular nature of the extremal solution in large dimensions close to the critical dimension is challenging. D\'avila et al. in \cite{DDGM} used a computer assisted proof to show that the extremal solution is singular in dimensions $13\leq N\leq 31$. Here by an improved Hardy-Rellich inequality which follows from the recent result of Ghoussoub-Moradifam \cite{GM} we overcome this difficulty and give a simple mathematical proof to show the extremal solution is singular in dimensions $N\geq13$.

Abstract:
We study the regularity of the extremal solution $u^*$ to the singular reaction-diffusion problem $-\Delta_p u = \lambda f(u)$ in $\Omega$, $u =0$ on $\partial \Omega$, where $1

p+2$ and $|\nabla u^*|^{p-1} \in L^{\frac{n}{n-(p'+1)}} (\Omega)$ if $n > p p'$.

Abstract:
We study the regularity of the extremal solution of the semilinear biharmonic equation $\bi u=\f{\lambda}{(1-u)^2}$, which models a simple Micro-Electromechanical System (MEMS) device on a ball $B\subset\IR^N$, under Dirichlet boundary conditions $u=\partial_\nu u=0$ on $\partial B$. We complete here the results of F.H. Lin and Y.S. Yang \cite{LY} regarding the identification of a "pull-in voltage" $\la^*>0$ such that a stable classical solution $u_\la$ with $0\la^*$. Our main result asserts that the extremal solution $u_{\lambda^*}$ is regular $(\sup_B u_{\lambda^*} <1)$ provided $ N \le 8$ while $u_{\lambda^*} $ is singular ($\sup_B u_{\lambda^*} =1$) for $N \ge 17$, in which case $1-C_0|x|^{4/3}\leq u_{\lambda^*} (x) \leq 1-|x|^{4/3}$ on the unit ball, where $ C_0:= <(\frac{\lambda^*}{\bar{\lambda}}>)^{1/3}$ and $ \bar{\lambda}:= \frac{8 (N-{2/3}) (N- {8/3})}{9}$. The singular character of the extremal solution for the remaining cases (i.e., when $9\leq N\leq 16$) requires a computer assisted proof and will not be addressed in this paper.

Abstract:
Finite time singularity formation in a fourth order nonlinear parabolic partial differential equation (PDE) is analyzed. The PDE is a variant of a ubiquitous model found in the field of Micro-Electro Mechanical Systems (MEMS) and is studied on a one-dimensional (1D) strip and the unit disc. The solution itself remains continuous at the point of singularity while its higher derivatives diverge, a phenomenon known as quenching. For certain parameter regimes it is shown numerically that the singularity will form at multiple isolated points in the 1D strip case and along a ring of points in the radially symmetric 2D case. The location of these touchdown points is accurately predicted by means of asymptotic expansions. The solution itself is shown to converge to a stable self-similar profile at the singularity point. Analytical calculations are verified by use of adaptive numerical methods which take advantage of symmetries exhibited by the underlying PDE to accurately resolve solutions very close to the singularity.

Abstract:
We consider the reaction-diffusion problem $-\Delta_g u = f(u)$ in $\mathcal{B}_R$ with zero Dirichlet boundary condition, posed in a geodesic ball $\mathcal{B}_R$ with radius $R$ of a Riemannian model $(M,g)$. This class of Riemannian manifolds includes the classical \textit{space forms}, i.e., the Euclidean, elliptic, and hyperbolic spaces. For the class of semistable solutions we prove radial symmetry and monotonicity. Furthermore, we establish $L^\infty$, $L^p$, and $W^{1,p}$ estimates which are optimal and do not depend on the nonlinearity $f$. As an application, under standard assumptions on the nonlinearity $\lambda f(u)$, we prove that the corresponding extremal solution $u^*$ is bounded whenever $n\leq9$. To establish the optimality of our regularity results we find the extremal solution for some exponential and power nonlinearities using an improved weighted Hardy inequality.

Abstract:
In this note, we investigate the regularity of extremal solution $u^*$ for semilinear elliptic equation $-\triangle u+c(x)\cdot\nabla u=\lambda f(u)$ on a bounded smooth domain of $\mathbb{R}^n$ with Dirichlet boundary condition. Here $f$ is a positive nondecreasing convex function, exploding at a finite value $a\in (0, \infty)$. We show that the extremal solution is regular in low dimensional case. In particular, we prove that for the radial case, all extremal solution is regular in dimension two.

Abstract:
We consider the fourth order problem $\Delta^{2}u=\lambda f(u)$ on a general bounded domain $\Omega$ in $R^{n}$ with the Navier boundary condition $u=\Delta u=0$ on $\partial \Omega$. Here, $\lambda$ is a positive parameter and $ f:[0,a_{f}) \rightarrow \Bbb{R}_{+} $ $ (0 < a_{f} \leqslant \infty)$ is a smooth, increasing, convex nonlinearity such that $ f(0) > 0 $ and which blows up at $ a_{f} $. Let $$0<\tau_{-}:=\liminf_{t\rightarrow a_{f}} \frac{f(t)f"(t)}{f'(t)^{2}}\leq \tau_{+}:=\limsup_{t\rightarrow a_{f}} \frac{f(t)f"(t)}{f'(t)^{2}}<2.$$ We show that if $u_{m}$ is a sequence of semistable solutions correspond to $\lambda_{m}$ satisfy the stability inequality $$ \sqrt{\lambda_{m}}\int_{\Omega}\sqrt{f'(u_{m})}\phi^{2}dx\leq \int_{\Omega}|\nabla\phi|^{2}dx, ~~\text{for all}~\phi\in H^{1}_{0}(\Omega),$$ then $\sup_{m} ||u_{m}||_{L^{\infty}(\Omega)}

Abstract:
We investigate the problem of entire solutions for a class of fourth order, dilation invariant, semilinear elliptic equations with power-type weights and with subcritical or critical growth in the nonlinear term. These equations define non compact variational problems and are characterized by the presence of a term containing lower order derivatives, whose strength is ruled by a parameter {\lambda}. We can prove existence of entire solutions found as extremal functions for some Rellich-Sobolev type inequalities. Moreover, when the nonlinearity is suitably close to the critical one and the parameter {\lambda} is large, symmetry breaking phenomena occur and in some cases the asymptotic behavior of radial and non radial ground states can be somehow described.

Abstract:
In this paper we compute the Leray Schauder degree for a fourth order elliptic boundary value problem with exponential nonlinearity and Navier boundary condition. This will be made by proving a Poincare'-Hopf type theorem. Moreover by using this result, together with some quantitative results about the formal set of barycenters, we are able to establish a direct and geometrically clear degree counting formula for our problem. We remark that this formula has been proven with complete different methods by Lin Wei and Wang by using blow-up type estimates.