In this paper, the authors study an optimal control problem for quasilinear elliptic partial differential equations with pointwise state constraints. Weak and strong optimality conditions of Pontryagin maximum principle type are derived. In proving these results, we penalized the state constraints and respectively use the Ekeland variational principle and an exact penalization method.

The generalized boundary value problem (BVP for short) for $n$-th order ordinary differential equations generates a mapping from a Banach space into another Banach space which mapping in most cases is the sum of a Fredholm mapping of index $0$ and of a completely continuous mapping. The properties of such mappings are very thoroughly studied in the paper, especially generic properties, surjectivity and the bifurcation problem. Applications to the generalized BVP are given and in the case when the corresponding ar homogeneous BVP is positive or self-adjoint some new results have been derived.

$\Omega\subset\mathbb{R}^n(n\geq7)$ be a bounded domain with smooth boundary. For $\lambda>0$, let $u_\lambda$ be a solution of $$ \begin{align} -\Delta u+\lambda u&=u^{n+2\over n-2}\quad\rm{in }\quad\Omega,\\ u&>0\quad\rm{in }\quad\Omega,\\ {\partial u\over\partial\nu}&=0\quad\rm{on }\quad\partial\Omega, \end{align} $$ whose energy is less than the first critical level. Here we study the blow up points and the $L^\infty$-estimates of $u_\lambda$ as $\lambda\to\infty$. We show that the blow up points are the critical points of the mean curvature on the boundary.

In this paper, we provide alternative simple proofs for some of the results in [12], with applications to nonlinear systems, and an extension of the necessary and sufficient condition for the Maximum Principle to some cooperative quasi-linear elliptic systems involving the $p$-Laplacian. We also show that under the same conditions, we have existence of positive solutions.

We present new asymptotic properties of solutions of some quasilinear heat equations with absorption or source including the equations $$ u_t = \nabla \cdot (u^{1/2} \nabla u) - u^{1/2}, \quad u_t = \nabla \cdot (u^{-1/2} \nabla u) - u^{1/2}, $$ $$ u_t = \nabla \cdot (u^{-1/2} \nabla u) + u^{3/2}, $$ $$ u_t = \nabla \cdot (u^{-4/(N+2)} \nabla u) + u^{(N+6)/(N+2)}, \quad x \in \Bbb R^N , \,\, N \ge 1. $$ We show that these equations admit explicit solutions which are nonsymmetric and nonmonotone in spatial variables near the extinction of blow-up time. The corresponding equations are shown to be reduced to finite dimensional dynamical systems on linear subspaces which are invariant under certain nonlinear reaction-diffusion operators.

In this paper we prove global $L^q$-regularity of solutions to a class of nonlinear degenerate elliptic equations.

We give a new (and unified) proof of a local result of existence and multiplicity for periodic solutions of the pendulum equation. We also show that, under a suitable assumption, there can be generically a finite number of geometrically distinct solutions.

Several versions of the classical maximum principle, which are valid for parabolic differential problems, are proved to hold for second order integro-differential problems.

We consider the sublinear Sturm-Liouville problems with power nonlinearity. We establish asymptotic formulas of eigenvalues as the $L^2$-norm and $L^\infty$-norm of the corresponding eigenfunctions tend to zero.

We give a principle of linearized stability of stationary solutions of semilinear Volterra integral equations in Banach spaces.