We consider a semilinear elliptic equation in R 2 with the nonlinear exponent approaching innnity. In contrast to the blow-up behavior of the corresponding problem in R n with n 3, the L 1 (R 2) norms of the solutions to the equation in R 2 remain bounded from below and above. After a careful study on the decay rates of several quantities, we prove that the normalized solutions will approach th...

Abstract. We discuss a general framework for the numerical solution of a family of semilinear elliptic problems whose leading differential operator is the Laplacian. A problem is first transformed to one on a standard domain via a conformal mapping. The boundary value problem on the standard domain is then reduced to an equivalent integral operator equation. We employ the Galerkin method to sol...

<p style='text-indent:20px;'>This paper deals with stability of solution map to a parametric control problem governed by semilinear elliptic equations finite unilateral constraints, where the objective functional is not convex. By using first-order necessary optimality conditions, we derive some sufficient conditions under which upper semicontinuous respect parameters.</p>

In this article, an optimal control problem subject to a semilinear elliptic equation and mixed control-state constraints is investigated. The problem data depends on certain parameters. Under an assumption of seperation of the active sets and a second-order sufficient optimality condition, Bouliganddifferentiability (B-differentiability) of the solutions with respect to the parameter is establ...

We consider an optimal control problem subject to a semilinear elliptic PDE together with its variational discretization. We provide a condition which allows to decide whether a solution of the necessary first order conditions is a global minimum. This condition can be explicitly evaluated at the discrete level. Furthermore, we prove that if the above condition holds uniformly with respect to t...

We construct examples of strictly convex functions f on (?1; 1) satisfying f 0 (?1) < n 2 < f 0 (1) such that the Dirichlet problem u 00 + f(u) = h(x) in 0; ], u(0) = u() = 0, has an innnite number of solutions, for any choice of h(x). Kaper and Kwong earlier have presented examples with ve solutions to settle a conjecture raised by Lazer and McKenna. Here, we also give a suucient condition for...

We consider a semilinear elliptic equation in R with the nonlinear exponent approaching infinity. In contrast to the blow-up behavior of the corresponding problem in Rn with n ≥ 3, the L(R) norms of the solutions to the equation in R remain bounded from below and above. After a careful study on the decay rates of several quantities, we prove that the normalized solutions will approach the funda...

In this work, we study the local wellposedness of solution to a nonlinear elliptic-dispersive coupled system which serves as model for Micro-Electro-Mechanical System (MEMS). A simple electrostatically actuated MEMS capacitor device consists two parallel plates separated by gas-filled thin gap. The modelling combines linear elliptic equation gas pressure with semilinear dispersive gap width. We...

We generalize previous uniqueness results on a semilinear elliptic equation with zero Dirichlet boundary condition and superlinear, subcritical nonlinearity. Our proof is based on a bifurcation approach and a Pohozaev type integral identity, which greatly simplifies the previous arguments.

The main purpose of this paper is to establish the existence of nontrivial solutions to semilinear polyharmonic equations with exponential growth at the subcritical or critical level. This growth condition is motivated by the Adams inequality [1] of Moser-Trudinger type. More precisely, we consider the semilinear elliptic equation (−∆) u = f(x, u), subject to the Dirichlet boundary condition u ...