We consider the uniqueness of the inverse problem for a semilinear elliptic differential equation with Dirichlet condition. The necessary and sufficient condition of a unique solution is obtained. We improved the results obtained by Isakov and Sylvester (1994) for the same problem.

We consider semilinear elliptic equations ∆u ± ρ(x)f(u) = 0, or more generally ∆u + φ(x, u) = 0, posed in RN (N ≥ 3). We prove that the existence of bounded positive entire solutions is closely related to the existence of bounded solution for ∆u + ρ(x) = 0 in RN . Many sufficient conditions which are invariant under the isometry group of RN are established. Our proofs use the standard barrier m...

We study a class of fractional semilinear elliptic equations and formulate the corresponding Calder\'on problem. determine nonlinearity from exterior partial measurements Dirichlet-to-Neumann map by using first order linearization Runge approximation property.

We study a new class of backward stochastic di€erential equations, which involves the integral with respect to a continuous increasing process. This allows us to give a probabilistic formula for solutions of semilinear partial di€erential equations with Neumann boundary condition, where the boundary condition itself is nonlinear. We consider both parabolic and elliptic equations.

This paper is devoted to the study of semilinear degenerate elliptic boundary value problems arising in combustion theory which obey the simple Arrhenius rate law and a general Newton law of heat exchange. We prove that ignition and extinction phenomena occur in the stable steady temperature profile at some critical values of a dimensionless rate of heat production. 2000 Mathematics Subject Cla...

Using minimax methods we study the existence and multiplicity of nontrivial solutions for a singular class of semilinear elliptic nonhomogeneous equation where the potentials can change sign and the nonlinearities may be unbounded in x and behaves like exp(αs2) when |s| → +∞. We establish the existence of two distinct solutions when the perturbation is suitable small.

We obtain upper bounds for the number of nodal domains of sign changing solutions of semilinear elliptic Dirichlet problems using suitable min-max descriptions. These are consequences of a generalization of Courant’s nodal domain theorem. The solutions need not to be isolated. We also obtain information on the Morse index of solutions and the location of suband supersolutions.

We study the numerical approximation of boundary optimal control problems governed by semilinear elliptic partial differential equations with pointwise constraints on the control. The analysis of the approximate control problems is carried out. The uniform convergence of discretized controls to optimal controls is proven under natural assumptions by taking piecewise constant controls. Finally, ...