We consider an optimal control problem for systems governed by a highly nonlinear second order elliptic partial differential equation, with control and state constraints. The problem is formulated in the classical and in the relaxed form, and various necessary conditions for optimality are given. For the numerical solution of these problems, we propose a penalized gradient projection method gen...

This survey article is concerned with two basic approximation concepts and their interrelation with the numerical solution of elliptic operator equations, namely nonlinear and adaptive approximation. On one hand, for nonlinear approximation based on wavelet expansions the best possible approximation rate, which a function can have for a given number of degrees of freedom, is characterized in te...

We establish the existence and multiplicity of solutions for a class of fourth-order superlinear elliptic problems under Navier conditions on the boundary. Here we do not use the Ambrosetti-Rabinowitz condition; instead we assume that the nonlinear term is a nonlinear function which is nonquadratic at infinity.

Via non-smooth critical point theory, we prove existence of entire positive solutions for a class of nonlinear elliptic problems with asymptotic p-Laplacian behaviour and subjected to natural growth conditions.

Keywords: Boundary value problems Differential-operator equations Banach-valued Besov spaces Operator-valued multipliers Interpolation of Banach spaces a b s t r a c t The boundary value problems for linear and nonlinear degenerate differential-operator equations in Banach-valued Besov spaces are studied. Several conditions for the separability of linear elliptic problems are given. Moreover, t...

We consider a singular elliptic perturbation of a Hammerstein integral equation with singular nonlinear term at the origin. The compactness of the solutions to the perturbed problem and, hence, the existence of a positive solution for the integral equation are proved. Moreover, these results are applied to nonlinear singular homogeneous Dirichlet problems.

Using a method that goes back to J. Saint Raymond, we prove the existence of infinitely many weak solutions of certain nonlinear elliptic problems defined on the SG. Mathematics Subject Classification (2010): 35J20, 28A80, 35J25, 35J60, 47J30, 49J52.

We study the uniqueness and stability of nonnegative solutions for classes of nonlinear elliptic Dirichlet problems on a ball, when the nonlinearity is monotone, negative at the origin, and either concave or convex.

In this paper, we first give an intersection theory for moduli problems for nonlinear elliptic operators with certain precompact space of solutions in differential geometry. Then we apply the theory to constructing Gromov-Witten invariants for general symplectic manifolds.