We classify stable and finite Morse index solutions to general semilinear elliptic equations posed in Euclidean space of dimension at most 10, or some unbounded domains.

Elliptic equations: 1. Harmonic functions 2. Perron’s method 3. Potential theory 4. Existence results; the method of suband supersolutions 5. Classical maximum principles for elliptic equations 6. More regularity, Schauder’s theory for general elliptic operators 7. The weak solution approach in one space dimension 8. Eigenfunctions for the Sturm-Liouville problem 9. Generalization to more dimen...

Non-existence and uniqueness results are proved for several local and non-local supercritical bifurcation problems involving a semilinear elliptic equation depending on a parameter. The domain is star-shaped and such that a Poincaré inequality holds but no other symmetry assumption is required. Uniqueness holds when the bifurcation parameter is in a certain range. Our approach can be seen, in s...

Abstract. Second-order sufficient optimality conditions are established for the optimal control of semilinear elliptic and parabolic equations with pointwise constraints on the control and the state. In contrast to former publications on this subject, the cone of critical directions is the smallest possible in the sense that the second-order sufficient conditions are the closest to the associat...

We show that, under so called controllable growth conditions, any weak solution in the energy class of the semilinear parabolic system ut(t, x) +Au(t, x) = f(t, x, u, . . . ,∇u), (t, x) ∈ (0, T )× Ω, is locally regular. Here, A is an elliptic matrix differential operator of order 2m. The result is proved by writing the system as a system with linear growth in u, . . . ,∇mu but with “bad” coeffi...

When an unbounded domain is inside a slab, existence of a positive solution is proved for the Dirichlet problem of a class of semilinear elliptic equations that are similar either to the singular Emden-Fowler equation or a sublinear elliptic equation. The result obtained can be applied to equations with coefficients of the nonlinear term growing exponentially. The proof is based on the super an...

In this paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are stable or more generally finite Morse index even locally stable.

In the present paper the Dirichlet problem for semilinear elliptic and parabolic equations in general form is considered. New condition guaranteeing the global solvability of this problem for a wide class of superlinear sources, including e u and |u|p−1u , p > 1 , is formulated. For sublinear case (for example ln(1+ |u|) or |u|p−1u , p < 1) this condition is automatically fulfilled. Our approac...