A semilinear elliptic equation with strong resonance at infinity and with a nonsmooth potential is studied. Using nonsmooth critical point theory and developing some abstract minimax principles which complement and extend results in the literature, two results on existence are obtained.

A semilinear elliptic equation, −∆u = λf(u), is studied in a ball with the Dirichlet boundary condition. For a closed subgroup G of the orthogonal group, it is proved that the number of non-radial G invariant solutions diverges to infinity as λ tends to ∞ if G is not transitive on the unit sphere.

In this article, we study the semilinear elliptic equation −∆u = |u|p(x)−2u, x ∈ R u ∈ D(R ), where N ≥ 3, p(x) = ( p, x ∈ Ω 2∗, x 6∈ Ω, with 2 < p < 2∗ := 2N/(N − 2), Ω ⊂ RN is a bounded set with nonempty interior. By using the Nehari manifold, we obtain a positive ground state solution.

We study the existence of positive radial solutions to the singular semilinear elliptic equation {−∆u = f (x, u) , in B u = 0, x ∈ ∂B. Throughout, our nonlinearity is allowed to change sign. The singularity may occur at u = 0 and |x | = 1. © 2005 Elsevier Ltd. All rights reserved. MSC: 34B15; 35J20

Abstract. We study the Cauchy problem for a semilinear stochastic partial differential equation driven by a finite-dimensional Wiener process. In particular, under the hypothesis that all the coefficients are sufficiently smooth and have bounded derivatives, we consider the equation in the context of power scale generated by a strongly elliptic differential operator. Application of semigroup ar...

We show that in dimension two or greater, a certain equivalence class of the scalar coefficient a(x, u) of the semilinear elliptic equation ∆u + a(x, u) = 0 is uniquely determined by the Dirichlet to Neumann map of the equation on a bounded domain with smooth boundary. We also show that the coefficient a(x, u) can be determined by the Dirichlet to Neumann map under some additional hypotheses.

In this article, we study the existence of infinitely many solutions for the semilinear elliptic equation −∆u+a(x)u = f(x, u) in a bounded domain of RN (N ≥ 3) with the Dirichlet boundary conditions, where the primitive of the nonlinearity f is either superquadratic at infinity or subquadratic at zero.

An optimal control problem for an semilinear elliptic equation is investigated, where pointwise constraints are given on the control and the state. The state constraints are of mixed (bottleneck) type, where associated Lagrange multipliers can assumed to be bounded and measurable functions. Based on this property, a second-order sufficient optimality condition is established that considers stro...

Semilinear elliptic optimal control problems with pointwise control and mixed control-state constraints are considered. Necessary and sufficient optimality conditions are given. The equivalence of the SQP method and Newton’s method for a generalized equation is discussed. Local quadratic convergence of the SQP method is proved.

Imperfect bifurcation phenomena are formulated in framework of analytical bifurcation theory on Banach spaces. In particular the perturbations of transcritical and pitchfork bifurcations at a simple eigenvalue are examined, and two-parameter unfoldings of singularities are rigorously established. Applications include semilinear elliptic equations, imperfect Euler buckling beam problem and pertu...