By means of the modified method test functions, we obtain sufficient conditions absence nontrivial solutions for some classes semilinear elliptic inequalities higher order and quasilinear containing nonhomogeneous terms (independent unknown function).

We study a semilinear fourth order elliptic problem with exponential nonlinearity. Motivated by a question raised in [Li], we partially extend known results for the corresponding second order problem. Several new difficulties arise and many problems still remain to be solved. We list the ones we feel particularly interesting in the final section. Mathematics Subject Classification: 35J65; 35J40.

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

The Gibbons conjecture stating the one-dimensional symmetry of certain solutions of semilinear elliptic equations has been proved by several authors. We show how attractivity properties of minimal propagating terraces of one-dimensional parabolic problems can be used in a proof of a version of this result and related statements.

We study local bifurcation from an eigenvalue with multiplicity greater than one for a class of semilinear elliptic equations. In particular, we obtain the exact number of bifurcation branches of non trivial solutions at every eigenvalue of a square and at the second eigenvalue of a cube. We also compute the Morse index of the solutions in those branches.

In this paper a class of semilinear elliptic optimal control problem with pointwise state and control constraints is studied. We show that sufficient second order optimality conditions for regularized problems with small regularization parameter can be obtained from a second order sufficient condition assumed for the unregularized problem. Moreover, error estimates with respect to the regulariz...

We solve elliptic semilinear boundary value problems in which the nonlinear term is superlinear. By weakening the hypotheses , we are able to include more equations than hitherto permitted. In particular, we do not require the superquadrac-ity condition imposed by most authors, and it is not assumed that the region is bounded.