TWO-SIDED ESTIMATES FOR POSITIVE SOLUTIONS OF SUPERLINEAR ELLIPTIC BOUNDARY VALUE PROBLEMS

We give two-sided estimates for positive solutions of the superlinear elliptic problem$-\unicode[STIX]{x1D6E5}u=a(x)|u|^{p-1}u$with zero Dirichlet boundary condition in a bounded Lipschitz domain. Our result improves the well-knowna priori$L^{\infty }$-estimate and provides information about the boundary decay rate of solutions.

Infinitely Many Solutions of Superlinear Elliptic Equation

Via the Fountain theorem, we obtain the existence of infinitely many solutions of the following superlinear elliptic boundary value problem:−Δu=f(x,u)inΩ,u=0on∂Ω, whereΩ⊂ℝN (N>2)is a bounded domain with smooth boundary andfis odd inuand continuous. There is no assumption near zero on the behavior of the nonlinearityf, andfdoes not satisfy the Ambrosetti-Rabinowitz type technical condition near infinity.

Infinitely Many Solutions of Elliptic Problems With Perturbed Symmetries in Symmetric Domains

We study superlinear elliptic boundary value problems with perturbed symmetries in domains which are invariant under the action of a group G. We give conditions on the growth of the nonlinearity which guarantee the existence of infinitely many G-invariant solutions. These conditions improve those obtained by Bahri and Lions (1988) and Bolle, Ghoussoub and Tehrani (2000) if the domain contains a G-orbit of large enough dimension.

Multiplicity results for semilinear elliptic boundary value problems in Besov and Triebel-Lizorkin spaces

The paper deals with superlinear elliptic boundary value problems depending on a parameter. Given appropriate hypotheses concerning the asymptotic behaviour of the nonlinearity, we prove lower bounds on the number of solutions. The results generalize a theorem due to Lazer and McKenna within the framework of quasi-Banach spaces of Besov and Triebel-Lizorkin spaces.