Multiple solutions for asymptotically linear resonant elliptic problems

Multiple Solutions for Asymptotically Linear Resonant Elliptic Problems

In this paper we establish the existence of multiple solutions for the semilinear elliptic problem (1.1) −∆u = g(x, u) in Ω, u = 0 on ∂Ω, where Ω ⊂ RN is a bounded domain with smooth boundary ∂Ω, a function g: Ω×R→ R is of class C1 such that g(x, 0) = 0 and which is asymptotically linear at infinity. We considered both cases, resonant and nonresonant. We use critical groups to distinguish the c...

Multiple Solutions for Nonlinear Discontinuous Strongly Resonant Elliptic Problems

We consider quasilinear strongly resonant problems with discontinuous right-hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points. We prove the existence of at least three nontrivial solutions. Our approach uses the nonsmooth critical point theory for locally Lipschitz functionals due to Chang (1981) and ...

Multiple Solutions for Strongly Resonant Nonlinear Elliptic Problems with Discontinuities

We examine a nonlinear strongly resonant elliptic problem driven by the p-Laplacian and with a discontinuous nonlinearity. We assume that the discontinuity points are countable and at them the nonlinearity has an upward jump discontinuity. We show that the problem has at least two nontrivial solutions without using a multivalued interpretation of the problem as it is often the case in the liter...

Bifurcation Problem for Biharmonic Asymptotically Linear Elliptic Equations

In this paper, we investigate the existence of positive solutions for the ellipticequation $Delta^{2},u+c(x)u = lambda f(u)$ on a bounded smooth domain $Omega$ of $R^{n}$, $ngeq2$, with Navier boundary conditions. We show that there exists an extremal parameter$lambda^{ast}>0$ such that for $lambda< lambda^{ast}$, the above problem has a regular solution butfor $lambda> lambda^{ast}$, the probl...

Multiplicity of Solutions for a Class of Fourth-Order Elliptic Problems with Asymptotically Linear Term