Positive solutions of asymptotically linear elliptic eigenvalue 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...

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

Asymptotic Expansion of Solutions to Nonlinear Elliptic Eigenvalue Problems

We consider the nonlinear eigenvalue problem −∆u+ g(u) = λ sinu in Ω, u > 0 in Ω, u = 0 on ∂Ω, where Ω ⊂ RN (N ≥ 2) is an appropriately smooth bounded domain and λ > 0 is a parameter. It is known that if λ 1, then the corresponding solution uλ is almost flat and almost equal to π inside Ω. We establish an asymptotic expansion of uλ(x) (x ∈ Ω) when λ 1, which is explicitly represented by g.

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...

Symmetry of Positive Solutions of Asymptotically Symmetric Parabolic Problems

In this paper we investigate symmetry properties of positive solution of quasilinear parabolic problems in the whole space. As the main result, we prove that if the problem converges exponentially to a symmetric one, then the solution converges to the space of symmetric functions. We also show, that this result does not hold true, if the convergence is not exponential.