Degenerate Semilinear Elliptic Problems Near Resonance With a Nonprincipal Eigenvalue

Abstract

Abstract. Using the minimax methods in critical point theory, we study the multiplicity of solutions for a class of degenerate Dirichlet problem in the case near resonance.

Multiplicity of Solutions for Neumann Problems for Semilinear Elliptic Equations

“…Here, it is worth pointing out that ( ∞ ) is weaker than ( 1 ) in [13] (or ( ) in [14]). More to the point, there are functions satisfying the assumptions of our main results in Section 2 and not satisfying the assumptions in [13,14]. For example, let ( , ) = / ln(1 + | |).…”

“…de Paiva and Massa in [13], especially, studied the semilinear elliptic boundary value problem in any spatial dimension and using variational techniques; they showed that a suitable perturbation will turn the almost resonant situation ( near to , i.e., near resonance with a nonprincipal eigenvalue) in a situation where the solutions are at least two. In [14], those results were extended to the degenerate elliptic equations in the bounded domain.…”

“…Motivated by the above idea, we have the goal in this paper of extending these results in [6,[12][13][14] to some elliptic equations with the Neumann boundary conditions. Here, it is worth pointing out that ( ∞ ) is weaker than ( 1 ) in [13] (or ( ) in [14]). More to the point, there are functions satisfying the assumptions of our main results in Section 2 and not satisfying the assumptions in [13,14].…”

The Neumann Problem for a Degenerate Elliptic System Near Resonance

“…Moreover, this result was extended to some equations and systems; see [6][7][8][9][10]. In particular, Massa and Rossato [11] studied a nondegenerate elliptic system and two solutions were obtained by using Galerkin techniques.…”

“…In recent decades, many kinds of perturbed problems were studied by many scholars, such as [1][2][3][4][5][6][7][8][9][10][11]. Here, we want to say that the authors in [5] studied the following Dirichlet boundary problem: −Δ = ± ( , ) + ℎ ( ) , ∈ Ω, = 0, ∈ Ω.…”