Semilinear elliptic problems near resonance with a nonprincipal eigenvalue

Abstract

We consider the Dirichlet problem for the equation − u = λu ± f (x, u) + h(x) in a bounded domain, where f has a sublinear growth and h ∈ L 2 . We find suitable conditions on f and h in order to have at least two solutions for λ near to an eigenvalue of − . A typical example to which our results apply is when f (x, u) behaves at infinity like a(x)|u| q−2 u, with M > a(x) > δ > 0, and 1 < q < 2.

Degenerate Semilinear Elliptic Problems Near Resonance With a Nonprincipal Eigenvalue

“…In the present paper, we extend the main results of [7] to the variational degenerate elliptic problem (1) by Local Saddle Point Theorem [12,8] and Mountain Pass Lemma. Our main results are the following theorems.…”

“…Results for higher eigenvalues were obtained in [9], [13] and [7]. Where [9] only considered the one-dimensional case via bifurcation from infinity and degree theory.…”

“…[13] used bifurcation theory to deal with the eigenvalues of odd multiplicity. Recently, in [7], de Paiva and Massa considered this problem in any spatial dimension, and proved that there exist at least two solutions near resonance with any nonprincipal eigenvalue.…”

The Neumann Problem for a Degenerate Elliptic System Near Resonance

“…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.…”

“…Here, we want to say that the authors in [5] studied the following Dirichlet boundary problem: −Δ = ± ( , ) + ℎ ( ) , ∈ Ω, = 0, ∈ Ω.…”