Existence and multiplicity results for some elliptic systems at resonance

Existence and multiplicity of solutions for asymptotically linear noncooperative elliptic systems

“…All papers mentioned above are based on bifurcation theory. Using variational methods, [23,14] has proved that there exist at least three solutions for semilinear elliptic equation near resonant at the first eigenvalue, subsequently, these results were extended to p-Laplacian equation in [18,13], and to cooperative systems in [20]. Results for higher eigenvalues were obtained in [12,16,21], where [12] used bifurcation from infinity and degree theory, but only for the one-dimensional case and making use of the fact that in this case all eigenvalues are simple.…”

Multiplicity of Solutions for Neumann Problems for Semilinear Elliptic Equations

“…In [10,11], these results were extended to the perturbed -Laplacian equation in . In [12], Ou and Tang extended above some results to some elliptic systems with the Dirichlet boundary conditions. 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.…”

“…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]).…”

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.…”