Infinitely many solutions for the Dirichlet problem involving the -Laplacian

Quasilinear elliptic problems in RN involving oscillatory nonlinearities

“…Our results complement not only the aforementioned papers ( [1,3,4,8,9,13], where the case p N has been treated) but also some results obtained on bounded domains where elliptic problems with oscillatory nonlinearities have been considered; Dirichlet problems were studied in [2,7,21,24], while Neumann type problems in [18,23]. The common feature of these works is that infinitely many solutions are obtained by means of various methods; for instance, sub-super solution arguments (see [21]); the general variational principle of Ricceri (see [7,18,23]); continuity of certain superposition operators (see [2,24]).…”

“…The common feature of these works is that infinitely many solutions are obtained by means of various methods; for instance, sub-super solution arguments (see [21]); the general variational principle of Ricceri (see [7,18,23]); continuity of certain superposition operators (see [2,24]). Note however that our proofs do not use any abstract argument apart from the compactness result of Lions [17] and the Palais' principle [22].…”

Infinitely many solutions for a differential inclusion problem inRN

“…Moreover, F ∞ = 2 N +p+3 p . Since L ∞ = 1, we may fix ∞ = 1 4 which verifies (4). For every k ∈ N let a k = e e (2k−1) −1 − 1 and b k = e e 2k −1 − 1.…”

“…By means of [14], Candito [5] studied (P ) (with Neumann boundary condition as well) when the nonlinearity f may possesses uncountable discontinuities. Through [16], Cammaroto et al [4] treated a version of (P ) subjected to Dirichlet boundary condition. The aforementioned papers [4,5,14,17] have the following common features: p > N; the domain is bounded; and, without any symmetry requirement on the nonlinearity (f in (P )), infinitely many solutions are guaranteed for the studied problems.…”

“…Through [16], Cammaroto et al [4] treated a version of (P ) subjected to Dirichlet boundary condition. The aforementioned papers [4,5,14,17] have the following common features: p > N; the domain is bounded; and, without any symmetry requirement on the nonlinearity (f in (P )), infinitely many solutions are guaranteed for the studied problems. These results were achieved by providing the nonlinearity with a suitable oscillatory behavior.…”