Quasilinear elliptic equations on $${\mathbb{R}^{N}}$$ with singular potentials and bounded nonlinearity

Abstract

We study the existence and multiplicity of nontrivial radial solutions of the quasilinear equationwith singular radial potentials V, Q and bounded nonlinearity f . The approaches used here are based on a compact embedding from W 1,p r (R N ; V ) into L 1 (R N ; Q) and minimax methods. A uniqueness result is given for f ≡ 1. (2000). 35J05 · 35J20 · 35J60 · 58C20. Mathematics Subject Classification

Quasilinear elliptic equations with singular potentials and bounded discontinuous nonlinearities

“…We will establish a new multiple critical point theorem for non-smooth functional which extends one theorem in [16] for smooth functional. The result of Theorem 1.2 is completely new and completes the statement in the end of [12]. We point out that this type of result is even new for the semilinear elliptic boundary value problem on bounded domains with discontinuous nonlinearities.…”

“…In [12], the following compact embedding theorem has been established. Theorem 1.1 (Theorem 2.1 in [12]). Assume (V) and (Q) with…”

“…In [13] the authors further explored the effects of the potentials and extend with 1 < q < p and then to study (P) with sub-p-linear nonlinearity. In [12] the compact embedding W 1,p r (R N ; V ) → L 1 (R N ; Q) (Theorem 1.1) was established and then was applied to the solvability of the quasilinear problem (P) with a bounded continuous nonlinearity f . Since there is no relation between L q (R N ; Q) and L s (R N ; Q) for any q and s, the embedding from W 1,p r (R N ; V ) into L q (R N ; Q) are independent from each other for q ≥ p, 1 < q < p and q = 1.…”

Compactness and existence results for the p-Laplace equation

“…The proof of Theorem 2.5 does not require β ∞ ≥ 0, but this condition is not a restriction of generality in stating the theorem. Indeed, under assumption (14), if (13) holds with β ∞ < 0, then it also holds with α ∞ and β ∞ replaced by α ∞ − β ∞ γ ∞ and 0 respectively, and this does not change the thesis (15), because…”

Positive solutions for a class of quasilinear problems with critical growth in ℝN

“…A standard argument shows that critical points of Ψ ε are solutions of (P ε ) (see [1,3,33]). Let N ε denote the Nehari manifold related to Ψ ε , given by…”