Quasilinear scalar field equations with competing potentials

Existence and multiple solutions for a critical quasilinear equation with singular potentials

“…However, the case q = 2(N − 2s/b)/(N − 2) was left open in [5] since for q = 2(N − 2s/b)/(N − 2) the compact embedding breaks down in general and in a sense it is a critical exponent problem as this q appears on the boundary of the embedding range. For general p ∈ (1, N), Lyberopoulos proved in [15] (see also [16]) that when b > 0, 1 < p < N and p(N −ps/b)/(N − p) < q < p * , (1.1) has a nonzero solution if V and K satisfy (A 1 ) and (A 2 ). When V and K are radially symmetric functions, existence of solutions for (1.1) can be obtained through some compact embedding theorems of weighted Sobolev spaces.…”

“…When V and K are radially symmetric functions, existence of solutions for (1.1) can be obtained through some compact embedding theorems of weighted Sobolev spaces. Again the critical case q = p(N −ps/b)/(N −p) was not treated in [15,16], We also mention [20,21,[23][24][25][26] in which cases, certain compact embedding was proved so the Euler-Lagrange functional corresponding to (1.1) satisfies the Palais-Smale condition and the nonzero solution of (1.1) can be obtained through the standard critical point theorems.…”

Quasilinear Scalar Field Equations Involving Critical Sobolev Exponents and Potentials Vanishing at Infinity

“…As another manifestation for the merit of this undertaking, also recall the already well-elucidated fact that the functional analytic treatment of (1) becomes, even in the subcritical case, intricate when lim inf |x|→∞ V (x) = 0; see e.g. [6,8,14,18,47,64] for p = 2 and [40,41] for p > 1.…”

The weak solution of quasilinear System concerning Sobolev Critical conditions in RN

“…Based on the Lions principle [5], we can pick subsequence{ 1 2 x R p J ∈ ∈ } and { , N q y R q ∈ ∈Κ }, which J and K are countable sets and real numbers…”