Quasilinear nonhomogeneous Schrödinger equation with critical exponential growth in R^n

Quasilinear Schrödinger equations involving critical exponents in $mathbb{textbf{R}}^2$

‎We study the existence of soliton solutions for a class of‎ ‎quasilinear elliptic equation in $mathbb{textbf{R}}^2$ with critical exponential growth‎. ‎This model has been proposed in the self-channeling of a‎ ‎high-power ultra short laser in matter‎.

Eigenvalue Problems for a Quasilinear Elliptic Equation on Rn

where λ ∈ R. Next, we state the general hypotheses which will be assumed throughout the paper. (E) Assume that N , p satisfy the following relation N > p > 1. (G) g is a smooth function, at least C1,α(RN ) for some α∈ (0,1), such that g ∈ L∞(RN ) and g(x) > 0, on Ω+, with measure of Ω+, |Ω+| > 0. Also there exist R0 sufficiently large and k > 0 such that g(x) <−k, for all |x| > R0. Generally, p...

Existence of standing wave solutions for coupled quasilinear Schrödinger systems with critical exponents in RN

This paper is concerned with the following quasilinear Schrödinger system in RN : { −ε2∆u + V1(x)u− ε2∆(u2)u = K1(x)|u| −2u + h1(x, u, v)u, −ε2∆v + V2(x)v− ε2∆(v2)v = K2(x)|v| −2v + h2(x, u, v)v, where N ≥ 3, Vi(x) is a nonnegative potential, Ki(x) is a bounded positive function, i = 1, 2. h1(x, u, v)u and h2(x, u, v)v are superlinear but subcritical functions. Under some proper conditions, min...

N−laplacian Equations in Rn with Critical Growth

We study the existence of nontrivial solutions to the following problem: { u ∈ W 1,N (R ), u ≥ 0 and −div(| ∇u |N−2 ∇u) + a (x) | u |N−2 u = f(x, u) in R (N ≥ 2), where a is a continuous function which is coercive, i.e., a (x) → ∞ as | x |→ ∞ and the nonlinearity f behaves like exp ( α | u |N/(N−1) ) when | u |→ ∞.

Ja n 20 04 Schrödinger equation with critical Sobolev exponent