Gibbons’ conjecture for quasilinear elliptic equations involving a gradient term

Abstract

Abstract We prove Gibbons’ conjecture for the quasilinear elliptic equation - Δ p ⁢ u + a ⁢ ( u ) ⁢ | ∇ ⁡ u | q = f ⁢ ( u ) in ⁢ ℝ N , -\Delta_{p}u+a(u)|\nabla u|^{q}=f(u)\quad\text{in }\mathbb{R}^{N}, where N ≥ 2 {N\geq 2} , 2 ⁢ N + 2 N + 2 < p < 2 {\frac{2N+2}{N+2}<p<2} , q ≥ 1 {q\geq 1} and a and f are Lipschitz continuous functions which satisfy some relevant conditions. This conjecture states that every weak solution u ∈ C 1 ⁢ ( ℝ N ) {u\in C^{1}(\mathbb{R}^{N})} of the equation with | u | ≤ 1 {|u|\leq 1} and lim x N → ± ∞ ⁡ u ⁢ ( x ′ , x N ) = ± 1 {\lim_{x_{N}\to\pm\infty}u(x^{\prime},x_{N})=\pm 1} , uniformly in x ′ ∈ ℝ N - 1 {x^{\prime}\in\mathbb{R}^{N-1}} , must depend only on x N {x_{N}} and ∂ ⁡ u ∂ ⁡ x N > 0 {\frac{\partial u}{\partial x_{N}}>0} in ℝ N {\mathbb{R}^{N}} . In particular, our result holds for a being non-decreasing on [ - 1 , - 1 + δ ] {[-1,-1+\delta]} and on [ 1 - δ , 1 ] {[1-\delta,1]} and f ⁢ ( u ) = | u | r ⁢ u ⁢ | 1 - u 2 | s ⁢ ( 1 - u 2 ) {f(u)=|u|^{r}u|1-u^{2}|^{s}(1-u^{2})} , where r , s , δ ≥ 0 {r,s,\delta\geq 0} . The main tool we use is an adaptation of the sliding method to the corresponding quasilinear elliptic operator.