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Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition

100%

Xiu Z., Chen C.

Annales Polonici Mathematici

2013

tom 109

nr 1

93-107

We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ $-div(|x|^{-ap} |∇u|^{p-2} ∇u) + h(x)|u|^{p-2}u = g(x)|u|^{r-2}u$, x ∈ Ω, ⎨ ⎩ $|x|^{-ap}|∇u|^{p-2} ∂u/∂ν = λf(x)|u|^{q-2}u$, x ∈ ∂Ω, where Ω is an exterior domain in $ℝ^N$, that is, $Ω = {ℝ^N}∖D$, where D is a bounded domain in $ℝ^N$ with smooth boundary ∂D(=∂Ω), and 0 ∈ Ω. Here λ > 0, 0 ≤ a < (N-p)/p, 1 < p< N, ∂/∂ν is the outward normal derivative on ∂Ω. By the variational method, we prove the existence of multiple solutions. By the test function method, we give a sufficient condition under which the problem has no nontrivial nonnegative solutions.

Positive solution for a quasilinear equation with critical growth in $ℝ^N$

81%

Chen L., Chen C., Xiu Z.

Annales Polonici Mathematici

2016

tom 116

nr 3

251-262

We study the existence of positive solutions of the quasilinear problem ⎧ $-Δ_N u + V(x)|u|^{N-2}u = f(u,|∇u|^{N-2}∇u)$, $x ∈ ℝ^N$, ⎨ ⎩ u(x) > 0, $x∈ ℝ^N$, where $Δ_N u = div(|∇u|^{N-2}∇u)$ is the N-Laplacian operator, $V:ℝ^N → ℝ$ is a continuous potential, $f:ℝ × ℝ^N → ℝ$ is a continuous function. The main result follows from an iterative method based on Mountain Pass techniques.

Ograniczanie wyników

2 Annales Polonici Mathematici

2 Chen C.

2 Xiu Z.

1 Chen L.

1 2016

1 2013

Wyniki wyszukiwania

Existence and nonexistence of solutions for a singular elliptic problem with a nonlinear boundary condition

Positive solution for a quasilinear equation with critical growth in $ℝ^N$