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Existence of global attractors in ${L}^{p}$ for $m$-Laplacian parabolic equation in ${ℝ}^{N}$.

Boundary Value Problems [electronic only]

Global existence, uniqueness, and asymptotic behavior of solution for $p$-Laplacian type wave equation.

Journal of Inequalities and Applications [electronic only]

Existence and nonexistence of positive solutions for singular $p$-Laplacian equation in ${ℝ}^{N}$.

Boundary Value Problems [electronic only]

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

Annales Polonici Mathematici

We consider the existence and nonexistence of solutions for the following singular quasi-linear elliptic problem with concave and convex nonlinearities: ⎧ $-{div\left(|x|}^{-ap}{|\nabla u|}^{p-2}{\nabla u\right)+h\left(x\right)|u|}^{p-2}u=g\left(x\right){|u|}^{r-2}u$, x ∈ Ω, ⎨ ⎩ ${|x|}^{-ap}{|\nabla u|}^{p-2}\partial u/\partial \nu =\lambda f\left(x\right){|u|}^{q-2}u$, x ∈ ∂Ω, where Ω is an exterior domain in ${ℝ}^{N}$, that is, $\Omega ={ℝ}^{N}\setminus D$, 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,...

${L}^{\infty }$ estimates of solution for $m$-Laplacian parabolic equation with a nonlocal term

Czechoslovak Mathematical Journal

In this paper, we consider the global existence, uniqueness and ${L}^{\infty }$ estimates of weak solutions to quasilinear parabolic equation of $m$-Laplacian type ${u}_{t}-{\mathrm{div}\left(|\nabla u|}^{m-2}{\nabla u\right)=u|u|}^{\beta -1}{\int }_{\Omega }{|u|}^{\alpha }\mathrm{d}x$ in $\Omega ×\left(0,\infty \right)$ with zero Dirichlet boundary condition in $\partial \Omega$. Further, we obtain the ${L}^{\infty }$ estimate of the solution $u\left(t\right)$ and $\nabla u\left(t\right)$ for $t>0$ with the initial data ${u}_{0}\in {L}^{q}\left(\Omega \right)$ $\left(q>1\right)$, and the case $\alpha +\beta .

Soliton solutions for quasilinear Schrödinger equation with critical exponential growth in ${ℝ}^{N}$

Applications of Mathematics

In this work, we study the existence of nonnegative and nontrivial solutions for the quasilinear Schrödinger equation $-{\Delta }_{N}u+b{|u|}^{N-2}u-{\Delta }_{N}\left({u}^{2}\right)u=h\left(u\right),\phantom{\rule{1.0em}{0ex}}x\in {ℝ}^{N},$ where ${\Delta }_{N}$ is the $N$-Laplacian operator, $h\left(u\right)$ is continuous and behaves as $exp\left(\alpha |u{|}^{N/\left(N-1\right)}\right)$ when $|u|\to \infty$. Using the Nehari manifold method and the Schwarz symmetrization with some special techniques, the existence of a nonnegative and nontrivial solution $u\left(x\right)\in {W}^{1,N}\left({ℝ}^{N}\right)$ with $u\left(x\right)\to 0$ as $|x|\to \infty$ is established.

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

Annales Polonici Mathematici

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

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