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

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

In this paper, we consider the global existence, uniqueness and ${L}^{\infty}$ estimates of weak solutions to quasilinear parabolic equation of $m$-Laplacian type ${u}_{t}-{\mathrm{div}\left(\right|\nabla u|}^{m-2}{\nabla u)=u|u|}^{\beta -1}{\int}_{\Omega}{\left|u\right|}^{\alpha}\mathrm{d}x$ in $\Omega \times (0,\infty )$ 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 ${u}_{0}\in {L}^{q}\left(\Omega \right)$ $(q>1)$, and the case $\alpha +\beta <m-1$.

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

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

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