Adams’ type inequality and application to a quasilinear nonhomogeneous equation with singular and vanishing radial potentials in $$ \mathbb {R}^4 $$ R 4

Abstract

In this paper, we establish some Adams' type inequality for weighted second-order Sobolev spaces in four dimensions. The weights are radial and can have a singular or decaying behavior. This inequality is used to study some nonhomogeneous quasilinear elliptic equation. Keywords Adams' inequality • Singular or decaying weights • Radial functions • Nonhomogeneous quasilinear elliptic equation • Exponential critical growth Mathematics Subject Classification 35A23 • 35B33 • 35J30 • 35J35 • 35J91 N −1 , |∇u| N is the Dirichlet norm, that is, |∇u| N := |∇u| N dx 1 N and ω N −1 is the measure of the unit sphere in R N. In addition, for any bounded domain , the supremum in (1.1) can be attained. This result has been generalized and extended in many B Sami Aouaoui

On a weighted Adams type inequality and an application to a biharmonic equation

“…In this paper, we improve and extend some results obtained in [30][31][32][33][34][35][36][37][38][39][40][43][44][45][46], in the sense that we have considered nonlinearities with critical exponential growth and potentials that can vanish at infinity. These features considered here are not treated in these previous works.…”

“…This notion was motivated by the Trudinger-Moser inequality (1.1). Sani [43], Aouaoui and Albuquerque [44], Miyagaki et al [45], and Yang [46] have applied these ideas to treat some fourth-order problems. In these works, following the ideas from [41,42] and motivated by Adams' inequality (1.2), Sani [43] studied a class of problems involving the biharmonic operator and a class of spherically symmetric potentials (or even coercive) and bounded from below by a positive constant.…”

“…In these works, following the ideas from [41,42] and motivated by Adams' inequality (1.2), Sani [43] studied a class of problems involving the biharmonic operator and a class of spherically symmetric potentials (or even coercive) and bounded from below by a positive constant. Aouaoui and Albuquerque [44] considered potentials and weights that are radial and can have singularity at the origin and can vanish at infinity. Miyagaki et al [45] studied the existence of ground state solution for fourth-order elliptic equations of the form…”

“…In these works, following the ideas from [41, 42] and motivated by Adams' inequality (), Sani [43] studied a class of problems involving the biharmonic operator and a class of spherically symmetric potentials (or even coercive) and bounded from below by a positive constant. Aouaoui and Albuquerque [44] considered potentials and weights that are radial and can have singularity at the origin and can vanish at infinity. Miyagaki et al [45] studied the existence of ground state solution for fourth‐order elliptic equations of the form $$$$ {\Delta}^2u-\Delta u+u=Q(x)\left({f}_1(u)+{f}_2(u)\right)\kern0.40em \mathrm{in}\kern0.40em {\mathrm{\mathbb{R}}}^4, $$$$ where $$$ {f}_1 $$$ is a continuous nonnegative function with polynomial growth at infinity, $$$ {f}_2 $$$ is a continuous nonnegative function with exponential growth, and $$$ Q $$$ is a positive bounded continuous function that can vanish at infinity in sense that if $$$ \left\{{A}_n\right\} $$$ is a sequence of Borel sets of $$$ \mathrm{\mathbb{R}} $$$ with $$$ {\sup}_{n\in \mathrm{\mathbb{N}}}\mid {A}_n\mid \le R $$$, for some $$$ R>0 $$$, then …”