Soufiane Maatouk | Searcharxiv

arXiv2020

Quasilinear elliptic problem without Ambrosetti-Rabinowitz condition involving a potential in Musielak-Sobolev spaces setting

In this paper, we consider the following quasilinear elliptic problem with potential $$(P) \begin{cases} -\mbox{div}(ϕ(x,|\nabla u|)\nabla u)+ V(x)|u|^{q(x)-2}u= f(x,u) & \ \ \mbox{ in }Ω, u=0 & \ \ \mbox{ on } \partialΩ, \end{cases}$$ where $Ω$ is a smooth bounded domain in $\mathbb{R}^{N}$ ($N\geq 2$), $V$ is a given function in a generalized Lebesgue space $L^{s(x)}(Ω)$, and $f(x,u)$ is a Carathéodory function satisfying suitable growth conditions. Using variational arguments, we study the existence of weak solutions for $(P)$ in the framework of Musielak-Sobolev spaces. The main difficulty here is that the nonlinearity $f(x,u)$ considered does not satisfy the well-known Ambrosetti-Rabinowitz condition.

math.AP↗

arXiv2019

Multiplicity of solutions for a class of elliptic problem of $p$-Laplacian type with a $p$-Gradient term

We consider the following problem $$(P) \begin{cases} -Δ_{p}u= c(x)|u|^{q-1}u+μ|\nabla u|^{p}+h(x) & \ \ \mbox{ in }Ω, u=0 & \ \ \mbox{ on } \partialΩ, \end{cases}$$ where $Ω$ is a bounded set in $\mathbb{R}^{N}$ ($N\geq 3$) with a smooth boundary, $1 0$, $μ\in \mathbb{R}^{*}$, and $c$ and $ h$ belong to $L^{k}(Ω)$ for some $k>\frac{N}{p}$. In this paper, we assume that $c\gneqq 0$ a.e. in $Ω$ and $h$ without sign condition, then we prove the existence of at least two bounded solutions under the condition that $\|c\|_{k}$ and $\|h\|_{k}$ are suitably small. For this purpose, we use the Mountain Pass theorem, on an equivalent problem to $(P)$ with variational structure. Here, the main difficulty is that the nonlinearity term considered does not satisfy Ambrosetti and Rabinowitz condition. The key idea is to replace the former condition by the \textbf{nonquadraticity condition at infinity}.

math.AP↗