Browsing by Subject "Nehari manifold"

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Access status: Open Access ,
Double phase problems: a survey of some recent results
(Wydawnictwa AGH, 2022) Papageorgiou, Nikolaos Socrates
We review some recent results on double phase problems. We focus on the relevant function space framework, which is provided by the generalized Orlicz spaces. We also describe the basic tools and methods used to deal with double phase problems, given that there is no global regularity theory for these problems.
Access status: Open Access ,
Multiplicity result for mixed local and nonlocal Kirchhoff problems involving critical growth
(Wydawnictwa AGH, 2025) Tripathi, Vinayak Mani
In this paper, we study the multiplicity of nonnegative solutions for the following nonlocal elliptic problem $\textcolor{white}\$ \begin{cases}M\Big(\ \int_{\mathbb{R}^N}|\nabla u|^2dx+\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\Big)\mathcal{L}(u) \\ = \lambda {f(x)}|u|^{p-2}u+|u|^{2^*-2}u &\text{ in }\Omega, \\ u=0 &\text{ on }\mathbb R^N\setminus \Omega, \qquad \qquad \end{cases} \textcolor{white}\$$ where $\Omega\subset\mathbb{R}^N$ is bounded domain with smooth boundary, $1\lt p\lt 2\lt 2^*=\frac{2N}{N-2}$, $N\geq 3$, $\lambda\gt 0$, $M$ is a Kirchhoff coefficient and $\mathcal{L}$ denotes the mixed local and nonlocal operator. The weight function $f\in L^{\frac{2^*}{2^*-p}}(\Omega)$ is allowed to change sign. By applying variational approach based on constrained minimization argument, we show the existence of at least two nonnegative solutions.