Eigenvalue type problem in s(.,.)s(.,.)-fractional Musielak-Sobolev spaces

In this paper, first we introduce the $s(.,.)$-fractional Musielak-Sobolev spaces $W^{s(x,y)}L_{\varPhi_{x,y}}(\Omega)$. Next, by means of Ekeland's variational principal, we show that there exists $\lambda_*>0$ such that any $\lambda\in(0, \lambda_*)$ is an eigenvalue for the following problem $$(\mathcal{P}_a) \left\{ \begin{array}{ll}\left( -\Delta\right)^{s(x,.)}_{a_{(x,.)}} u = \lambda |u|^{q(x)-2}u &\quad {\rm in}\ \Omega, \\ \qquad\quad u = 0 &\quad {\rm in }\ \mathbb{R}^N\setminus \Omega, \end{array} \right.$$ where $\Omega$ is a bounded open subset of $\mathbb{R}^N$ with $C^{0,1}$-regularity and bounded boundary.Comment: arXiv admin note: text overlap with arXiv:2203.01756, arXiv:2007.1104

Some characterization results in the calculus of variations in the degenerate case

Abstract In this article, we prove an approximation result in weighted Sobolev spaces and we give an application of this approximation result to a necessary condition in the calculus of variations. Mathematics Subject Classification: 46E3

Existence of solutions for some nonlinear elliptic unilateral problems with measure data

In this paper, we prove the existence of an entropy solution to unilateral problems associated to the equations of the type: Au + H(x, u, ∇u) − divφ(u) = µ ∈ L 1 (Ω) + W −1,p ′ (x) (Ω), where A is a Leray-Lions operator acting from W 1,p(x) 0 (Ω) into its dual W −1,p(x) (Ω), the nonlinear term H(x, s, ξ) satisfies some growth and the sign conditions and φ(u) ∈ C 0 (R, R N)

Existence of solutions for quasilinear degenerate elliptic equations

In this paper, we study the existence of solutions for quasilinear degenerate elliptic equations of the form $A(u)+g(x,u,abla u)=h$, where $A$ is a Leray-Lions operator from $W_0^{1,p}(Omega,w)$ to its dual. On the nonlinear term $g(x,s,xi)$, we assume growth conditions on $xi$, not on $s$, and a sign condition on $s$

Quasilinear degenerate elliptic unilateral problems

We will be concerned with the existence result of a degenerate elliptic unilateral problem of the form Au+H(x,u,∇u)=f, where A is a Leray-Lions operator from W1,p(Ω,w) into its dual. On the nonlinear lower-order term H(x,u,∇u), we assume that it is a Carathéodory function having natural growth with respect to |∇u|, but without assuming the sign condition. The right-hand side f belongs to L1(Ω)