Existence result for a nonlinear elliptic problem by topological degree in Sobolev spaces with variable exponent. (English) Zbl 07836874
Summary: The aim of this paper is to establish the existence of solutions for a nonlinear elliptic problem of the form
\[
\begin{cases}
\begin{aligned}
&A(u) = f \quad && \text{in} \quad \Omega\\
&u = 0 \quad && \text{on} \quad {\partial \Omega}
\end{aligned}
\end{cases}
\]
where \(A(u) =-\operatorname{div} a(x, u, \nabla u)\) is a Leray-Lions operator and \(f \in W^{-1,p^{\prime} (.)} (\Omega)\) with \(p(x) \in (1, \infty)\).
Our technical approach is based on topological degree method and the theory of variable exponent Sobolev spaces.
Our technical approach is based on topological degree method and the theory of variable exponent Sobolev spaces.
MSC:
| 35J25 | Boundary value problems for second-order elliptic equations |
| 35J60 | Nonlinear elliptic equations |
| 47H11 | Degree theory for nonlinear operators |
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
Keywords:
nonlinear elliptic problem; Sobolev spaces with variable exponent; topological degree; normalising mapsReferences:
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