This paper considers the existence of entropy solutions for some generalized elliptic p(u)-Laplacian problem with Fourier boundary conditions, when the variable exponent p is a real continuous function and we have dependency on the solution u. We get the results by assuming the right-hand sidefunction f to be an integrable function, and by using the regularization approach combined with the theoryof Sobolev spaces with variable exponents.

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In this paper, we study the existence of entropy solutions for some generalized elliptic p(u)-Laplacian problem when p(u) is a local quantity. We get the results by assuming the right-hand side function f to be an integrable function, and by using the regularization approach combined with the theory of Sobolev spaces with variable exponents.

In this paper, we study the $p$-Laplacian problems in the case where $p$ depends on the solution itself. We consider two situations, when $p$ is a local and nonlocal quantity. By using a singular perturbation technique, we prove the existence of weak solutions for the problem associated to the following equation $$ \begin{cases} -\mathrm{d}\mathrm{i}\mathrm{v}(|\nabla u|^{p(u)-2}\nabla u)+|u|^{p(u)-2}u=f & \mbox{in}\; \Omega\\ u=0& \mbox{on}\; \partial\Omega , \end{cases}$$ and also for its nonlocal version. The main goal of this paper is to extend the results established by M. Chipot and H. B. de Oliveira (Math. Ann., 2019, 375, 283-306).