The sub-supersolution method for a nonhomogeneous elliptic equation involving Lebesgue generalized spaces

Abstract

In this paper, a nonhomogeneous elliptic equation of the form $$\begin{aligned}& - \mathcal{A}\bigl(x, \vert u \vert _{L^{r(x)}}\bigr) \operatorname{div}\bigl(a\bigl( \vert \nabla u \vert ^{p(x)}\bigr) \vert \nabla u \vert ^{p(x)-2} \nabla u\bigr) \\& \quad =f(x, u) \vert \nabla u \vert ^{\alpha (x)}_{L^{q(x)}}+g(x, u) \vert \nabla u \vert ^{ \gamma (x)}_{L^{s(x)}} \end{aligned}$$ − A ( x , | u | L r ( x ) ) div ( a ( | ∇ u | p ( x ) ) | ∇ u | p ( x ) − 2 ∇ u ) = f ( x , u ) | ∇ u | L q ( x ) α ( x ) + g ( x , u ) | ∇ u | L s ( x ) γ ( x ) on a bounded domain Ω in ${\mathbb{R}}^{N}$ R N ($N >1$ N > 1 ) with $C^{2}$ C 2 boundary, with a Dirichlet boundary condition is considered. Using the sub-supersolution method, the existence of at least one positive weak solution is proved. As an application, the existence of at least one solution of a generalized version of the logistic equation and a sublinear equation are shown.

Existence of infinitely many solutions for an anisotropic equation using genus theory

“…Another generalization of Laplace operator is the nonhomogeneous differential operator , called ‐Laplacian. There is a wide range of applications, including mathematical biology, 18 biophysics, 19 and quantum and plasma physics, 20 (see earlier research 21–25 and the references therein).…”

Existence of radial weak solutions to Steklov problem involving Leray–Lions type operator

“…Finally, we assume that h ∈ L p � (⋅) (Ω) is a positive continuous function such that |h| p � (⋅) is small enough. Faria et al [4] have studied the existence of positive solutions for the following nonlinear elliptic problems under Dirichlet boundary condition Their approach relies on the method of sub-supersolution and nonlinear regularity theory (see [5,16,17] for sub-supersolution methods). Hai Ha et al [6] have proved the existence of infinitely many solutions for a generalized p(⋅)-Laplace equation involving Leray-Lions operators where Ω is a bounded domain in ℝ N with a Lipchitz boundary Ω ; a ∶ Ω × ℝ N → ℝ N and f ∶ Ω × ℝ → ℝ are Carathéodory functions with suitable growth conditions.…”

Solutions to a $$(p_1, \ldots ,p_n)$$-Laplacian Problem with Hardy Potentials

“…positive in an open set. In 2019, Behboudi et al [2] verified the existence of two weak solutions for the following problem where 2 ≤ q < p < N (one can see [9,10,12,13,17,[20][21][22][23][24][25]33] and references therein for the importance of study of these kinds of problems).…”