On a class of p(x)-Laplacian-like Dirichlet problem depending on three real parameters

Abstract

This research establishes the existence of weak solution for a Dirichlet boundary value problem involving the p(x)-Laplacian-like operator depending on three real parameters, originated from a capillary phenomena, of the following form: $$\begin{aligned} \displaystyle \left\{ \begin{array}{ll} \displaystyle -\Delta ^{l}_{p(x)}u+\delta \vert u\vert ^{\alpha (x)-2}u=\mu g(x, u)+\lambda f(x, u, \nabla u) &{} \mathrm {i}\mathrm {n}\ \Omega ,\\ \\ u=0 &{} \mathrm {o}\mathrm {n}\ \partial \Omega , \end{array}\right. \end{aligned}$$ - Δ p ( x ) l u + δ | u | α ( x ) - 2 u = μ g ( x , u ) + λ f ( x , u , ∇ u ) i n Ω , u = 0 o n ∂ Ω , where $$\Delta ^{l}_{p(x)}$$ Δ p ( x ) l is the p(x)-Laplacian-like operator, $$\Omega $$ Ω is a smooth bounded domain in $$\mathbb {R}^{N}$$ R N , $$\delta ,\mu $$ δ , μ , and $$\lambda $$ λ are three real parameters, and $$p(\cdot ),\alpha (\cdot )\in C_{+}(\overline{\Omega })$$ p ( · ) , α ( · ) ∈ C + ( Ω ¯ ) and g, f are Carathéodory functions. Under suitable nonstandard growth conditions on g and f and using the topological degree for a class of demicontinuous operator of generalized $$(S_{+})$$ ( S + ) type and the theory of variable-exponent Sobolev spaces, we establish the existence of a weak solution for the above problem.

Nonlocal τ(m)‐Laplacian‐like problem with logarithmic nonlinearity and without Ambrosetti–Rabinowitz condition on compact Riemannian manifolds

“…It is widely known that the Ambrosetti-Rabinowitz condition ((AR) for short) in Ambrosetti and Rabinowitz [9], that is, there exist 𝜂 > 0 and A > 0, such that 0 < Ψ(m, t) ≤ t 𝜂 𝜓(m, t), for |t ≥ A and m ∈ 𝔐, is an important condition to ensure the boundedness of the Palais-Smale (PS) sequence of an energy functional and crucial in the application of the critical point theory. This is why, in the last few years, some authors of previous works [10][11][12][13][14][15] have tried to get rid of the (AR) condition.…”

“…It is widely known that the Ambrosetti–Rabinowitz condition ( $$$ (AR) $$$ for short) in Ambrosetti and Rabinowitz [9], that is, there exist $$$ \eta &amp;gt;0 $$$ and $$$ A&amp;gt;0 $$$, such that $$$$ 0&amp;lt;\Psi \left(m,t\right)\le \frac{t}{\eta}\psi \left(m,t\right),\kern0.4em \mathrm{for}\kern0.4em \mid t\ge A\kern0.4em \mathrm{and}\kern0.4em m\in \mathfrak{M}, $$$$ is an important condition to ensure the boundedness of the Palais–Smale (PS) sequence of an energy functional and crucial in the application of the critical point theory. This is why, in the last few years, some authors of previous works [10–15] have tried to get rid of the $$$ (AR) $$$ condition.…”

Existence of Weak Solutions to a Class of Nonlinear Degenerate Parabolic Equations in Weighted Sobolev Space Mohamed El Ouaarabi, Chakir Allalou and Said Melliani

“…Browder [6] constructed a topological degree for operators of type (S + ) in reflexive Banach spaces. For more informations about the history of this theory, the reader can refer to [1,4,5,8,11,12].…”

On a new p(x)-Kirchhoff type problems with p(x)-Laplacian-like operators and Neumann boundary conditions

“…In the one hand, we have the physical motivation; since the p(x)-Laplacian-like operators and p(x)-Laplacian operators has been used to model the steady-state solutions of reactiondiffusion problems, that arise in biophysic, plasma-physic and in the study of chemical reactions. In the other hand, these operators provide a useful paradigm for describing the behaviour of strongly anisotropic materials, whose hardening properties are linked to the exponent governing the growth of the gradient change radically with the point (see [1,6,10,21,24] and the references given there).…”