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

Abstract

AbstractThis 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.