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This article presents several sufficient conditions for the existence of at least one weak solution and infinitely many weak solutions for the following Neumann problem, originated from a capillary phenomena,{−   d  i  v    (    (   1  +|   ∇  u    |    p  (  x  )1  +   |   ∇  u    |    2  p  (  x  ))    |   ∇  u    |    p  (  x  )  −  2    ∇  u   )   +  α  (  x  )   |   u    |    p  (  x  )  −  2    u=  λ  f  (  x  ,  u  )  inΩ  ,∂  u    ∂  ν    =  0  on∂  Ω$$\begin{equation*} \left\{\begin{array}{ll} -{\rm div}\bigg(\bigg(1+\frac{|\nabla u|^{p(x)}}{\sqrt{1+|\nabla u|^{2p(x)}}}\bigg) |\nabla u|^{p(x)-2}\nabla u\bigg)+\alpha(x)|u|^{p(x)-2}u\\=\lambda f(x,u) \mbox{in}\,\,\Omega,\\ \frac{\partial u}{\partial \nu}=0\mbox{on}\,\,\partial \Omega \end{array}\right. \end{equation*}$$

Variational Approaches to P(X) -Laplacian-Like Problems with Neumann Condition Originated from a Capillary Phenomena