Existence and multiplicity results for a kind of double phase problems with mixed boundary value conditions

Abstract

<p>In this article, we study a double phase variable exponents problem with mixed boundary value conditions of the form</p><p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \left\lbrace \begin{aligned} D(u) +\vert u \vert ^{p(x)-2} u + b(x) \vert u \vert ^{q(x)-2}u & = f(x,u) \ \ \ \ \text{ in } \Omega,\\ u& = 0 \quad \quad \quad \ \text{ on } \Lambda _1, \\ \left( \vert \nabla u \vert ^{p(x)-2} u + b(x) \vert \nabla u \vert ^{q(x)-2} u \right) \cdot \nu & = g(x,u) \quad \ \text{ on } \Lambda _2 . \end{aligned} \right. $\end{document} </tex-math></disp-formula></p><p>First of all, using the mountain pass theorem, we establish that this problem admits at least one nontrivial weak solution without assuming the Ambrosetti–Rabinowitz condition. In addition, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the Fountain theorem with the Cerami condition.</p>