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

In this article, we study a double phase variable exponents problem with mixed boundary value conditions of the form

$$\{\begin{array}{rl}D(u)+|u{|}^{p(x)-2}u+b(x)|u{|}^{q(x)-2}u& =f(x,u)\text{in}\mathrm{\Omega },\\ u& =0\text{on}{\mathrm{\Lambda }}_1,\\ \left(|\mathrm{\nabla }u{|}^{p(x)-2}u+b(x)|\mathrm{\nabla }u{|}^{q(x)-2}u\right)\cdot \nu & =g(x,u)\text{on}{\mathrm{\Lambda }}_2.\end{array}$$

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.