In this paper, we consider the following double phase problem with a nonlinear boundary condition $\left\{\begin{array}{cc}-\text{div}\left({|\nabla u|}^{p-2}\nabla u+\mu \left(x\right){|\nabla u|}^{q-2}\nabla u\right)=f(x,u)-{\left|u\right|}^{p-2}u-\mu \left(x\right){\left|u\right|}^{q-2}u\hfill & \text{in}\Omega ,\hfill \\ \left({|\nabla u|}^{p-2}\nabla u+\mu \left(x\right){|\nabla u|}^{q-2}\nabla u\right)\cdot \nu =g(x,u)\hfill & \text{on}\partial \Omega .\hfill \end{array}\right.$First of all, we prove the existence of a solution and infinitely many solutions for this problem with superlinear nonlinearity (without A–R condition). In addition, under the sublinear assumptions on $f$ and $g$, we establish the existence of infinitely many solutions via Clark’s theorem for the aforementioned problem.

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具有非线性边界条件的双相问题的存在性和多重性结果

在本文中，我们考虑以下具有非线性边界条件的双相问题 $\left\{\begin{array}{cc}-\text{div}\left({|\nabla ü|}^{p-2}\nabla ü+\mu （X）{|\nabla ü|}^{q-2}\nabla ü\right)=F（X，ü）-{\left|ü\right|}^{p-2}ü-\mu （X）{\left|ü\right|}^{q-2}ü\hfill & \text{在}\Omega ，\hfill \\ \left({|\nabla ü|}^{p-2}\nabla ü+\mu （X）{|\nabla ü|}^{q-2}\nabla ü\right)\cdot \nu =G（X，ü）\hfill & \text{在}\partial \Omega 。\hfill \end{array}\right.$首先，我们证明了具有超线性非线性（无A–R条件）的该问题的解和无穷多个解的存在。另外，在亚线性假设下$F$ 和 $G$，我们通过克拉克定理针对上述问题建立了许多解的存在。