On the existence and multiplicity of solutions for double phase equations with Robin boundary conditions

This paper studies the following new class of nonlinear double-phase equations with a defined Robin boundary condition (the nonlinear disturbances that drive them to obey a suitable condition at the origin and on the boundary) {cases} -div(A(∇ u)) - div(B(∇ u)) = μ f(x, u) & in Ω, \\ (A(∇ u) + B(∇ u)) · η + b(x) |u|ᵖ⁽ˣ⁾⁻² u + d(x) |u|q(x)-2 u = g(x, u) & on ∂Ω, {cases} , where μ > 0 , Ω ⊂ R^N ( N ≥ 2 ) is a bounded open domain with smooth boundary ∂Ω, η is the outer unit normal vector on ∂Ω, b(·) and d(·) are positive and continuous functions on ∂Ω. At first, via a generalized mountain-pass approach with Critical point theory, we prove that this problem with superlinear nonlinearity has a solution and infinitely many solutions. Furthermore, we rigorously verify the existence of infinitely many solutions by imposing sufficient constraints on the functions ƒ and g. Our results comprehensively expand and generalize specific recent contributions within the existing literature.