Multiplicity of Radial Solutions of Quasilinear Problems with Minimum and Maximum

Abstract

Abstract We show the existence and multiplicity of radial solutions for the problems with minimum and maximum involving mean curvature operators in the Minkowski space: { div ⁡ ( ϕ N ⁢ ( ∇ ⁡ v ) ) = F ⁢ ( v ) ⁢ ( | x | )   for a.e. ⁢ R 1 < | x | < R 2 , x ∈ ℝ N , N ≥ 2 , min ⁡ { v ⁢ ( x ) ∣ R 1 ≤ | x | ≤ R 2 } = A , max ⁡ { v ⁢ ( x ) ∣ R 1 ≤ | x | ≤ R 2 } = B , $\left\{\begin{aligned} &\displaystyle\operatorname{div}(\phi_{N}(\nabla v))=F(% v)(\lvert x\rvert)\quad\text{for a.e. }R_{1}<\lvert x\rvert<R_{2},\,x\in% \mathbb{R}^{N},\,N\geq 2,\\ &\displaystyle\min\bigl{\{}v(x)\mid R_{1}\leq\lvert x\rvert\leq R_{2}\bigr{\}}% =A,\quad\max\bigl{\{}v(x)\mid R_{1}\leq\lvert x\rvert\leq R_{2}\bigr{\}}=B,% \end{aligned}\right.$ where ϕ N ⁢ ( z ) = z / 1 - | z | 2 ${\phi_{N}(z)=z/\sqrt{1-\lvert z\rvert^{2}}}$ , z ∈ ℝ N ${z\in\mathbb{R}^{N}}$ , R 1 , R 2 , A , B ∈ ℝ ${R_{1},R_{2},A,B\in\mathbb{R}}$ are constants satisfying 1 < R 1 < R 2 - 1 ${1<R_{1}<R_{2}-1}$ and A < B ${A<B}$ ; | ⋅ | ${\lvert\cdot\rvert}$ denotes the Euclidean norm in ℝ N ${\mathbb{R}^{N}}$ , and F : C 1 ⁢ [ R 1 , R 2 ] → L 1 ⁢ [ R 1 , R 2 ] ${F:C^{1}[R_{1},R_{2}]\to L^{1}[R_{1},R_{2}]}$ is an unbounded operator. By using the Leray–Schauder degree theory and the Borsuk theorem, we prove that the problem has at least two different radial solutions.