Multiple solutions for a quasilinear Schrödinger equation involving critical Hardy–Sobolev exponent with Robin boundary condition

2021 ◽  
pp. 1-17
Author(s):  
Yin Deng ◽  
Gao Jia
Keyword(s):  
Boundary Condition ◽  
Schrödinger Equation ◽  
Multiple Solutions ◽  
Schrodinger Equation ◽  
Robin Boundary Condition ◽  
Quasilinear Schrödinger Equation ◽  
Sobolev Exponent ◽  
Robin Boundary

scholarly journals Multiple solutions to a quasilinear Schrödinger equation with Robin boundary condition

2020 ◽  
Vol 5 (4) ◽  
pp. 3825-3839
Author(s):  
Yin Deng ◽  
◽  
Gao Jia ◽  
Fanglan Li ◽  
◽  
...  
Keyword(s):  
Boundary Condition ◽  
Multiple Solutions ◽  
Robin Boundary Condition ◽  
Dinger Equation ◽  
Robin Boundary

Multiple solutions for an elliptic system with indefinite Robin boundary conditions

2017 ◽  
Vol 8 (1) ◽  
pp. 603-614 ◽  
Author(s):  
Pablo Amster
Keyword(s):  
Boundary Condition ◽  
Elliptic System ◽  
Multiple Solutions ◽  
Robin Boundary Condition ◽  
Robin Boundary Conditions ◽  
Precise Calculation ◽  
Scalar Operator ◽  
Point Operator ◽  
Robin Boundary ◽  
Linear Scalar

Abstract Multiplicity of solutions is proved for an elliptic system with an indefinite Robin boundary condition under an assumption that links the linearisation at 0 and the eigenvalues of the associated linear scalar operator. Our result is based on a precise calculation of the topological degree of a suitable fixed point operator over large and small balls.

scholarly journals Infinitely many solutions of degenerate quasilinear Schrödinger equation with general potentials

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Yan Meng ◽  
Xianjiu Huang ◽  
Jianhua Chen
Keyword(s):  
Schrödinger Equation ◽  
Variational Methods ◽  
Schrodinger Equation ◽  
Infinitely Many Solutions ◽  
Type Condition ◽  
Nontrivial Solutions ◽  
Quasilinear Schrödinger Equation ◽  
Sobolev Exponent

AbstractIn this paper, we study the following quasilinear Schrödinger equation: $$ -\operatorname{div}\bigl(a(x,\nabla u)\bigr)+V(x) \vert x \vert ^{-\alpha p^{*}} \vert u \vert ^{p-2}u=K(x) \vert x \vert ^{- \alpha p^{*}}f(x,u) \quad \text{in } \mathbb{R}^{N}, $$ − div ( a ( x , ∇ u ) ) + V ( x ) | x | − α p ∗ | u | p − 2 u = K ( x ) | x | − α p ∗ f ( x , u ) in  R N , where $N\geq 3$ N ≥ 3 , $1< p< N$ 1 < p < N , $-\infty <\alpha <\frac{N-p}{p}$ − ∞ < α < N − p p , $\alpha \leq e\leq \alpha +1$ α ≤ e ≤ α + 1 , $d=1+\alpha -e$ d = 1 + α − e , $p^{*}:=p^{*}(\alpha ,e)=\frac{Np}{N-dp}$ p ∗ : = p ∗ ( α , e ) = N p N − d p (critical Hardy–Sobolev exponent), V and K are nonnegative potentials, the function a satisfies suitable assumptions, and f is superlinear, which is weaker than the Ambrosetti–Rabinowitz-type condition. By using variational methods we obtain that the quasilinear Schrödinger equation has infinitely many nontrivial solutions.

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