scholarly journals Singular Finsler Double Phase Problems with Nonlinear Boundary Condition

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Csaba Farkas ◽  
Alessio Fiscella ◽  
Patrick Winkert
Keyword(s):  
Boundary Condition ◽  
Weak Solution ◽  
Variational Methods ◽  
Nonlinear Boundary ◽  
Nonlinear Boundary Condition ◽  
Euclidean Norm ◽  
Phase Problem ◽  
Double Phase ◽  
Truncation Techniques ◽  
Double Phase Problem

Abstract In this paper, we study a singular Finsler double phase problem with a nonlinear boundary condition and perturbations that have a type of critical growth, even on the boundary. Based on variational methods in combination with truncation techniques, we prove the existence of at least one weak solution for this problem under very general assumptions. Even in the case when the Finsler manifold reduces to the Euclidean norm, our work is the first one dealing with a singular double phase problem and nonlinear boundary condition.

scholarly journals Solutions for parametric double phase Robin problems

2021 ◽  
Vol 121 (2) ◽  
pp. 159-170 ◽  
Author(s):  
Nikolaos S. Papageorgiou ◽  
Calogero Vetro ◽  
Francesca Vetro
Keyword(s):  
Boundary Condition ◽  
Existence Theorems ◽  
Robin Boundary Condition ◽  
Phase Problem ◽  
Double Phase ◽  
Robin Boundary ◽  
Double Phase Problem

We consider a parametric double phase problem with Robin boundary condition. We prove two existence theorems. In the first the reaction is ( p − 1 )-superlinear and the solutions produced are asymptotically big as λ → 0 + . In the second the conditions on the reaction are essentially local at zero and the solutions produced are asymptotically small as λ → 0 + .

scholarly journals Existence results for double phase problems depending on Robin and Steklov eigenvalues for the p-Laplacian

2021 ◽  
Vol 11 (1) ◽  
pp. 304-320
Author(s):  
Said El Manouni ◽  
Greta Marino ◽  
Patrick Winkert
Keyword(s):  
Nonlinear Boundary ◽  
Eigenvalue Problems ◽  
Nonlinear Boundary Condition ◽  
Existence Results ◽  
Double Phase ◽  
Steklov Eigenvalue ◽  
Variational Tools ◽  
Steklov Eigenvalues ◽  
Truncation Techniques ◽  
Steklov Eigenvalue Problems

Abstract In this paper we study double phase problems with nonlinear boundary condition and gradient dependence. Under quite general assumptions we prove existence results for such problems where the perturbations satisfy a suitable behavior in the origin and at infinity. Our proofs make use of variational tools, truncation techniques and comparison methods. The obtained solutions depend on the first eigenvalues of the Robin and Steklov eigenvalue problems for the p-Laplacian.

Asymptotically vanishing nodal solutions for critical double phase problems

2020 ◽  
pp. 1-12
Author(s):  
Zhenhai Liu ◽  
Nikolaos S. Papageorgiou
Keyword(s):  
Sobolev Space ◽  
Combined Effects ◽  
Phase Problem ◽  
Nodal Solutions ◽  
Unbalanced Growth ◽  
Critical Term ◽  
Double Phase ◽  
Truncation Techniques ◽  
Sign Changing Solutions ◽  
Double Phase Problem

We consider a Dirichlet double phase problem with unbalanced growth. In the reaction we have the combined effects of a critical term and of a locally defined Carathéodory perturbation. Using cut-off functions and truncation techniques we bypass the critical term and deal with a coercive problem. Using this auxillary problem, we show that the original Dirichlet equation has a whole sequence of nodal (sign-changing) solutions which converge to zero in the Musielak–Orlice–Sobolev space and in L ∞ .

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