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 ∞ .

scholarly journals Multiplicity of Nodal Solutions for a Class of Double-Phase Problems

2020 ◽  
Vol 2020 ◽  
pp. 1-7
Author(s):  
Jie Yang ◽  
Haibo Chen ◽  
Senli Liu
Keyword(s):  
Growth Conditions ◽  
Phase Problem ◽  
Nodal Solutions ◽  
Double Phase ◽  
Double Phase Problem ◽  
Global Growth

We consider the following parametric double-phase problem: −div∇up−2∇u+ax∇uq−2∇u=λfx,u in Ω,u=0,on ∂Ω.. We do not impose any global growth conditions to the nonlinearity fx,u, which refer solely to its behavior in a neighborhood of u=0. And we will show that they suffice for the multiplicity of signed and nodal solutions of the double-phase problem above when λ is large enough.

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 Anisotropic singular double phase Dirichlet problems

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nikolaos S. Papageorgiou ◽  
Vicenţiu D. Rǎdulescu ◽  
Youpei Zhang
Keyword(s):  
Positive Solutions ◽  
Singular Term ◽  
Independent Interest ◽  
Phase Problem ◽  
Type Result ◽  
Dirichlet Problems ◽  
Double Phase ◽  
Variational Tools ◽  
Superlinear Perturbation ◽  
Double Phase Problem

<p style='text-indent:20px;'>We consider an anisotropic double phase problem with a reaction in which we have the competing effects of a parametric singular term and a superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies on <inline-formula><tex-math id="M1">\begin{document}$ \mathring{\mathbb{R}}_+ = (0, +\infty) $\end{document}</tex-math></inline-formula>. Our approach uses variational tools together with truncation and comparison techniques as well as several general results of independent interest about anisotropic equations, which are proved in the Appendix.</p>

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