Double Phase Problems Research Articles

Lipschitz truncation method for parabolic double-phase systems and applications

We discuss a Lipschitz truncation technique for parabolic double-phase problems of p-Laplace type in order to prove energy estimates and uniqueness results for the Dirichlet problem. Moreover, we show existence for a non-homogeneous double-phase problem. The Lipschitz truncation method is based on a Whitney-type covering result and a related partition of unity in the intrinsic geometry for the double-phase problem.

Hölder regularity results for parabolic nonlocal double phase problems

Hölder regularity results for parabolic nonlocal double phase problems

The Existence of a Solution to a Class of Fractional Double Phase Problems

The Existence of a Solution to a Class of Fractional Double Phase Problems

Infinitely many positive solutions for a double phase problems involving the double phase operator

In this paper we study a double phase problem which involves the double phase operator, and the nonlinear term has an oscillatory behavior. By using variational methods and the theory of the Musielak-Orlicz-Sobolev space, we establish the existence of infinitely many solutions whose $W_0^{1,H}(\Omega)$-norms tend to zero (to infinity, respectively) whenever the nonlinearity oscillates at zero (at infinity, respectively).

Singular double phase problems with convection

We consider a Dirichlet problem driven by the double phase differential operator and a parametric reaction which has the combined effects of a singular term and of a convective perturbation. Using nonlinear operators of monotone type, truncation and comparison techniques, and fixed point theory, we show that for all small values of the parameter, the problem has a bounded positive solution.

Blow-up and global existence for a new class of parabolic p(x,⋅)-Kirchhoff equation involving double phase operator

In this paper, we are concerned with study solutions for double phase problems involving a Kirchhoff function, and nonlinearity with subcritical growth:{ut+M([u]p,q,as)Lp,q,asu=|u|r−2u,inU×[0,T),u(x,t)=0in∂U×[0,T),u(x,0)=u0(x),inU, where, Lp,q,ss is double phase operator, and M is a Kirchhoff function. Combining the Faedo-Galerking approximation with Gronwall's inequality, the existence of a unique weak solution is established. More interestingly, we study the global existence and finite time blow of solution when the initial energy is sub-critical and supercritical. Finally, the stabilization of the solution with positive initial energy is established based on Komornik's integral inequality.

Double Phase Variable Exponent Problems with Nonlinear Matrices Diffusion

This work tackles a class of double phase elliptic problems with variable exponents and matrices diffusion. Under suitable assumptions on the data, we use critical point theory to establish both the existence and uniqueness of weak solutions to the double phase problem under consideration.

Positive Solutions for Convective Double Phase Problems

Positive Solutions for Convective Double Phase Problems

Gradient Estimates in the Whole Space for the Double Phase Problems by the Maximal Function Method

Gradient Estimates in the Whole Space for the Double Phase Problems by the Maximal Function Method

Regularity results for quasiminima of a class of double phase problems

Regularity results for quasiminima of a class of double phase problems

Concentration of solutions for non-autonomous double-phase problems with lack of compactness

The present paper is devoted to the study of the following double-phase equation -div(|∇u|p-2∇u+με(x)|∇u|q-2∇u)+Vε(x)(|u|p-2u+με(x)|u|q-2u)=f(u)inRN,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} -\ ext {div}(|\ abla u|^{p-2}\ abla u+\\mu _{\\varepsilon }(x)|\ abla u|^{q-2}\ abla u)+V_{\\varepsilon }(x)(|u|^{p-2}u+\\mu _{\\varepsilon }(x)|u|^{q-2}u)=f(u)\\quad \ ext{ in }\\quad \\mathbb {R}^{N}, \\end{aligned}$$\\end{document}where N≥2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N\\ge 2$$\\end{document}, 1<p<q<N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<q<N$$\\end{document}, q<p∗\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q<p^{*}$$\\end{document} with p∗=NpN-p\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p^{*}=\\frac{Np}{N-p}$$\\end{document}, μ:RN→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu :\\mathbb {R}^{N}\\rightarrow \\mathbb {R}$$\\end{document} is a continuous non-negative function, με(x)=μ(εx)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu _{\\varepsilon }(x)=\\mu (\\varepsilon x)$$\\end{document}, V:RN→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V:\\mathbb {R}^{N}\\rightarrow \\mathbb {R}$$\\end{document} is a positive potential satisfying a local minimum condition, Vε(x)=V(εx)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$V_{{{\\,\\mathrm{\\varepsilon }\\,}}}(x)=V({{\\,\\mathrm{\\varepsilon }\\,}}x)$$\\end{document}, and the nonlinearity f:R→R\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$f:\\mathbb {R}\\rightarrow \\mathbb {R}$$\\end{document} is a continuous function with subcritical growth. Under natural assumptions on μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document}, V and f, by using penalization methods and Lusternik–Schnirelmann theory we first establish the multiplicity of solutions, and then, we obtain concentration properties of solutions.

A new kind of double phase problems governed by anisotropic matrices diffusion

A new kind of double phase problems governed by anisotropic matrices diffusion

A class of double phase problem without Ambrosetti-Rabinowitz-type growth condition: infinitely many solutions

This paper concerns with a class of double phase problem without Ambrosetti-Rabinowitz-type growth condition. Under reasonable hypotheses, we establish the existence of infinitely many solutions by using the variant fountain theorems due to Zou \cite{71}.

Two solutions for Dirichlet double phase problems with variable exponents

Abstract This paper is devoted to the study of a double phase problem with variable exponents and Dirichlet boundary condition. Based on an abstract critical point theorem, we establish existence results under very general assumptions on the nonlinear term, such as a subcritical growth and a superlinear condition. In particular, we prove the existence of two bounded weak solutions with opposite energy sign and we state some special cases in which they turn out to be nonnegative.

Normalized homoclinic solutions of discrete nonlocal double phase problems

The aim of this paper is to discuss the existence of normalized solutions to the following nonlocal double phase problems driving by the discrete fractional Laplacian: [Formula: see text] where [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] if [Formula: see text], [Formula: see text] if [Formula: see text], and [Formula: see text]([Formula: see text] or [Formula: see text], [Formula: see text] or [Formula: see text]) is the discrete fractional [Formula: see text]-Laplacian. By variational methods, we discuss the existence of non-negative normalized homoclinic solutions under the conditions that the nonlinear term satisfies sublinear growth or superlinear growth conditions. In particular, we establish the compactness of the associated energy functional of the problem without weights. Our paper is the first time to deal with the existence of normalized solutions for discrete double phase problems.

Sequences of nodal solutions for critical double phase problems with variable exponents

In this paper, we study a double phase problem with both variable exponents. Such problem has a reaction consisting of a Carathéodory perturbation defined only locally and of a critical term. The presence of the critical term does not permit to use results of the critical point theory for the corresponding energy functional. Consequently, using suitable cut-off functions and truncation techniques we focus on an auxiliary coercive problem on which, differently from our main problem, we can act with variational tools. In this way, we are able to produce a sequence of sign-changing solutions to our main problem converging to 0 in L∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^{\\infty }$$\\end{document} and in the Musielak–Orlicz Sobolev space.

Multiplicity and concentration properties of solutions for double-phase problem in fractional modular spaces

Multiplicity and concentration properties of solutions for double-phase problem in fractional modular spaces

Liouville-type theorems for double-phase problems involving the Grushin operator

In this paper, we are concerned with the double-phase problem involving the Grushin operator in the whole space \mathbb{R}^{N}=\mathbb{R}^{N_1}\times\mathbb{R}^{N_2} - \operatorname{div}_{G} (|\nabla_{G} u|^{p-2}\nabla_{G} u + w(z) |\nabla_{G} u|^{q-2}\nabla_{G} u) = f(z)|u|^{r-1}u, where \nabla_{G} is the Grushin gradient, \Delta_{G} is the Grushin operator, q\geq p \geq 2 , r&gt;q-1 and w, f \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N}) are two nonnegative functions satisfying some growth conditions at infinity. Our purpose is to establish some Liouville-type theorems for stable weak solutions or for weak solutions which are stable outside a compact set of the equation above.

Gradient higher integrability for double phase problems on metric measure spaces

We study local and global higher integrability properties for quasiminimizers of a class of double phase integrals characterized by nonstandard growth conditions. We work purely on a variational level in the setting of a metric measure space with a doubling measure and a Poincaré inequality. The main novelty is an intrinsic approach to double phase Sobolev-Poincaré inequalities.

Existence and multiplicity results for a kind of double phase problems with mixed boundary value conditions

&lt;p&gt;In this article, we study a double phase variable exponents problem with mixed boundary value conditions of the form&lt;/p&gt;&lt;p&gt;&lt;disp-formula&gt; &lt;label/&gt; &lt;tex-math id="FE1"&gt; \begin{document}$ \left\lbrace \begin{aligned} D(u) +\vert u \vert ^{p(x)-2} u + b(x) \vert u \vert ^{q(x)-2}u &amp;amp; = f(x,u) \ \ \ \ \text{ in } \Omega,\\ u&amp;amp; = 0 \quad \quad \quad \ \text{ on } \Lambda _1, \\ \left( \vert \nabla u \vert ^{p(x)-2} u + b(x) \vert \nabla u \vert ^{q(x)-2} u \right) \cdot \nu &amp;amp; = g(x,u) \quad \ \text{ on } \Lambda _2 . \end{aligned} \right. $\end{document} &lt;/tex-math&gt;&lt;/disp-formula&gt;&lt;/p&gt;&lt;p&gt;First of all, using the mountain pass theorem, we establish that this problem admits at least one nontrivial weak solution without assuming the Ambrosetti–Rabinowitz condition. In addition, we give a result on the existence of an unbounded sequence of nontrivial weak solutions by employing the Fountain theorem with the Cerami condition.&lt;/p&gt;