Double Phase Problems Research Articles

Calderón-Zygmund estimates for non-uniformly elliptic equations with discontinuous nonlinearities on nonsmooth domains

We investigate a general class of the so-called double phase problems on bounded domains. We establish an optimal global Calderón-Zygmund theory for such a non-uniformly elliptic problem under the assumptions that the coefficients have a small BMO semi-norm and the domain is sufficiently flat. Our regularity results not only cover discontinuous coefficients but also non-smooth domains going beyond the Lipschitz category.

Double phase problems: a survey of some recent results

We review some recent results on double phase problems. We focus on the relevant function space framework, which is provided by the generalized Orlicz spaces. We also describe the basic tools and methods used to deal with double phase problems, given that there is no global regularity theory for these problems.

Quasilinear double phase problems in the whole space via perturbation methods

We are concerned with the following double phase problems in the whole space $$ \begin{aligned} -{\rm div}(|\nabla u|^{p-2}\nabla u + & \mu(x)|\nabla u|^{q-2}\nabla u) \\ & +|u|^{p-2}u+\mu(x)|u|^{q-2}u= f(x,u)\;{\rm in}\; \mathbb{R}^N. \end{aligned}$$ The nonlinearity is super-linear but does not satisfy the Ambrosetti-Rabinowitz type condition. The main difficulty is that weak limits of $(PS)$ sequences are not always weak solutions of the problem. To overcome this difficulty, we add {a} potential term and, using the mountain pass theorem, we get weak solutions $u_\lambda$ of the perturbed equations. First, we prove that $u_\lambda\rightharpoonup u$ as $\lambda\rightarrow 0$. Then, via a vanishing lemma, we get that $u$ is a nontrivial solution of the original problem.

Regularizing effect in some Mingione’s double phase problems with very singular data

<abstract><p>In this paper we study the existence of solutions of the Dirichlet problem associated to the following nonlinear PDE</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} { } -{{{\rm{\;div}}}}\big(a(x)\,\nabla u|\nabla u|^{p-2}\big) -{{{\rm{\;div}}}}\big( |u|^{(r-1)\lambda+1}\nabla u|\nabla u|^{\lambda-2}\big) = f \end{equation*} $\end{document} </tex-math></disp-formula></p> <p>where $ 1 < \lambda \leq p $, $ r > 1 $ and $ f \in L^1(\Omega) $.</p></abstract>

Existence and multiplicity of eigenvalues for some double-phase problems involving an indefinite sign reaction term

We study the following class of double-phase nonlinear eigenvalue problems − div ⁡ [ ϕ ( x , | ∇ u | ) ∇ u + ψ ( x , | ∇ u | ) ∇ u ] = λ f ( x , u ) in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain from R N and the potential functions ϕ and ψ have ( p 1 ( x ) ; p 2 ( x ) ) variable growth. The primitive of the reaction term of the problem (the right-hand side) has indefinite sign in the variable u and allows us to study functions with slower growth near + ∞ , that is, it does not satisfy the Ambrosetti–Rabinowitz condition. Under these hypotheses we prove that for every parameter λ ∈ R + ∗ , the problem has an unbounded sequence of weak solutions. The proofs rely on variational arguments based on energy estimates and the use of Fountain Theorem.

On a class of double phase problem with nonlinear boundary conditions

<abstract><p>The existence of nontrivial solutions of the double phase problem with nonlinear boundary value condition is an important quasilinear problem: we use variational techniques and sum decomposition of a space $ W_0^{1, \xi}(\Omega) $ to prove the existence of infinitely many solutions of the problem considered. Moreover, our conditions are suitable and different from those considered previously.</p></abstract>

A multiplicity result for double phase problem in the whole space

<abstract><p>In the present paper, we discuss the solutions of the following double phase problem</p> <p><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ -{\rm div}(|\nabla u|^{^{p-2}}\nabla u+ \mu(x) |\nabla u|^{^{q-2}}\nabla u)+ |u|^{^{p-2}}u+\mu(x)|u|^{^{q-2}}u = f(x, u), \;x\in \mathbb{R}^N, $\end{document} </tex-math></disp-formula></p> <p>where $ N \geq2 $, $ 1 < p < q < N $ and $ 0\leq\mu\in C^{^{0, \alpha}}(\mathbb{R}^N), \; \alpha\in(0, 1] $. Based on the theory of the double phase Sobolev spaces $ W^{^{1, H}}(\mathbb{R}^N) $, we prove the existence of at least two non-trivial weak solutions.</p></abstract>

Multiple solutions to the double phase problems involving concave-convex nonlinearities

<abstract><p>This paper is concerned with several existence results of multiple solutions for Schrödinger-type problems involving the double phase operator for the case of a combined effect of concave-convex nonlinearities. The first one is to discuss that our problem has infinitely many large energy solutions. Second, we obtain the existence of a sequence of infinitely many small energy solutions to the given problem. To establish such multiplicity results, we employ the fountain theorem and the dual fountain theorem as the primary tools, respectively. In particular we give the existence result of small energy solutions on a new class of nonlinear term.</p></abstract>

Bounded nonnegative weak solutions to anisotropic parabolic double phase problems with variable growth

We study the homogeneous Dirichlet problem for a class of nonlinear anisotropic parabolic double phase equations with nonstandard growth conditions. We prove the existence of a unique nonnegative weak solution in suitable Orlicz-Sobolev spaces, and derive the global boundedness of the weak solution.

Least Energy Solutions with Sign Information for Parametric Double Phase Problems

Least Energy Solutions with Sign Information for Parametric Double Phase Problems

Parametric superlinear double phase problems with singular term and critical growth on the boundary

In this paper, we study quasilinear elliptic equations driven by the double phase operator along with a reaction that has a singular and a parametric superlinear term and with a nonlinear Neumann boundary condition of critical growth. Based on a new equivalent norm for Musielak–Orlicz Sobolev spaces and the Nehari manifold along with the fibering method, we prove the existence of at least two weak solutions, provided the parameter is sufficiently small.

Existence of least energy nodal solutions for a double-phase problem with nonlocal nonlinearity

In this paper, we study the following double-phase problem with nonlocal nonlinearity where is a bounded domain, , 1<p<q<N, , is the Riesz potential of order . By using the constrained variational method and Brouwer degree theory, we prove the existence of least energy nodal solutions.

Combined effects of singular and superlinear nonlinearities in singular double phase problems in [formula omitted

This paper is concerned with multiplicity results for parametric singular double phase problems in RN via the Nehari manifold approach. It is shown that the problem under consideration has at least two nontrivial weak solutions provided the parameter is sufficiently small. The idea is to split the Nehari manifold into three disjoint parts minimizing the energy functional on two of them. The third set turns out to be the empty set for small values of the parameter.

Irregular obstacle problems for Orlicz double phase

An irregular obstacle problem with a non-uniformly elliptic operator in divergence form of (G,H)-growth is studied. We provide local Calderón-Zygmund type estimates for an Orlicz double phase problem by proving that the gradient of a solution is integrable as both the gradient of the obstacle function and the associated nonhomogeneous term in the divergence under sharp conditions on the nonlinearity.

Existence and multiplicity of solutions for a quasilinear double phase problem on the whole space

In this paper, we consider a quasilinear double phase problem on the whole space. Some new criteria to guarantee the existence of ground states for the considered problem is established using the mountain pass theorem. Our results generalize some existing results.

Nonvariational and singular double phase problems for the Baouendi-Grushin operator

In this paper we introduce a new double phase Baouendi-Grushin type operator with variable coefficients. We give basic properties of the corresponding functions space and prove a compactness result. In the second part, using topological argument, we prove the existence of weak solutions of some nonvariational problems in which this new operator is present. The present paper extends and complements some of our previous contributions related to double phase anisotropic variational integrals.

Infinitely many solutions for double phase problem with unbounded potential in [formula omitted

We consider a double phase problem in RN−div(|∇u|p−2∇u+a(x)|∇u|q−2∇u)+V(x)(|u|p−2u+a(x)|u|q−2u)=f(x,u)with an unbounded potential V and reaction term f, which does not satisfy the Ambrosetti–Rabinowitz condition. A new functional setting was provided for this problem. Using the Fountain and Dual Fountain Theorem with Cerami condition, we obtain some existence of infinitely many solutions. Our result extends some recent work in the literature.

Concentration of solutions for fractional double-phase problems: critical and supercritical cases

This paper is concerned with concentration and multiplicity properties of solutions to the following fractional problem with unbalanced growth and critical or supercritical reaction:{(−Δ)psu+(−Δ)qsu+V(εx)(|u|p−2u+|u|q−2u)=h(u)+|u|r−2u in RN,u∈Ws,p(RN)∩Ws,q(RN),u>0, in RN,} where ε is a positive parameter, 0<s<1, 2⩽p<q<N/s, (−Δ)ts(t∈{p,q}) is the fractional t-Laplace operator, while V:RN↦R and h:R↦R are continuous functions. The analysis developed in this paper covers both critical and supercritical cases, that is, we assume that either r=qs⁎:=Nq/(N−sq) or r>qs⁎. The main results establish the existence of multiple positive solutions as well as related concentration properties. In the first case, due to the strong influence of the critical term, the result holds true for “high perturbations” of the subcritical nonlinearity. In the second framework, the result holds true for “low perturbations” of the supercritical nonlinearity. The concentration properties are achieved by combining topological and variational methods, provided that ε is small enough and in close relationship with the set where the potential V attains its minimum.

Singular Finsler Double Phase Problems with Nonlinear Boundary Condition

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.

A Multiplicity Theorem for Superlinear Double Phase Problems

We consider a nonlinear Dirichlet problem driven by the double phase differential operator and with a superlinear reaction which need not satisfy the Ambrosetti–Rabinowitz condition. Using the Nehari manifold, we show that the problem has at least three nontrivial bounded solutions: nodal, positive and by the symmetry of the behaviour at +∞ and −∞ also negative.