Double Phase Problems Research Articles

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

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.

Existence and Nonexistence of Solutions for the Double Phase Problem

Existence and Nonexistence of Solutions for the Double Phase Problem

Parabolic equations in Musielak - Orlicz spaces with discontinuous in time N-function

We consider a parabolic PDE with Dirichlet boundary condition and monotone operator A with non-standard growth controlled by an N-function depending on time and spatial variable. We do not assume continuity in time for the N-function. Using an additional regularization effect coming from the equation, we establish the existence of weak solutions and in the particular case of isotropic N-function, we also prove their uniqueness. This general result applies to equations studied in the literature like p(t,x)-Laplacian and double-phase problems.

Fractional differentiability for a class of double phase problems with measure data

This paper is concerned with a borderline case of double phase problems with a finite Radon measure on the right-hand side. We obtain sharp fractional regularity estimates for such non-uniformly elliptic problems.

Existence of two solutions for double-phase problems with a small perturbation

ABSTRACT In this paper, we study the existence of at least two nontrivial solutions for a class of double-phase problems with a small perturbation. We provide one new criterion to ensure the existence of two solutions. Recent results in the literature are extended and significantly improved.

An existence result for singular Finsler double phase problems

In this paper, we study a class of singular double phase problems defined on Minkowski spaces in terms of Finsler manifolds and with right-hand sides that allow a certain type of critical growth for such problems. Under very general assumptions and based on variational tools we prove the existence of at least one nontrivial weak solution for such a problem. This is the first work dealing with a Finsler double phase operator even in the nonsingular case.

Existence and multiplicity results for double phase problem with nonlinear boundary condition

In this paper, we consider the following double phase problem with a nonlinear boundary condition −div|∇u|p−2∇u+μ(x)|∇u|q−2∇u=f(x,u)−|u|p−2u−μ(x)|u|q−2uinΩ,|∇u|p−2∇u+μ(x)|∇u|q−2∇u⋅ν=g(x,u)on∂Ω.First of all, we prove the existence of a solution and infinitely many solutions for this problem with superlinear nonlinearity (without A–R condition). In addition, under the sublinear assumptions on f and g, we establish the existence of infinitely many solutions via Clark’s theorem for the aforementioned problem.

Multiple Solutions for Double Phase Problems with Hardy Type Potential

In this paper, we are concerned with the singular elliptic problems driven by the double phase operator and and Dirichlet boundary conditions. In view of the variational approach, we establish the existence of at least one nontrivial solution and two distinct nontrivial solutions under some general assumptions on the nonlinearity f. Here we use Ricceri’s variational principle and Bonanno’s three critical points theorem in order to overcome the lack of compactness.

On a double phase problem with sublinear and superlinear nonlinearities

In this paper, we study a class of double-phase problems with sub-critical growth. Some new criteria to guarantee that the existence and multiplicity results for the considered problem is established by using variational methods and critical point theory.

Symmetry and monotonicity of singular solutions of double phase problems

We consider positive singular solutions of PDEs arising from double phase functionals. Exploiting a rather new version of the moving plane method originally developed by Sciunzi, we prove symmetry and monotonicity properties of such solutions.

Existence of solutions for double-phase problems by topological degree

The double-phase problem with a reaction term depending on the gradient is considered in this paper. Using the topological degree theory for a class of demicontinuous operators, we prove the existence of at least one solution of such problem. Our assumptions are suitable and different from those studied previously.

A double phase equation with convection

We consider a double phase problem with a gradient dependent reaction (convection). Using the theory of nonlinear operators of monotone type, we show the existence of a nontrivial, positive, bounded solution.

Anisotropic singular double phase Dirichlet problems

<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>

MULTIPLICITY OF SOLUTIONS FOR DOUBLE PHASE EQUATIONS WITH CONCAVE-CONVEX NONLINEARITIES

This paper is devoted to the study of the L ∞-bound of solutions to a double-phase problem with concave-convex nonlinearities by applying the De Giorgi's iteration method and the localization method. Employing this and a variant of Ekeland's variational principle, we provide the existence of at least two distinct nontrivial solutions belonging to L ∞-space when the convex term does not satisfy the Ambrosetti-Rabinowitz condition in general. In addition, our problem has a sequence of multiple small energy solutions whose L ∞-norms converge to zero. To achieve this result, we utilize the modified functional method and the dual fountain theorem as the main tools.

Sign changing solution for a double phase problem with nonlinear boundary condition via the Nehari manifold

In this paper we study quasilinear elliptic equations driven by the so-called double phase operator and with a nonlinear boundary condition. Due to the lack of regularity, we prove the existence of multiple solutions by applying the Nehari manifold method along with truncation and comparison techniques and critical point theory. In addition, we can also determine the sign of the solutions (one positive, one negative, one nodal). Moreover, as a result of independent interest, we prove for a general class of such problems the boundedness of weak solutions.

Anisotropic double-phase problems with indefinite potential: multiplicity of solutions

We consider an anisotropic double-phase problem plus an indefinite potential. The reaction is superlinear. Using variational tools together with truncation, perturbation and comparison techniques and critical groups, we prove a multiplicity theorem producing five nontrivial smooth solutions, all with sign information and ordered. In this process we also prove two results of independent interest, namely a maximum principle for anisotropic double-phase problems and a strong comparison principle for such solutions.

Double phase problems with variable growth and convection for the Baouendi\u2013Grushin operator

In this paper we study a class of quasilinear elliptic equations with double phase energy and reaction term depending on the gradient. The main feature is that the associated functional is driven by the Baouendi–Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. We first establish some new qualitative properties of a differential operator introduced recently by Bahrouni et al. (Nonlinearity 32(7):2481–2495, 2019). Next, under quite general assumptions on the convection term, we prove the existence of stationary waves by applying the theory of pseudomonotone operators. The analysis carried out in this paper is motivated by patterns arising in the theory of transonic flows.

Singular Double Phase Problems with Convection

We consider a nonlinear Dirichlet problem driven by the sum of a $p$ -Laplacian and of a $q$ -Laplacian (double phase equation). In the reaction we have the combined effects of a singular term and of a gradient dependent term (convection) which is locally defined. Using a mixture of variational and topological methods, together with suitable truncation and comparison techniques, we prove the existence of a positive smooth solution.

Robin double-phase problems with singular and superlinear terms

We consider a nonlinear Robin problem driven by the sum of p-Laplacian and q-Laplacian (i.e. the (p,q)-equation). In the reaction there are competing effects of a singular term and a parametric perturbation λf(z,x), which is Carathéodory and (p−1)-superlinear at x∈R, without satisfying the Ambrosetti–Rabinowitz condition. Using variational tools, together with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter λ>0 varies.

Multiplicity results for double phase problems in RN

We consider a double phase problem in RN with a q−1-superlinear subcritical Carathéodory reaction term, which does not satisfy the Ambrosetti–Rabinowitz condition. Using variational methods based on the topological degree and critical point theory together with the Nehari manifold method and deformation lemma, we find three nontrivial solutions: among them, the first one is the positive ground state solution, the second one is the negative ground state solution, and the third one is the least energy sign-changing solution, which changes sign only once. We also discuss the existence of infinitely many solutions for even energy functional and that of radial solutions.