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Patrick Winkert

Publications and source records attributed to Patrick Winkert.

At least 37 records · Page 2Linked to original sources
arXiv2022

On a class of critical double phase problems

In this paper we study a class of double phase problems involving critical growth, namely $-\text{div}\big(|\nabla u|^{p-2} \nabla u+ μ(x) |\nabla u|^{q-2} \nabla u\big)=λ|u|^{\vartheta-2}u+|u|^{p^*-2}u$ in $Ω$ and $u= 0$ on $\partialΩ$, where $Ω\subset \mathbb{R}^N$ is a bounded Lipschitz domain, $1<\vartheta 0$, respectively. Based on variational and topological tools such as truncation arguments and genus theory, we show the existence of $λ^*>0$ such that the problem above has infinitely many weak solutions with negative energy values for any $λ\in (0,λ^*)$.

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arXiv2022

Double phase obstacle problems with multivalued convection and mixed boundary value conditions

In this paper, we consider a mixed boundary value problem with a double phase partial differential operator, an obstacle effect and a multivalued reaction convection term. Under very general assumptions, an existence theorem for the mixed boundary value problem under consideration is proved by using a surjectivity theorem for multivalued pseudomonotone operators together with the approximation method of Moreau-Yosida. Then, we introduce a family of the approximating problems without constraints corresponding to the mixed boundary value problem. Denoting by $\mathcal S$ the solution set of the mixed boundary value problem and by $\mathcal S_n$ the solution sets of the approximating problems, we establish the following convergence relation \begin{align*} \emptyset\neq w\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n=s\text{-}\limsup\limits_{n\to\infty}{\mathcal S}_n\subset \mathcal S, \end{align*} where $w$-$\limsup_{n\to\infty}\mathcal S_n$ and $s$-$\limsup_{n\to\infty}\mathcal S_n$ stand for the weak and the strong Kuratowski upper limit of $\mathcal S_n$, respectively.

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arXiv2022

Double phase implicit obstacle problems with convection and multivalued mixed boundary value conditions

In this paper we consider a mixed boundary value problem with a nonhomogeneous, nonlinear differential operator (called double phase operator), a nonlinear convection term (a reaction term depending on the gradient), three multivalued terms and an implicit obstacle constraint. Under very general assumptions on the data, we prove that the solution set of such implicit obstacle problem is nonempty (so there is at least one solution) and weakly compact. The proof of our main result uses the Kakutani-Ky Fan fixed point theorem for multivalued operators along with the theory of nonsmooth analysis and variational methods for pseudomonotone operators.

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arXiv2022

A new class of double phase variable exponent problems: Existence and uniqueness

In this paper we introduce a new class of quasilinear elliptic equations driven by the so-called double phase operator with variable exponents. We prove certain properties of the corresponding Musielak-Orlicz Sobolev spaces (an equivalent norm, uniform convexity, Radon-Riesz property with respect to the modular) and the properties of the new double phase operator (continuity, strict monotonicity, (S$_+$)-property). In contrast to the known constant exponent case we are able to weaken the assumptions on the data. Finally we show the existence and uniqueness of corresponding elliptic equations with right-hand sides that have gradient dependence (so-called convection terms) under very general assumptions on the data. As a result of independent interest, we also show the density of smooth functions in the new Musielak-Orlicz Sobolev space even when the domain is unbounded.

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arXiv2022

Existence of solutions for singular double phase problems via the Nehari manifold method

In this paper we study quasilinear elliptic equations driven by the double phase operator and a right-hand side which has the combined effect of a singular and of a parametric term. Based on the fibering method by using the Nehari manifold we are going to prove the existence of at least two weak solutions for such problems when the parameter is sufficiently small.

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arXiv2022

The boundedness and Hölder continuity of weak solutions to elliptic equations involving variable exponents and critical growth

In this paper we prove the boundedness and Hölder continuity of quasilinear elliptic problems involving variable exponents for a homogeneous Dirichlet and a nonhomogeneous Neumann boundary condition, respectively. The novelty of our work is the fact that we allow critical growth even on the boundary and so we close the gap in the papers of Fan-Zhao [Nonlinear Anal. 36 (1999), no. 3, 295--318.] and Winkert-Zacher [Discrete Contin. Dyn. Syst. Ser. S 5 (2012), no. 4, 865--878.] in which the critical cases are excluded. Our approach is based on a modified version of De Giorgi's iteration technique along with the localization method. As a consequence of our results, the $C^{1,α}$-regularity follows immediately.

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arXiv2021

On double phase Kirchhoff problems with singular nonlinearity

In this paper, we study multiplicity results for double phase problems of Kirchhoff type with right-hand sides that include a parametric singular term and a nonlinear term of subcritical growth. Under very general assumptions on the data, we prove the existence of at least two weak solutions that have different energy sign. Our treatment is based on the fibering method in form of the Nehari manifold. We point out that we cover both the non-degenerate as well as the degenerate Kirchhoff case in our setting.

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arXiv2021

Combined effects of singular and superlinear nonlinearities in singular double phase problems in $\mathbb{R}^N$

This paper is concerned with multiplicity results for parametric singular double phase problems in $\mathbb{R}^N$ 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.

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arXiv2021

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.

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arXiv2021

Singular Finsler double phase problems with nonlinear boundary condition

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.

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arXiv2021

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

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 existence of the obtained solutions depends on the first eigenvalues of the Robin and Steklov eigenvalue problems for the $p$-Laplacian.

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arXiv2021

Singular Dirichlet $(p,q)$-equations

We consider a nonlinear Dirichlet problem driven by the $(p,q)$-Laplacian and with a reaction having the combined effects of a singular term and of a parametric $(p-1)$-superlinear perturbation. We prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $λ>0$ varies. Moreover, we prove the existence of a minimal positive solution $u^*_λ$ and study the monotonicity and continuity properties of the map $λ\to u^*_λ$.

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arXiv2021

On a class of singular anisotropic $(p,q)$-equations

We consider a Dirichlet problem driven by the anisotropic $(p,q)$-Laplacian and with a reaction that has the competing effects of a singular term and of a parametric superlinear perturbation. Based on variational tools along with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter varies.

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arXiv2021

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.

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arXiv2021

A multiplicity theorem for anisotropic Robin equations

In this paper we consider an anisotropic Robin problem driven by the $p(x)$-Laplacian and a superlinear reaction. Applying variational tools along with truncation and comparison techniques as well as critical groups, we prove that the problem has at least five nontrivial smooth solutions to be ordered and with sign information: two positive, two negative and the fifth nodal.

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arXiv2021

Positive solutions for singular anisotropic $(p,q)$-equations

In this paper we consider a Dirichlet problem driven by an anisotropic $(p,q)$-differential operator and a parametric reaction having the competing effects of a singular term and of a superlinear perturbation. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter moves. Moreover, we prove the existence of a minimal positive solution and determine the monotonicity and continuity properties of the minimal solution map.

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arXiv2020

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.

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