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Patrick Winkert

Publications and source records attributed to Patrick Winkert.

At least 19 recordsLinked to original sources
arXiv2026

H\"{o}lder regularity for logarithmic double phase problems

We investigate boundedness and regularity properties of weak solutions to a class of generalized logarithmic double phase equations with variable exponents. The considered operators arise from Musielak-Orlicz type energies of logarithmic double phase type and exhibit nonstandard growth features. Under general structural assumptions, we derive a priori boundedness estimates in the subcritical setting and establish boundedness of weak solutions also in the presence of critical growth terms. In addition, we prove global H\"{o}lder continuity up to the boundary by means of the De Giorgi iteration scheme, localization arguments, and the frozen functional technique. The obtained results extend several existing regularity results for double phase and related nonstandard growth problems.

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arXiv2026

Mixed double phase equations with local and nonlocal operators

In this paper, we study a new class of mixed double phase problems that combine local and nonlocal operators. We consider two different models. The first model is driven by the fractional $p$-Laplacian together with a local double phase operator, while the second model involves the local $p$-Laplacian coupled with a fractional double phase operator. In order to describe the interaction between local and nonlocal effects within the double phase framework, we introduce an appropriate variational setting based on classical and fractional Musielak-Orlicz Sobolev spaces. Within this setting, we establish several existence and multiplicity results for weak solutions by means of variational and topological techniques. In particular, for the problem driven by the fractional $p$-Laplacian and a local double phase operator, we prove the existence of a nonnegative solution using the Nehari manifold method in the presence of concave-convex nonlinearities. We also investigate the associated Brezis-Nirenberg type problem and obtain the existence of infinitely many solutions via genus theory. For the problem governed by the local $p$-Laplacian and a fractional double phase operator, we show the existence of at least two nontrivial constant sign solutions by exploiting the variational structure of the associated energy functional. Furthermore, in the subcritical case, we prove the existence of a least energy sign-changing solution by combining the Poincar\'{e}-Miranda existence theorem with the quantitative deformation lemma.

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arXiv2025

Critical double phase problems involving sandwich-type nonlinearities

In this paper we study problems with critical and sandwich-type growth represented by \begin{align*} -\operatorname{div}\Big(|\nabla u|^{p-2}\nabla u + a(x)|\nabla u|^{q-2}\nabla u\Big)= \lambda w(x)|u|^{s-2}u+\theta B\left(x,u\right) \quad \text{in } \Omega,\quad u= 0 \quad\text{on } \partial \Omega, \end{align*} where $\Omega\subset\mathbb{R}^N$ is a bounded domain with Lipschitz boundary $\partial\Omega$, $1<p<s<q<N$, $\frac{q}{p}<1+\frac{1}{N}$, $0\leq a(\cdot)\in C^{0,1}(\overline{\Omega})$, $\lambda$, $\theta$ are real parameters, $w$ is a suitable weight and $B\colon \overline{\Omega}\times \mathbb{R}\to\mathbb{R}$ is given by \begin{align*} B(x,t) :=b_0(x)|t|^{p^*-2}t+b(x)|t|^{q^*-2}t, \end{align*} where $r^*:=Nr/(N-r)$ for $r\in\{p,q\}$. Here the right-hand side combines the effect of a critical term given by $B(\cdot,\cdot)$ and a sandwich-type perturbation with exponent $s \in (p,q)$. Under different values of the parameters $\lambda$ and $\theta$, we prove the existence and multiplicity of solutions to the problem above. For this, we mainly exploit different variational methods combined with topological tools, like a new concentration-compactness principle, a suitable truncation argument and the Krasnoselskii's genus theory, by considering very mild assumptions on the data $a(\cdot)$, $b_0(\cdot)$ and $b(\cdot)$.

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arXiv2025

Logarithmic double phase problems with generalized critical growth

In this paper we study logarithmic double phase problems with variable exponents involving nonlinearities that have generalized critical growth. We first prove new continuous and compact embedding results in order to guarantee the well-definedness by studying the Sobolev conjugate function of our generalized $N$-function. In the second part we prove the concentration compactness principle for Musielak-Orlicz Sobolev spaces having logarithmic double phase modular function structure. Based on this we are going to show multiplicity results for the problem under consideration for superlinear and sublinear growth, respectively.

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arXiv2024

Degenerate singular Kirchhoff problems in Musielak-Orlicz spaces

In this paper we study quasilinear elliptic Kirchhoff equations driven by a non-homogeneous operator with unbalanced growth and right-hand sides that consist of sub-linear, possibly singular, and super-linear reaction terms. Under very general assumptions we prove the existence of at least two solutions for such problems by using the fibering method along with an appropriate splitting of the associated Nehari manifold. In contrast to other works our treatment is very general, with much easier and shorter proofs as it was done in the literature before. Furthermore, the results presented in this paper cover a large class of second-order differential operators like the $p$-Laplacian, the $(p,q)$-Laplacian, the double phase operator, and the logarithmic double phase operator.

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arXiv2024

Double phase problems with variable exponents depending on the solution and the gradient in the whole space $\mathbb{R}^N$

In this paper, we establish continuous and compact embeddings for a new class of Musielak-Orlicz Sobolev spaces in unbounded domains driven by a double phase operator with variable exponents that depend on the unknown solution and its gradient. Using these embeddings and an abstract critical point theorem, we prove the existence and multiplicity of weak solutions for such problems associated with this new operator in the whole space $\mathbb{R}^d$. This work can be seen as a continuation of the recent paper by Bahrouni--Bahrouni--Missaoui--R\u{a}dulescu \cite{Bahrouni-Bahrouni-Missaoui-Radulescu-2024}.

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arXiv2024

On singularly perturbed $(p, N )$-Laplace Schr\"{o}dinger equation with logarithmic nonlinearity

This article focuses on the study of the existence, multiplicity and concentration behavior of ground states as well as the qualitative aspects of positive solutions for a $(p, N)$-Laplace Schr\"{o}dinger equation with logarithmic nonlinearity and critical exponential nonlinearity in the sense of Trudinger-Moser in the whole Euclidean space $\mathbb{R}^N$. Through the use of smooth variational methods, penalization techniques, and the application of the Lusternik-Schnirelmann category theory, we establish a connection between the number of positive solutions and the topological properties of the set in which the potential function achieves its minimum values.

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arXiv2024

Degenerate Kirchhoff problems with nonlinear Neumann boundary condition

In this paper we consider degenerate Kirchhoff-type equations of the form \[-\phi(\Xi(u)) \left(\mathcal{A}(u)-|u|^{p-2}u\right) = f(x,u)\quad \text{in } \Omega,\] \[\phantom{aaiaaaaaaaaa}\phi (\Xi(u)) \mathcal{B}(u) \cdot \nu = g(x,u) \quad \text{on } \partial\Omega,\] where $\Omega\subseteq \mathbb{R}^N$, $N\geq 2$, is a bounded domain with Lipschitz boundary $\partial\Omega$, $\mathcal{A}$ denotes the double phase operator given by \begin{align*} \mathcal{A}(u)=\operatorname{div} \left(|\nabla u|^{p-2}\nabla u + \mu(x) |\nabla u|^{q-2}\nabla u \right)\quad \text{for }u\in W^{1,\mathcal{H}}(\Omega), \end{align*} $\nu(x)$ is the outer unit normal of $\Omega$ at $x \in \partial\Omega$, \[\mathcal{B}(u)=|\nabla u|^{p-2}\nabla u + \mu(x) |\nabla u|^{q-2}\nabla u,\] \[\phantom{aaaiaaaa}\Xi(u)= \int_\Omega \left(\frac{|\nabla u|^p+|u|^p}{p}+\mu(x) \frac{|\nabla u|^q}{q}\right)\,\mathrm{d} x,\] $1 0$ and $\zeta \geq 1$, and $f\colon\Omega\times\mathbb{R}\to\mathbb{R}$, $g\colon\partial\Omega\times\mathbb{R}\to\mathbb{R}$ are Carath\'{e}odory functions that grow superlinearly and subcritically. We prove the existence of a nodal ground state solution to the problem above, based on variational methods and minimization of the associated energy functional $\mathcal{E}\colon W^{1,\mathcal{H}}(\Omega) \to\mathbb{R}$ over the constraint set \[\mathcal{C}=\Big\{u \in W^{1,\mathcal{H}}(\Omega)\colon u^{\pm}\neq 0,\, \left\langle \mathcal{E}'(u),u^+ \right\rangle= \left\langle \mathcal{E}'(u),-u^- \right\rangle=0 \Big\},\] whereby $\mathcal{C}$ differs from the well-known nodal Nehari manifold due to the nonlocal character of the problem.

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arXiv2024

Elliptic $p$-Laplacian systems with nonlinear boundary condition

In this paper we study quasilinear elliptic systems given by \begin{equation*} \begin{aligned} -\Delta_{p_1}u_1 & =-|u_1|^{p_1-2}u_1 \quad && \text{in } \Omega,\newline -\Delta_{p_2}u_2 & =-|u_2|^{p_2-2}u_2 \quad && \text{in } \Omega,\newline |\nabla u_1|^{p_1-2}\nabla u_1 \cdot \nu &=g_1(x,u_1,u_2) && \text{on } \partial\Omega,\newline |\nabla u_2|^{p_2-2}\nabla u_2 \cdot \nu &=g_2(x,u_1,u_2) && \text{on } \partial\Omega, \end{aligned} \end{equation*} where $\nu(x)$ is the outer unit normal of $\Omega$ at $x \in \partial\Omega$, $\Delta_{p_i}$ denotes the $p_i$-Laplacian and $g_i\colon \partial\Omega \times\mathbb{R}\times\mathbb{R}\to\mathbb{R}$ are Carath\'{e}odory functions that satisfy general growth and structure conditions for $i=1,2$. In the first part we prove the existence of a positive minimal and a negative maximal solution based on an appropriate construction of sub- and supersolution along with a certain behavior of $g_i$ near zero related to the first eigenvalue of the $p_i$-Laplacian with Steklov boundary condition. The second part is related to the existence of a third nontrivial solution by imposing a variational structure, that is, $(g_1,g_2)=\nabla g$ with a smooth function $(s_1,s_2)\mapsto g(x,s_1,s_2)$. By using the variational characterization of the second eigenvalue of the Steklov eigenvalue problem for the $p_i$-Laplacian together with the properties of the related truncated energy functionals, which are in general nonsmooth, we show the existence of a nontrivial solution whose components lie between the components of the positive minimal and the negative maximal solution.

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arXiv2023

Least energy sign-changing solution for degenerate Kirchhoff double phase problems

In this paper we study the following nonlocal Dirichlet equation of double phase type \begin{align*} -\psi \left [ \int_\Omega \left ( \frac{|\nabla u |^p}{p} + \mu(x) \frac{|\nabla u|^q}{q}\right)\,\mathrm{d} x\right] \mathcal{G}(u) = f(x,u)\quad \text{in } \Omega, \quad u = 0\quad \text{on } \partial\Omega, \end{align*} where $\mathcal{G}$ is the double phase operator given by \begin{align*} \mathcal{G}(u)=\operatorname{div} \left(|\nabla u|^{p-2}\nabla u + \mu(x) |\nabla u|^{q-2}\nabla u \right)\quad u\in W^{1,\mathcal{H}}_0(\Omega), \end{align*} $\Omega\subseteq \mathbb{R}^N$, $N\geq 2$, is a bounded domain with Lipschitz boundary $\partial\Omega$, $1 0$ and $\vartheta \geq 1$, and $f\colon\Omega\times\mathbb{R}\to\mathbb{R}$ is a Carath\'{e}odory function that grows superlinearly and subcritically. We prove the existence of two constant sign solutions (one is positive, the other one negative) and of a sign-changing solution which turns out to be a least energy sign-changing solution of the problem above. Our proofs are based on variational tools in combination with the quantitative deformation lemma and the Poincar\'{e}-Miranda existence theorem.

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arXiv2023

On logarithmic double phase problems

In this paper we introduce a new logarithmic double phase type operator of the form\begin{align*}\mathcal{G}u:=-\operatorname{div}\left(|\nabla u|^{p(x)-2}\nabla u+\mu(x)\left[\log(e+|\nabla u|)+\frac{|\nabla u|}{q(x)(e+|\nabla u|)}\right]|\nabla u|^{q(x)-2} \nabla u \right),\end{align*}where $\Omega\subseteq\mathbb{R}^N$, $N\geq 2$, is a bounded domain with Lipschitz boundary $\partial\Omega$, $p,q\in C(\overline{\Omega})$ with $1<p(x)\leq q(x)$ for all $x\in\overline{\Omega}$ and $0\leq\mu(\cdot)\in L^1(\Omega)$. First, we prove that the logarithmic Musielak-Orlicz Sobolev spaces $W^{1,\mathcal{H}_{\log}}(\Omega)$ and $W^{1, \mathcal{H}_{\log}}_0(\Omega)$ with $\mathcal{H}_{\log}(x,t)=t^{p(x)}+\mu(x)t^{q(x)}\log(e+t)$ for $(x,t)\in \overline{\Omega}\times [0,\infty)$ are separable, reflexive Banach spaces and $W^{1,\mathcal{H}_{\log}}_0(\Omega)$ can be equipped with an equivalent norm. We also prove several embedding results for these spaces and the closedness of these spaces under truncations. In addition we show the density of smooth functions in $W^{1,\mathcal{H}_{\log}}(\Omega)$ even in the case of an unbounded domain by supposing Nekvinda's decay condition on $p(\cdot)$. The second part is devoted to the properties of the operator and it turns out that it is bounded, continuous, strictly monotone, of type (S$_+$), coercive and a homeomorphism. As a result of independent interest we also present a new version of Young's inequality for the product of a power-law and a logarithm. In the last part of this work we consider equations driven by our new operator with superlinear right-hand sides. We prove multiplicity results for this type of equation, in particular about sign-changing solutions, by making use of a suitable variation of the corresponding Nehari manifold together with the quantitative deformation lemma and the Poincar\'e-Miranda existence theorem.

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arXiv2023

Superlinear elliptic equations with unbalanced growth and nonlinear boundary condition

In this paper we first introduce an innovative equivalent norm in the Musielak-Orlicz Sobolev spaces in a very general setting and we then present a new result on the boundedness of the solutions of a wide class of nonlinear Neumann problems, both of independent interest. Moreover, we study a variable exponent double phase problem with a nonlinear boundary condition and prove the existence of multiple solutions under very general assumptions on the nonlinearities. To be more precise, we get constant sign solutions (nonpositive and nonnegative) via a mountain-pass approach and a sign-changing solution by using an appropriate subset of the corresponding Nehari manifold along with the Brouwer degree and the Quantitative Deformation Lemma.

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arXiv2022

Nehari manifold approach for superlinear double phase problems with variable exponents

In this paper we consider quasilinear elliptic equations driven by the variable exponent double phase operator with superlinear right-hand sides. Under very general assumptions on the nonlinearity, we prove a multiplicity result for such problems whereby we show the existence of a positive solution, a negative one and a solution with changing sign. The sign-changing solution is obtained via the Nehari manifold approach and, in addition, we can also give information on its nodal domains.

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arXiv2022

Existence of ground state solutions for a Choquard double phase problem

In this paper we study quasilinear elliptic equations driven by the double phase operator involving a Choquard term of the form \begin{align*} -\mathcal{L}_{p,q}^{a}(u) + |u|^{p-2}u+ a(x) |u|^{q-2}u = \left( \int_{\mathbb{R}^N} \frac{F(y, u)}{|x-y|^\mu}\,\mathrm{d} y\right)f(x,u) \quad\text{in } \mathbb{R}^N, \end{align*} where $\mathcal{L}_{p,q}^{a}$ is the double phase operator given by \begin{align*} \mathcal{L}_{p,q}^{a}(u):= \operatorname{div}\big(|\nabla u|^{p-2}\nabla u + a(x) |\nabla u|^{q-2}\nabla u \big), \quad u\in W^{1,\mathcal{H}}(\mathbb{R}^N), \end{align*} $0<\mu<N$, $1<p<N$, $p<q<p+ \frac{\alpha p}{N}$, $0 \leq a(\cdot)\in C^{0,\alpha}(\mathbb{R}^N)$ with $\alpha \in (0,1]$ and $f\colon\mathbb{R}^N\times\mathbb{R}\to\mathbb{R}$ is a continuous function that satisfies a subcritical growth. Based on the Hardy-Littlewood-Sobolev inequality, the Nehari manifold and variational tools, we prove the existence of ground state solutions of such problems under different assumptions on the data.

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arXiv2022

Infinitely many solutions to Kirchhoff double phase problems with variable exponents

In this work we deal with elliptic equations driven by the variable exponent double phase operator with a Kirchhoff term and a right-hand side that is just locally defined in terms of very mild assumptions. Based on an abstract critical point result of Kajikiya (2005) and recent a priori bounds for generalized double phase problems by the authors (2022), we prove the existence of a sequence of nontrivial solutions whose $L^\infty$-norms converge to zero.

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arXiv2022

The sub-supersolution method for variable exponent double phase systems with nonlinear boundary conditions

In this paper we study quasilinear elliptic systems driven by variable exponent double phase operators involving fully coupled right-hand sides and nonlinear boundary conditions. The aim of our work is to establish an enclosure and existence result for such systems by means of trapping regions formed by pairs of sup- and supersolutions. Under very general assumptions on the data we then apply our result to get infinitely many solutions. Moreover, we also discuss the case when we have homogeneous Dirichlet boundary conditions and present some existence results for this kind of problem.

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arXiv2022

New embedding results for double phase problems with variable exponents and a priori bounds for corresponding generalized double phase problems

In this paper we present new embedding results for Musielak-Orlicz Sobolev spaces of double phase type. Based on the continuous embedding of $W^{1,\mathcal{H}}(\Omega)$ into $L^{\mathcal{H}_*}(\Omega)$, where $\mathcal{H}_*$ is the Sobolev conjugate function of $\mathcal{H}$, we present much stronger embeddings as known in the literature. Based on these results, we consider generalized double phase problems involving such new type of growth with Dirichlet and nonlinear boundary condition and prove appropriate boundedness results of corresponding weak solutions based on the De Giorgi iteration along with localization arguments.

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arXiv2022

Multi-valued variational inequalities for variable exponent double phase problems: comparison and extremality results

We prove existence and comparison results for multi-valued variational inequalities in a bounded domain $\Omega$ of the form \begin{equation*} u\in K\,:\, 0 \in Au+\partial I_K(u)+\mathcal{F}(u)+\mathcal{F}_\Gamma(u)\quad\text{in }W^{1,\mathcal{H}}(\Omega)^*, \end{equation*} where $A\colon W^{1, \mathcal{H}}(\Omega) \to W^{1, \mathcal{H}}(\Omega)^*$ given by \begin{equation*} Au:=-\text{div}\left(|\nabla u|^{p(x)-2} \nabla u+ \mu(x) |\nabla u|^{q(x)-2} \nabla u\right) \end{equation*} for $u \in W^{1, \mathcal{H}}(\Omega)$, is the double phase operator with variable exponents and $W^{1, \mathcal{H}}(\Omega)$ is the associated Musielak-Orlicz Sobolev space. First, an existence result is proved under some weak coercivity condition. Our main focus aims at the treatment of the problem under consideration when coercivity fails. To this end we establish the method of sub-supersolution for the multi-valued variational inequality in the space $W^{1, \mathcal{H}}(\Omega)$ based on appropriately defined sub- and supersolutions, which yields the existence of solutions within an ordered interval of sub-supersolution. Moreover, the existence of extremal solutions will be shown provided the closed, convex subset $K$ of $W^{1, \mathcal{H}}(\Omega)$ satisfies a lattice condition. As an application of the sub-supersolution method we are able to show that a class of generalized variational-hemivariational inequalities with a leading double phase operator are included as a special case of the multi-valued variational inequality considered here. Based on a fixed point argument, we also study the case when the corresponding Nemytskij operators $\mathcal{F}, \mathcal{F}_\Gamma$ need not be continuous. At the end, we give a nontrivial example of the construction of sub- and supersolutions related to the problem above.

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