Patrick Winkert
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View article: The effect of the weight function on the number of solutions for double phase problems in $$\mathbb {R}^N$$
The effect of the weight function on the number of solutions for double phase problems in $$\mathbb {R}^N$$ Open
In this paper we deal with quasilinear elliptic equations of the form $$\begin{aligned} -\operatorname {div}\left( |\nabla u|^{p-2}\nabla u+a(\varepsilon x)|\nabla u|^{q-2}\nabla u \right) +|u|^{p-2} u+a(\varepsilon x)|u|^{q-2}u&=f(u) \end…
View article: On singularly perturbed ( <i>p</i> , <i>N</i> )-Laplace Schrödinger equation with logarithmic nonlinearity
On singularly perturbed ( <i>p</i> , <i>N</i> )-Laplace Schrödinger equation with logarithmic nonlinearity Open
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 Schrödinger equation with logarithmic nonlinearit…
View article: Least energy sign-changing solution for logarithmic double phase problems with nonlinear boundary condition
Least energy sign-changing solution for logarithmic double phase problems with nonlinear boundary condition Open
In this paper we study logarithmic double phase problems with superlinear right-hand sides and nonlinear Neumann boundary condition. In particular, we show that the problem under consideration has a least energy sign-changing solution. The…
View article: Multiplicity results for logarithmic double phase problems via Morse theory
Multiplicity results for logarithmic double phase problems via Morse theory Open
In this paper, we study elliptic equations of the form where is the logarithmic double phase operator given by is Euler's number, , , is a bounded domain with Lipschitz boundary , , with , and . Under mild assumptions on the nonlinearity w…
View article: Critical double phase problems involving sandwich-type nonlinearities
Critical double phase problems involving sandwich-type nonlinearities Open
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)= λw(x)|u|^{s-2}u+θB\left(x,u\right) \quad \text{in } …
View article: Multi-valued variational inequalities for variable exponent double phase problems: comparison and extremality results
Multi-valued variational inequalities for variable exponent double phase problems: comparison and extremality results Open
We prove existence and comparison results for multi-valued variational inequalities in a bounded domain $$\Omega $$ of the form $$\begin{aligned} u\in K{:}\, 0 \in Au+\partial I_K(u)+{\mathcal {F}}(u)+{\mathcal {F}}_\Gamma (u)\quad \text…
View article: Logarithmic double phase problems with generalized critical growth
Logarithmic double phase problems with generalized critical growth Open
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-def…
View article: Degenerate singular Kirchhoff problems in Musielak-Orlicz spaces
Degenerate singular Kirchhoff problems in Musielak-Orlicz spaces Open
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 gen…
View article: Double phase problems with variable exponents depending on the solution and the gradient in the whole space $\mathbb{R}^N$
Double phase problems with variable exponents depending on the solution and the gradient in the whole space $\mathbb{R}^N$ Open
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 grad…
View article: Least energy sign-changing solution for degenerate Kirchhoff double phase problems
Least energy sign-changing solution for degenerate Kirchhoff double phase problems Open
In this paper we study the following nonlocal Dirichlet equation of double phase type−ψ[∫Ω(|∇u|pp+μ(x)|∇u|qq)dx]G(u)=f(x,u)in Ω,u=0on ∂Ω, where G is the double phase operator given byG(u)=div(|∇u|p−2∇u+μ(x)|∇u|q−2∇u)u∈W01,H(Ω), Ω⊆RN, N≥2, …
View article: On singularly perturbed $(p, N )$-Laplace Schrödinger equation with logarithmic nonlinearity
On singularly perturbed $(p, N )$-Laplace Schrödinger equation with logarithmic nonlinearity Open
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 Schrödinger equation with logarithmic nonlinearity…
View article: Multiple Positive Solutions for Quasilinear Elliptic Problems in Expanding Domains
Multiple Positive Solutions for Quasilinear Elliptic Problems in Expanding Domains Open
In this paper we prove the existence of multiple positive solutions for a quasilinear elliptic problem with unbalanced growth in expanding domains by using variational methods and the Lusternik–Schnirelmann category theory. Based on the pr…
View article: Preface to the First Edition
Preface to the First Edition Open
The aim of this book is to present the foundations of modern Nonlinear Functional Analysis and equip the reader with all the necessary tools to continue with theoretical and/or applied research in the field.Nonlinear Functional Analysis is…
View article: Two solutions for Dirichlet double phase problems with variable exponents
Two solutions for Dirichlet double phase problems with variable exponents Open
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 nonlin…
View article: Sequences of nodal solutions for critical double phase problems with variable exponents
Sequences of nodal solutions for critical double phase problems with variable exponents Open
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 perm…
View article: Elliptic p-Laplacian systems with nonlinear boundary condition
Elliptic p-Laplacian systems with nonlinear boundary condition Open
In this paper we study quasilinear elliptic systems given by−Δp1u1=−|u1|p1−2u1in Ω,−Δp2u2=−|u2|p2−2u2in Ω,|∇u1|p1−2∇u1⋅ν=g1(x,u1,u2)on ∂Ω,|∇u2|p2−2∇u2⋅ν=g2(x,u1,u2)on ∂Ω, where ν(x) is the outer unit normal of Ω at x∈∂Ω, Δpi denotes the pi…
View article: Degenerate Kirchhoff problems with nonlinear Neumann boundary condition
Degenerate Kirchhoff problems with nonlinear Neumann boundary condition Open
In this paper we consider degenerate Kirchhoff-type equations of the form \[-ϕ(Ξ(u)) \left(\mathcal{A}(u)-|u|^{p-2}u\right) = f(x,u)\quad \text{in } Ω,\] \[\phantom{aaiaaaaaaaaa}ϕ(Ξ(u)) \mathcal{B}(u) \cdot ν= g(x,u) \quad \text{on } \part…
View article: Nonlocal Double Phase Implicit Obstacle Problems with Multivalued Boundary Conditions
Nonlocal Double Phase Implicit Obstacle Problems with Multivalued Boundary Conditions Open
<p>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), th…
View article: Multiple sign-changing solutions for superlinear (p,q)-equations in symmetrical expanding domains
Multiple sign-changing solutions for superlinear (p,q)-equations in symmetrical expanding domains Open
In this paper we study quasilinear elliptic equations defined on symmetrical expanding domains driven by the (p,q)-Laplacian and with a superlinear right-hand side. Based on the Lusternik-Schnirelmann category we prove the existence of at …