Nonhomogeneous multiparameter problems in Orlicz–Sobolev spaces

This paper is concerned with the existence and multiplicity of solutions for a class of problems involving the Φ-Laplacian operator with general assumptions on the nonlinearities, which include both semipositone cases and critical concave convex problems. The research is based on the subsupersolution technique combined with a truncation argument and an application of the Mountain Pass Theorem. The results in this paper improve and complement some recent contributions to this field.     

Quasilinear elliptic problems with combined critical Sobolev–Hardy terms

In this paper, by investigating the effect of the subcritical terms and the coefficients of the singular terms, some existence results for quasilinear elliptic problems involving combined critical Sobolev–Hardy terms are obtained via variational methods.     

Strongly nonlinear elliptic boundary value problems in Musielak–Orlicz spaces

Monatshefte für Mathematik - In this paper we prove an existence result of solutions for some strongly nonlinear elliptic problems with lower order term and $$L^1$$ -data in...     

Composition operators in Orlicz–Sobolev spaces

We obtain necessary and sufficient conditions for a homeomorphism of domains in a Euclidean space to generate a bounded embedding operator of the Orlicz–Sobolev spaces defined by a special class of N-functions.     

Interpolation operators in Orlicz–Sobolev spaces

We study classical interpolation operators for finite elements, like the Scott–Zhang operator, in the context of Orlicz–Sobolev spaces. Furthermore, we show estimates for these operators with respect to quasi-norms which appear in the study of systems of p-Laplace type.     

On the entropy solution to an elliptic problem in anisotropic Sobolev–Orlicz spaces

For a certain class of anisotropic elliptic equations with the right-hand side from L ₁ in an arbitrary unbounded domains, the Dirichlet problem with an inhomogeneous boundary condition is considered. The existence and uniqueness of the entropy solution in anisotropic Sobolev–Orlicz spaces are proven.     

Quasiconvex variational functionals in Orlicz–Sobolev spaces

We prove a C ^1,α partial regularity result for minimizers of variational integrals of the type $$ J[u]:=\int\limits_\Omega f(\nabla u){\rm d}x, \, \, u:\Omega\subset \mathbb{R}^n \to \mathbb{R}^N, $$ where the integrand f is strictly quasiconvex and satisfies suitable growth conditions in terms of Young functions.     

Eigenvalue problems in Orlicz–Sobolev spaces for rapidly growing operators in divergence form

Eigenvalue problems involving the p-Laplacian and rapidly growing operators in divergence form are studied in an Orlicz–Sobolev setting. An asymptotic analysis of these problems leads to a full characterization of the spectrum of an exponential type perturbation of the Laplace operator.     

Solutions for a quasilinear elliptic equation in Musielak–Sobolev spaces

Critical point theory is used to show the existence of weak solutions to a quasilinear elliptic differential equation under the functional framework of the Musielak–Sobolev spaces in a bounded smooth domain with Dirichlet boundary condition.     

Multiple solutions for a nonlocal problem in Orlicz–Sobolev spaces

Using the mountain pass theorem combined with the minimum principle, we obtain a multiplicity result for a nonlocal problem in Orlicz–Sobolev spaces. To our knowledge, this is the first contribution to the study of nonlocal problems in this class of spaces.     

A Caffarelli–Kohn–Nirenberg inequality in Orlicz–Sobolev spaces and applications

A generalization of the classical Caffarelli–Kohn–Nirenberg inequality is obtained in the setting of Orlicz–Sobolev spaces. As applications, we prove a compact embedding result, and we establish the existence of weak solutions of the Dirichlet problem for a nonhomogeneous and degenerate/singular elliptic PDE.     

Solutions for the quasilinear elliptic problems involving critical Hardy–Sobolev exponents

In this article, we study the quasilinear elliptic problem involving critical Hardy–Sobolev exponents and Hardy terms. By variational methods and analytic techniques, we obtain the existence of sign–changing solutions to the problem.     

Sobolev–Orlicz inequalities,ultracontractivity and spectra of time changed Dirichlet forms

Let be a regular Dirichlet form on L ²(X,m), μ a positive Radon measure charging no sets of zero capacity and Φ an N-function. We prove that the Sobolev-Orlicz inequality(SOI) for every is equivalent to a capacitary-type inequality. Further we show that if is continuously embedded into L ²(X,μ), the latter one implies some integrability condition, which is nothing else but the classical uniform integrability condition if μ is finite. We also prove that a SOI for yields a Nash-type inequality and if further μ = m and Φ is admissible, it yields the ultracontractivity of the corresponding semigroup. After, in the spirit of SOIs, we derive criteria for to be compactly embedded into L ²(μ), provided μ is finite. As an illustration of the theory, we shall relate the compactness of the latter embedding to the discreteness of the spectrum of the time changed Dirichlet form and shall derive lower bounds for its eigenvalues in term of Φ. This work has been supported by the Deutsche Forschungsgemeinschaft.     

On a non-homogeneous eigenvalue problem involving a potential: An Orlicz–Sobolev space setting

In this paper we study a non-homogeneous eigenvalue problem involving variable growth conditions and a potential V. The problem is analyzed in the context of Orlicz–Sobolev spaces. Connected with this problem we also study the optimization problem for the particular eigenvalue given by the infimum of the Rayleigh quotient associated to the problem with respect to the potential V when V lies in a bounded, closed and convex subset of a certain variable exponent Lebesgue space.     

Solutions for semilinear elliptic problems with critical Sobolev–Hardy exponents in RN

Let $N\ge 3,0\le s<2,0\le \mu <{\left(\frac{N-2}{2}\right)}^2$ and $2^1\left(s\right)≔\frac{2(N-s)}{N-2}$ be the critical Sobolev–Hardy exponents. Via variational methods and the analytic technique, we prove the existence of a nontrivial solution to the singular semilinear problem $-\Delta u-\mu \frac{u}{{\left|x\right|}^2}+u=\frac{{\left|u\right|}^{2^1\left(s\right)-2}}{{\left|x\right|}^s}u+f\left(u\right),u\in H_r^1\left({\mathbb{R}}^N\right)$, for $N\ge 4,0\le \mu \le \bar{\mu }-1$ and suitable functions $f\left(u\right)$.