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This paper is concerned with a class of nonhomogeneous quasilinear elliptic system driven by the locally symmetric potential and the small continuous perturbations in the whole‐space ℝN$$ {\mathrm{\mathbb{R}}}^N $$. By a variant of Clark's theorem without the global symmetric condition and Moser's iteration technique, we obtain the existence of multiple solutions when the nonlinear term satisfies some growth conditions only in a circle with center 0, and the perturbation term is any continuous function with a small parameter and no any growth hypothesis. [ABSTRACT FROM AUTHOR]

This paper aims to show that there exists a weak solution to the following quasilinear system driven by the M-Laplacian 0.1 (- Δ m 1) u = F u (x , u , v) i n Ω , (- Δ m 2) v = F v (x , u , v) i n Ω , u = v = 0 i n ∂ Ω , where Ω is a bounded open subset in R N and (- Δ m) is the M-Laplacian operator. Here we consider the non-reflexive case taking into account the Orlicz and Orlicz-Sobolev Space. The non-reflexive case occurs when the N-function M ¯ does not verify the Δ 2 -condition. We consider an approximated quasilinear elliptic problem driven by the M ϵ -Laplacian and using the Mountain Pass Theorem to obtain the existence of a nontrivial and nonnegative solution for the above system in reflexive case. By tending ϵ → 0 we get the solution in the non-reflexive case. [ABSTRACT FROM AUTHOR]

This paper concerns with the study of existence of non-trivial solutions to the following quasilinear equation: - div (Φ ′ (| ∇ u |) ∇ u | ∇ u | ) + V (| x |) Φ ′ (| u |) u | u | = Q (| x |) f (x , u) , x ∈ R 2 , where Φ is a C 1 -Young function, V and Q are positive radial potentials, and the nonlinearity f possesses maximal exponential growth established by Trudinger–Moser inequalities on Orlicz–Sobolev spaces. Using variational methods, we prove the existence of a weak non-trivial solution [ABSTRACT FROM AUTHOR]

In this paper, we demonstrate the existence of a unique weak solution for a nonlocal Kirchhoff problem under Dirichlet boundary conditions, involving the fractional φ-Laplacian operator. Our major outcome is acquired by applying variational approaches and critical points theory. In addition, we analyse the spectrum and the eigenvalues associated to this problem. At the end and under some assumptions, we give an application to the previous problems. [ABSTRACT FROM AUTHOR]

We prove that for each p ∈ (1 , ∞) the energy functional associated with the Dirichlet system - div (ϕ p (∇ u) ∇ u) = ∂ 1 F (u , v) in Ω , - div (ϕ p (∇ v) ∇ v) = ∂ 2 F (u , v) in Ω , u = v = 0 on ∂ Ω , admits at least one global, nonnegative minimizer (u p , v p) ∈ W 0 Φ p (Ω) × W 0 Φ p (Ω) which converges uniformly on Ω ¯ to (d Ω , d Ω) , as p → ∞ . Here Φ p (t) : = ∫ 0 t s ϕ p (s) d s and d Ω stands for the distance function to the boundary ∂ Ω . [ABSTRACT FROM AUTHOR]

This paper is devoted to dealing with a kind of new Kirchhoff-type problem in RN that involves a general double-phase variable exponent elliptic operator ϕ. Specifically, the operator ϕ has behaviors like |τ|q(x)−2 τ if |τ| is small and like |τ| p(x)−2 τ if |τ| is large, where 1 < p(x) < q(x) < N. By applying some new analytical tricks, we first establish existence results of solutions for this kind of Kirchhoff-double-phase problem based on variational methods and critical point theory. In particular, we also replace the classical Ambrosetti–Rabinowitz type condition with four different superlinear conditions and weaken some of the assumptions in the previous related works. Our results generalize and improve the ones in [Q. H. Zhang, V. D. Rădulescu, J. Math. Pures Appl., 118 (2018), 159–203.] and other related results in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we consider the existence of periodic solutions of Lagrangian systems of the form d dt(∇Φ(u̇(t))) + Vu(t,u(t)) + λGu(t,u(t)) = 0, where Φ: ℝn → [0,∞) is a 풢-function in the sense of Trudinger, V,G: ℝ × ℝn → ℝ are C1-smooth, T-periodic in the time variable t and λ is a real parameter. We prove the existence of a T-periodic solution uλ: ℝ → ℝn for any sufficiently small |λ|, and show that the found solutions converge to a T-periodic solution of the unperturbed system if λ tends to 0. Let us stress that our theorem only makes a natural assumption on the potential V and no additional assumption on G except that it is continuously differentiable. Its proof requires to work in a rather unusual (mixed) Orlicz–Sobolev space setting, which bears several challenges. [ABSTRACT FROM AUTHOR]

In this note we provide an overview of some existence (with sign information) and regularity results for differential equations, in which the method of sub and supersolutions plays an important role. We list some classical results and then we focus on the Dirichlet problem, for problems driven by a general differential operator, depending on (x , u , ∇ u) , and with a convective term f. Our framework is that of Orlicz–Sobolev spaces. We also present several examples. [ABSTRACT FROM AUTHOR]

In this paper, we study a class of strongly degenerate problems with critical exponential growth in R N , N ≥ 2 . We do not assume ellipticity condition on the operator and thus the maximum principle given by Lieberman (Commun Partial Differ Equ 16:311–361, 1991) can not be accessed. Therefore, a careful and delicate analysis is necessary and some ideas can not be applied in our scenario. The arguments developed in this paper are variational and our main result completes the study made in the current literature about the subject. Moreover, when N = 2 or N = 3 the solutions model the slow steady-state flow of a fluid of Prandtl-Eyring type. [ABSTRACT FROM AUTHOR]

This paper explores strongly parabolic-elliptic systems within Orlicz–Sobolev spaces. It introduces the concept of capacity solutions and emphasizes the establishment of existence and regularity of solutions through rigorous proofs. Specifically, it addresses the existence of capacity solutions for a strongly nonlinear coupled system without reliance on the Δ 2 -condition for the N-function. This system, akin to a modified thermistor problem, concerns the determination of variables representing the temperature within a conductor and the associated electrical potential. [ABSTRACT FROM AUTHOR]

In this paper we investigate the existence of positive solution for a class of quasilinear problem on an Orlicz-Sobolev space that can be nonreflexive - ΔΦu + V(x)ϕ(|u|)u = K(x)f(u)in RN, where N ≥ 2, V, K are nonnegative continuous functions and f is a continuous function with a quasicritical growth. Here we extend the Hardy-type inequalities presented in [3] to nonreflexive Orlicz spaces. Through inequalities together with a variational method for non-differentiable functionals we will obtain a ground state solution. We analyze also the problem with V=0. [ABSTRACT FROM AUTHOR]

In this paper, we study the following quasi-linear elliptic equation { − d i v (ϕ ( ∣ ∇ u ∣ ) ∇ u) = λ ψ ( ∣ u ∣ ) u + φ ( ∣ u ∣ ) u , i n Ω , u = 0 , o n ∂ Ω , where Ω ⊂ ℝN is a bounded domain, λ > 0 is a parameter. The function ψ(∣t∣)t is the subcritical term, and ϕ(∣t∣)t is the critical Orlicz-Sobolev growth term with respect to φ. Under appropriate conditions on φ, ψ and ϕ, we prove the existence of infinitely many weak solutions for quasi-linear elliptic equation, for λ ∈ (0, λ0), where λ0 > 0 is a fixed constant. [ABSTRACT FROM AUTHOR]

We are concerned with the existence and regularity of the solutions to the Dirichlet problem, for a class of quasilinear elliptic equations driven by a general differential operator, depending on (x , u , ∇ u) , and with a convective term f. The assumptions on the members of the equation are formulated in terms of Young's functions, therefore we work in the Orlicz-Sobolev spaces. After establishing some auxiliary properties, that seem new in our context, we present two existence and two regularity results. We conclude with several examples. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and uniqueness of renormalized solution for nonlinear multivalued elliptic problems β (u) - d i v (a (x , ∇ u) + F (u)) ∋ f in the framework of Orlicz spaces and L 1 -data. [ABSTRACT FROM AUTHOR]

An existence result for quasilinear elliptic systems with nonstandard growth/coercive conditions, W - 1 L M ¯ -data and no sign condition is proved. [ABSTRACT FROM AUTHOR]

Let d Ω denote the distance function to the boundary of the smooth, bounded domain Ω ⊂ R N and let f : R → R be a continuous function whose primitive t ↦ ∫ 0 t f (s) d s is strictly increasing in the interval [ 0 , d Ω ∞ ]. For each p > N let Φ p be an N-function satisfying p Φ p (t) ≤ t Φ p ′ (t) for all t > 0. We prove that if p is sufficiently large and 0 < λ < λ ∗ , where λ ∗ does not depend on p, then the equation - Δ Φ p u = λ f (u) has a nonnegative weak solution u p ≢ 0 in the Orlicz-Sobolev space W 0 1 , Φ p (Ω). Further, u p converges uniformly on Ω ¯ to d Ω , as p → ∞. [ABSTRACT FROM AUTHOR]

Based on the maximal monotone mapping theory and applying the Schauder fixed point theorem, we prove the existence and the uniqueness of weak periodic solution for nonlinear parabolic-elliptic equations in Orlicz-Sobolev spaces, with growth nonlinearity in gradient associated with some appropriate N-functions. [ABSTRACT FROM AUTHOR]

We investigate the existence and multiplicity of weak solutions for a nonlinear Kirchhoff type quasilinear elliptic system on the whole space RN{{\mathbb{R}}}^{N}. We assume that the nonlinear term satisfies the locally super-(m1,m2)\left({m}_{1},{m}_{2}) condition, that is, lim∣(u,v)∣→+∞F(x,u,v)∣u∣m1+∣v∣m2=+∞{\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x∈Gx\in G, where GG is a domain in RN{{\mathbb{R}}}^{N}, which is weaker than the well-known Ambrosseti-Rabinowitz condition and the naturally global restriction, lim∣(u,v)∣→+∞F(x,u,v)∣u∣m1+∣v∣m2=+∞{\mathrm{lim}}_{| \left(u,v)| \to +\infty }\frac{F\left(x,u,v)}{| u{| }^{{m}_{1}}+| v{| }^{{m}_{2}}}=+\infty for a.e. x∈RNx\in {{\mathbb{R}}}^{N}. We obtain that the system has at least one weak solution by using the classical mountain pass theorem. To a certain extent, our theorems extend the results of Tang et al. [Nontrivial solutions for Schrodinger equation with local super-quadratic conditions, J. Dynam. Differ. Equ. 31 (2019), no. 1, 369–383]. Moreover, under the aforementioned naturally global restriction, we obtain that the system has infinitely many weak solutions of high energy by using the symmetric mountain pass theorem, which is different from those results of Wang et al. [Existence and multiplicity of solutions for a class of quasilinear elliptic systems in Orlicz-Sobolev spaces, J. Nonlinear Sci. Appl. 10 (2017), no. 7, 3792–3814] even if we consider the system on the bounded domain with Dirichlet boundary condition.

We establish some existence and regularity results to the Dirichlet problem, for a class of quasilinear elliptic equations involving a partial differential operator, depending on the gradient of the solution. Our results are formulated in the Orlicz–Sobolev spaces and under general growth conditions on the convection term. The sub‐ and supersolutions method is a key tool in the proof of the existence results. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence and multiplicity of weak solutions to a general type of Kirchhoff equations in magnetic fractional Orlicz-Sobolev spaces. Specifically, we appeal to Critical Point Theory to prove the existence of non-trivial solutions under the so-called Ambrosetti-Rabinowitz condition. We also state the existence of ground-state solutions. Moreover, multiplicity results which yield the existence of an unbounded sequence of solutions are also provided. Finally, we show existence under a weak-type Ambrosetti-Rabinowitz condition formulated in the framework of Orlicz spaces. [ABSTRACT FROM AUTHOR]

We study nonlinear measure data elliptic problems involving the operator of generalized Orlicz growth. Our framework embraces reflexive Orlicz spaces, as well as natural variants of variable exponent and double-phase spaces. Approximable and renormalized solutions are proven to exist and coincide for arbitrary measure datum and to be unique when for a class of data being diffuse with respect to a relevant nonstandard capacity. A capacitary characterization of diffuse measures is provided. [ABSTRACT FROM AUTHOR]

In this paper, we obtain inequalities involving the Taylor polynomial and weak derivatives of a function in an Orlicz–Sobolev type space. Moreover, we show that any such function can be expanded in a finite Taylor series almost everywhere. As a consequence, we prove that the coefficients of any extended best polynomial LΦ$L^\Phi$‐approximation of a function on a ball almost everywhere converge to the weak derivatives of such a function when the radius tends to 0. Lastly, we get a mean convergence result of such coefficients. [ABSTRACT FROM AUTHOR]

We study existence and regularity of weak solutions for a class of boundary value problems, whose form is − div ( log (1 + | ∇ u |) | ∇ u | m (x) ∇ u) + u | ∇ u | log (1 + | ∇ u |) = f (x) , in Ω u = 0 , on ∂ Ω where both the principal part and the lower order term have a logarithmic growth with respect to the gradient of the solutions. We prove that the solutions, due to the regularizing effect given by the lower order term, belong to the Orlicz–Sobolev space generated by the function s log (1 + | s |) even for L 1 (Ω) data. [ABSTRACT FROM AUTHOR]

Let \Omega be a smooth, bounded N-dimensional domain. For each p>N, let \Phi _{p} be an N-function satisfying p\Phi _{p}(t)\leq t\Phi _{p}^{\prime }(t) for all t>0, and let I_{p} be the energy functional associated with the equation -\Delta _{\Phi _{p}}u=f(u) in the Orlicz-Sobolev space W_{0}^{1,\Phi _{p}}(\Omega). We prove that I_{p} admits at least one global, nonnegative minimizer u_{p} which, as p\rightarrow \infty, converges uniformly on \overline {\Omega } to the distance function to the boundary \partial \Omega. [ABSTRACT FROM AUTHOR]

We prove the Hölder continuity of a homeomorphism f defined on a bounded domain Ω ⊂ ℝ n with Lipschitz boundary. The homeomorphism f is assumed to belong to a certain Orlicz–Sobolev class and to satisfy a distortion condition near the boundary. [ABSTRACT FROM AUTHOR]

Studying elliptic measure data problem with strongly nonlinear operator whose growth is described by the means of fully anisotropic N‐function, we prove the uniqueness for a broad class of measures. In order to provide it, the framework of capacities in fully anisotropic Orlicz–Sobolev spaces is developed and the capacitary characterization of a bounded measure is given. Moreover, we give an example of an anisotropic Young function Φ, such that |ξ|p≲Φ(ξ)≲|ξ|plogα(1+|ξ|), with arbitrary p ≥ 1, α > 0, but so irregularly growing that the Orlicz–Sobolev‐type space generated by Φ indispensably requires fully anisotropic tools to be handled. [ABSTRACT FROM AUTHOR]

The asymptotic behavior of the sequence $\{v_n\}$ of nonnegative solutions for a class of inhomogeneous problems settled in Orlicz–Sobolev spaces with prescribed Dirichlet data on the boundary of domain $\Omega$ is analysed. We show that $\{v_n\}$ converges uniformly in $\Omega$ as $n\rightarrow\infty$, to the distance function to the boundary of the domain.

Abstract In this paper, we study some results on the existence and multiplicity of solutions for a class of nonlocal quasilinear elliptic systems. In fact, we prove the existence of precise intervals of positive parameters such that the problem admits multiple solutions. Our approach is based on variational methods.

Multiscale periodic homogenization is extended to an Orlicz-Sobolev setting. It is shown by the reiteraded periodic two-scale convergence method that the sequence of minimizers of a class of highly oscillatory minimizations problems involving convex functionals, converges to the minimizers of a homogenized problem with a suitable convex function.

This paper concerns the study of the asymptotic behavior of the solutions to a family of fractional type problems on a bounded domain, satisfying homogeneous Dirichlet boundary conditions. The family of differential operators includes the fractional p n -Laplacian when p n → ∞ as a particular case, tough it could be extended to a function of the Hölder quotient of order s, whose primitive is an Orlicz function satisfying appropriated growth conditions. The limit equation involves the Hölder infinity Laplacian. [ABSTRACT FROM AUTHOR]

Recently, the existence of at least two weak solutions for a Kirchhoff–type problem has been studied in [M. Makvand Chaharlang and A. Razani, Two weak solutions for some Kirchhoff-type problem with Neumann boundary condition, Georgian Math. J. 28 2021, 3, 429–438]. Here, the existence of infinitely many solutions for nonlocal Kirchhoff-type systems including Dirichlet boundary conditions in Orlicz–Sobolev spaces is studied by using variational methods and critical point theory. [ABSTRACT FROM AUTHOR]

In this paper, we establish the existence of weak solution in Orlicz-Sobolev space for the following Kirchhoff type probelm { − M (∫ Ω Φ (| ∇ u |) d x) d i v (a (| ∇ u |) ∇ u) = f (x , u) i n Ω , u = 0 o n ∂ Ω , { − M (∫ Ω Φ (| ∇ u |) d x) d i v (a (| ∇ u |) ∇ u) = f (x , u) i n Ω , u = 0 o n ∂ Ω , where Ω Ω is a bounded subset in R N R N , N ≥ 1 N ≥ 1 with Lipschitz boundary ∂ Ω. ∂ Ω. The used technical approach is mainly based on Leray-Shauder's non linear alternative. [ABSTRACT FROM AUTHOR]

In this paper we prove the existence and multiplicity of solutions for a large class of quasilinear problems on a nonreflexive Orlicz-Sobolev space. Here, we use the variational methods developed by Szulkin combined with some properties of the weak topology. [ABSTRACT FROM AUTHOR]

In this paper we present a short proof of the following classification Theorem for g −harmonic functions in half-spaces. Assume that u is a nonnegative solution to Δgu = 0 in {xn > 0} that continuously vanishes on the flat boundary {xn = 0}. Then, modulo normalization, u(x) = xn in {xn ≥ 0}. Our proof depends on a recent quantitative version of the Hopf-Oleı̆nik Lemma proven by the authors in Braga and Moreira (Adv. Math.334, 184–242, 2018). Moreover, in this paper, we show how to adapt the proofs in the literature to extend Carleson Estimate, Boundary Harnack Inequality and Schwartz Reflection Principle to the context of nonnegative g −harmonic functions. These results are also ingredients for the proof of the main result. [ABSTRACT FROM AUTHOR]

Abstract Using variational arguments, we establish the existence of two nontrivial solutions for quasilinear elliptic problems in Orlicz–Sobolev spaces, where the nonlinear terms exhibit the combined effects of concave and convex without the Ambrosetti–Rabinowitz type condition.

We study the existence of solutions for the A-Laplace system of the form {-- Δu = f in Ω, u = 0 on ∂ Ω, where -- ΔAu: = -- div(a(|Du|)Du). The existence result is proved by using the technique of Young measures. [ABSTRACT FROM AUTHOR]

In this article we prove the existence of at least two weak solutions for a Kirchhoff-type problem by using the minimum principle, the mountain pass theorem and variational methods in Orlicz–Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the existence of positive solutions to a nonlinear elliptic inclusion problem driven by an elliptic differential operator with Dirichlet boundary and a multivalued term depending on the solution and its gradient in reflexive Orlicz–Sobolev spaces by using a linear functional analysis method and a subsolution–supersolution method. We provide some sufficient conditions to ensure that a subsolution and a supersolution exist. An existence theorem for positive solutions of the problem is established. [ABSTRACT FROM AUTHOR]

Based on a variational principle for smooth functionals defined on reflexive Banach spaces, the existence of at least one weak solution for a non-homogeneous Neumann problem in an appropriate Orlicz–Sobolev space is discussed. [ABSTRACT FROM AUTHOR]

In this paper, the author studies the existence of the minimal nonnegative solutions of some elliptic variational inequalities in Orlicz-Sobolev spaces on bounded or unbounded domains. She gets some comparison results between different solutions as tools to pass to the limit in the problems and to show the existence of the minimal solutions of the variational inequalities on bounded domains or unbounded domains. In both cases, coercive and noncoercive operators are handled. The sufficient and necessary conditions for the existence of the minimal nonnegative solution of the noncoercive variational inequality on bounded domains are established. [ABSTRACT FROM AUTHOR]

In this article, we prove modular and norm Pólya–Szegö inequalities in general fractional Orlicz–Sobolev spaces by using the polarization technique. We introduce a general framework which includes the different definitions of these spaces in the literature, and we establish some of its basic properties such as the density of smooth functions. As a corollary, we prove a Rayleigh–Faber–Krahn type inequality for Dirichlet eigenvalues under nonlocal nonstandard growth operators. [ABSTRACT FROM AUTHOR]

In this paper, we present our results (see our papers), which concern the existence of the renormalized solutions for equations of the type:

In this paper, we study some results on the existence and multiplicity of solutions for a class of nonlocal quasilinear elliptic systems. In fact, we prove the existence of precise intervals of positive parameters such that the problem admits multiple solutions. Our approach is based on variational methods. [ABSTRACT FROM AUTHOR]