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Dušan D. Repovš

Publications and source records attributed to Dušan D. Repovš.

At least 55 records · Page 3Linked to original sources
arXiv2021

The Nehari manifold approach for singular equations involving the p(x)-Laplace operator

We study the following singular problem involving the p$(x)$-Laplace operator $Δ_{p(x)}u= div(|\nabla u|^{p(x)-2}\nabla u)$, where $p(x)$ is a nonconstant continuous function, \begin{equation} \nonumber {(\rm P_λ)} \left\{\begin{aligned} - Δ_{p(x)} u & = a(x)|u|^{q(x)-2}u(x)+ \frac{λb(x)}{u^{δ(x)}} \quad\mbox{in}\,Ω,\\ u &>0 \quad\mbox{in}\,Ω, \\ u & =0 \quad\mbox{on}\,\partialΩ.\end{aligned} \right. \end{equation} Here, $Ω$ is a bounded domain in $\mathbb{R}^{N\geq2}$ with $C^2$-boundary, $λ$ is a positive parameter, $a(x), b(x) \in C(\overlineΩ)$ are positive weight functions with compact support in $Ω$, and $δ(x),$ $p(x),$ $q(x) \in C(\overlineΩ)$ satisfy certain hypotheses ($A_{0}$) and ($A_{1}$). We apply the Nehari manifold approach and some new techniques to establish the multiplicity of positive solutions for problem ${(\rm P_λ)}$.

math.AP↗
arXiv2021

Nonvariational and singular double phase problems for the Baouendi-Grushin operator

In this paper we introduce a new double phase Baouendi-Grushin type operator with variable coefficients. We give basic properties of the corresponding functions space and prove a compactness result. In the second part, using topological argument, we prove the existence of weak solutions of some nonvariational problems in which this new operator is present. The present paper extends and complements some of our previous contributions related to double phase anisotropic variational integrals.

math.AP↗
arXiv2021

Nodal solutions for double phase Kirchhoff problems with vanishing potentials

We consider the following $(p, q)$-Laplacian Kirchhoff type problem \begin{align*} \begin{split} &-\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{p}\, dx \right)Δ_{p}u - \left(c+d\int_{\mathbb{R}^{3}}|\nabla u|^{q}\, dx \right ) Δ_{q}u + V(x) (|u|^{p-2}u + |u|^{q-2}u)= K(x) f(u) \quad \mbox{ in } \mathbb{R}^{3}, \end{split} \end{align*} where $a, b, c, d>0$ are constants, $\frac{3}{2}< p< q<3$, $V: \mathbb{R}^{3}\rightarrow \mathbb{R}$ and $K: \mathbb{R}^{3}\rightarrow \mathbb{R}$ are positive continuous functions allowed vanishing behavior at infinity, and $f$ is a continuous function with quasicritical growth. Using a minimization argument and a quantitative deformation lemma we establish the existence of nodal solutions.

math.AP↗
arXiv2021

Multiplicity and concentration results for a $(p, q)$-Laplacian problem in $\mathbb{R}^{N}$

In this paper we study the multiplicity and concentration of positive solutions for the following $(p, q)$-Laplacian problem: \begin{equation*} \left\{ \begin{array}{ll} -Δ_{p} u -Δ_{q} u +V(\varepsilon x) \left(|u|^{p-2}u + |u|^{q-2}u\right) = f(u) &\mbox{ in } \mathbb{R}^{N}, \\ u\in W^{1, p}(\mathbb{R}^{N})\cap W^{1, q}(\mathbb{R}^{N}), \quad u>0 \mbox{ in } \mathbb{R}^{N}, \end{array} \right. \end{equation*} where $\varepsilon>0$ is a small parameter, $1< p<q<N$, $Δ_{r}u=\mbox{div}(|\nabla u|^{r-2}\nabla u)$, with $r\in \{p, q\}$, is the $r$-Laplacian operator, $V:\mathbb{R}^{N}\rightarrow \mathbb{R}$ is a continuous function satisfying the global Rabinowitz condition, and $f:\mathbb{R}\rightarrow \mathbb{R}$ is a continuous function with subcritical growth. Using suitable variational arguments and Ljusternik-Schnirelmann category theory, we investigate the relation between the number of positive solutions and the topology of the set where $V$ attains its minimum for small $\varepsilon$.

math.AP↗
arXiv2021

Existence results for integro-differential equations with reflection

We prove several important results concerning existence and uniqueness of pseudo almost automorphic (paa) solutions with measure for integro-differential equations with reflection. We use the properties of almost automorphic functions with measure and the Banach fixed point theorem, and we discuss two linear and nonlinear cases. We conclude with an example and some observations.

math.AP↗
arXiv2021

Anisotropic $(p,q)$-equations with gradient dependent reaction

We consider a Dirichlet problem driven by the anisotropic $(p,q)$-Laplacian and a reaction with gradient dependence (convection). The presence of the gradient in the source term excludes from consideration a variational approach in dealing with the qualitative analysis of this problem with unbalanced growth. Using the frozen variable method and eventually a fixed point theorem, the main result of this paper establishes that the problem has a positive smooth solution.

math.AP↗
arXiv2021

Positive solutions for singular double phase problems

We study the existence of positive solutions for a class of double phase Dirichlet equations which have the combined effects of a singular term and of a parametric superlinear term. The differential operator of the equation is the sum of a $p$-Laplacian and of a weighted $q$-Laplacian ($q 0$, the equation has at least two positive solutions.

math.AP↗
arXiv2020

Multiple solutions of double phase variational problems with variable exponent

This paper deals with the existence of multiple solutions for the quasilinear equation $-\mathrm{div}\,\mathbf{A}(x,\nabla u)| u| ^{α(x)-2}u=f(x,u)$ in $ \mathbb{R} ^{N}$, which involves a general variable exponent elliptic operator $\mathbf{ A}$ in divergence form. The problem corresponds to double phase anisotropic phenomena, in the sense that the differential operator has behaviors like $ | ξ| ^{q(x)-2}ξ$ for small $| ξ| $ and like $| ξ| ^{p(x)-2}ξ$ for large $ | ξ| $, where $1<α(\cdot )\leq p(\cdot )<q(\cdot )<N$. Our aim is to approach variationally the problem by using the tools of critical points theory in generalized Orlicz-Sobolev spaces with variable exponent. Our results extend the previous works Azzollini, d'Avenia, and Pomponio (2014) and Chorfi and Rădulescu (2016), from the case when exponents $p$ and $q$ are constant, to the case when $p(\cdot )$ and $% q(\cdot )$ are functions. We also substantially weaken some of their hypotheses overcome the lack of compactness by using the weighting method.

math.AP↗
arXiv2020

Positive solutions for the Robin $p$-Laplacian plus an indefinite potential

We consider a nonlinear elliptic equation driven by the Robin $p$-Laplacian plus an indefinite potential. In the reaction we have the competing effects of a strictly $(p-1)$-sublinear parametric term and of a $(p-1)$-linear and nonuniformly nonresonant term. We study the set of positive solutions as the parameter $λ>0$ varies. We prove a bifurcation-type result for large values of the positive parameter $λ$. Also, we show that for all admissible $λ>0$, the problem has a smallest positive solution $\overline{u}_λ$ and we study the monotonicity and continuity properties of the map $λ\mapsto\overline{u}_λ$.

math.AP↗
arXiv2020

Robin double-phase problems with singular and superlinear terms

We consider a nonlinear Robin problem driven by the sum of $p$-Laplacian and $q$-Laplacian (i.e. the $(p,q)$-equation). In the reaction there are competing effects of a singular term and a parametric perturbation $λf(z,x)$, which is Carathéodory and $(p-1)$-superlinear at $x\in\mathbb{R},$ without satisfying the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques, we prove a bifurcation-type result describing the changes in the set of positive solutions as the parameter $λ>0$ varies.

math.AP↗
arXiv2020

Anisotropic equations with indefinite potential and competing nonlinearities

We consider a nonlinear Dirichlet problem driven by a variable exponent $p$-Laplacian plus an indefinite potential term. The reaction has the competing effects of a parametric concave (sublinear) term and of a convex (superlinear) perturbation (an anisotropic concave-convex problem). We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the positive parameter $λ$ varies. Also, we prove the existence of minimal positive solutions.

math.AP↗
arXiv2020

On nonlinear Schrödinger equations on the hyperbolic space

We study existence of weak solutions for certain classes of nonlinear Schrödinger equations on the Poincaré ball model $\mathbb{B}^N$, $N\geq 3$. By using the Palais principle of symmetric criticality and suitable group theoretical arguments, we establish the existence of a nontrivial (weak) solution.

math.AP↗
arXiv2020

Existence results for some problems on Riemannian manifolds

By using variational techniques we provide new existence results for Yamabe-type equations with subcritical perturbations set on a compact $d$-dimensional ($d\geq 3$) Riemannian manifold without boundary. As a direct consequence of our main theorems, we prove the existence of at least one solution to the following singular Yamabe-type problem $$ \left\lbrace \begin{array}{ll} -Δ_g w + α(σ)w = μK(σ) w^\frac{d+2}{d-2} +λ\left( w^{r-1} + f(w)\right), \quad σ\in\mathcal{M} &\\ &\\ w\in H^2_α(\mathcal{M}), \quad w>0 \ \ \mbox{in} \ \ \mathcal{M} & \end{array} \right.$$ where, as usual, $Δ_g$ denotes the Laplace-Beltrami operator on $(\mathcal{M},g)$, $α, K:\mathcal{M}\to\mathbb{R}$ are positive (essentially) bounded functions, $r\in(0,1)$, and $f:[0,+\infty)\to[0,+\infty)$ is a subcritical continuous function. Restricting ourselves to the unit sphere ${\mathbb{S}}^d$ via the stereographic projection, we also solve some parametrized Emden-Fowler equations in the Euclidean case.

math.AP↗
arXiv2020

Low perturbations for a class of nonuniformly elliptic problems

We introduce and study a new functional which was motivated by our paper on the Caffarelli-Kohn-Nirenberg inequality with variable exponent (Bahrouni, Rădulescu and Repovš, Nonlinearity 31 (2018), 1518-1534). We also study the eigenvalue problem for equations involving this new functional.

math.AP↗
arXiv2020

On existence of PI-exponents of unital algebras

We construct a family of unital non-associative algebras $\{T_α\vert~ 2<α\in\mathbb R\}$ such that $\underline{exp}(T_α)=2$, whereas $α\le\overline{exp}(T_α)\leα+1$. In particular, it follows that ordinary PI-exponent of codimension growth of algebra $T_α$ does not exist for any $α> 2$. This is the first example of a unital algebra whose PI-exponent does not exist.

math.RA↗
arXiv2020

Existence and multiplicity of solutions for double-phase Robin problems

We consider a double phase Robin problem with a Carathéodory nonlinearity. When the reaction is superlinear but without satisfying the Ambrosetti-Rabinowitz condition, we prove an existence theorem. When the reaction is resonant, we prove a multiplicity theorem. Our approach is Morse theoretic, using the notion of homological local linking.

math.AP↗