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Dušan Repovš

Publications and source records attributed to Dušan Repovš.

At least 37 records · Page 2Linked to original sources
arXiv2017

Nonlinear elliptic equations on Carnot groups

This article concerns a class of elliptic equations on Carnot groups depending on one real positive parameter and involving a subcritical nonlinearity (for the critical case we refer to G. Molica Bisci and D. Repovš, Yamabe-type equations on Carnot groups, Potential Anal. 46:2 (2017), 369-383; arXiv:1705.10100 [math.AP]). As a special case of our results we prove the existence of at least one nontrivial solution for a subelliptic equation defined on a smooth and bounded domain $D$ of the Heisenberg group $\mathbb{H}^n=\mathbb{C}^n\times \mathbb{R}$. The main approach is based on variational methods.

math.AP↗
arXiv2017

Existence and symmetry of solutions for critical fractional Schrödinger equations with bounded potentials

This paper is concerned with the following fractional Schrödinger equations involving critical exponents: \begin{eqnarray*} (-Δ)^αu+V(x)u=k(x)f(u)+λ|u|^{2_α^{*}-2}u\quad\quad \mbox{in}\ \mathbb{R}^{N}, \end{eqnarray*} where $(-Δ)^α$ is the fractional Laplacian operator with $α\in(0,1)$, $N\geq2$, $λ$ is a positive real parameter and $2_α^{*}=2N/(N-2α)$ is the critical Sobolev exponent, $V(x)$ and $k(x)$ are positive and bounded functions satisfying some extra hypotheses. Based on the principle of concentration compactness in the fractional Sobolev space and the minimax arguments, we obtain the existence of a nontrivial radially symmetric weak solution for the above-mentioned equations without assuming the Ambrosetti-Rabinowitz condition on the subcritical nonlinearity.

math.AP↗
arXiv2016

Nontrivial solutions of superlinear nonlocal problems

We study the question of the existence of infinitely many weak solutions for nonlocal equations of fractional Laplacian type with homogeneous Dirichlet boundary data, in presence of a superlinear term. Starting from the well-known Ambrosetti-Rabinowitz condition, we consider different growth assumptions on the nonlinearity, all of superlinear type. We obtain three different existence results in this setting by using the Fountain Theorem, which extend some classical results for semilinear Laplacian equations to the nonlocal fractional setting.

math.AP↗
arXiv2016

On a $p(\cdot)$-biharmonic problem with no-flux boundary condition

The study of fourth order partial differential equations has flourished in the last years, however, a $p(\cdot)$-biharmonic problem with no-flux boundary condition has never been considered before, not even for constant $p$. This is an important step further, since surfaces that are impermeable to some contaminants are appearing quite often in nature, hence the significance of such boundary condition. By relying on several variational arguments, we obtain the existence and the multiplicity of weak solutions to our problem. We point out that, although we use a mountain pass type theorem in order to establish the multiplicity result, we do not impose an Ambrosetti-Rabinowitz type condition, nor a symmetry condition, on our nonlinearity $f$.

math.AP↗
arXiv2016

Algebraic systems with Lipschitz perturbations

By using variational methods, the existence of infinitely many solutions for a nonlinear algebraic system with a parameter is established in presence of a perturbed Lipschitz term. Our goal was achieved requiring an appropriate behavior of the nonlinear term $f$, either at zero or at infinity, without symmetry conditions.

math.AP↗
arXiv2016

Existence of solutions for $p$-Laplacian discrete equations

This work is devoted to the study of the existence of at least one (non-zero) solution to a problem involving the discrete $p$-Laplacian. As a special case, we derive an existence theorem for a second-order discrete problem, depending on a positive real parameter $α$, whose prototype is given by $$ \left\{ \begin{array}{l} -Δ^2u({k-1})=αf(k,u(k)),\quad \forall\; k \in {\mathbb{Z}}[1,T] \\ {u(0)=u({T+1})=0.}\\ \end{array} \right. $$ Our approach is based on variational methods in finite-dimensional setting.

math.AP↗
arXiv2016

Existence results for nonlinear elliptic problems on fractal domains

Some existence results for a parametric Dirichlet problem defined on the Sierpiński fractal are proved. More precisely, a critical point result for differentiable functionals is exploited in order to prove the existence of a well determined open interval of positive eigenvalues for which the problem admits at least one non-trivial weak solution.

math.AP↗
arXiv2016

On doubly nonlocal fractional elliptic equations

This work is devoted to the study of the existence of solutions to nonlocal equations involving the fractional Laplacian. These equations have a variational structure and we find a nontrivial solution for them using the Mountain Pass Theorem. To make the nonlinear methods work, some careful analysis of the fractional spaces involved is necessary. In addition, we require rather general assumptions on the local operator. As far as we know, this result is new and represent a fractional version of a classical theorem obtained working with Laplacian equations.

math.AP↗
arXiv2016

Nonlinear Neumann problems driven by a nonhomogeneous differential operator

We study a nonlinear parametric Neumann problem driven by a nonhomogeneous quasilinear elliptic differential operator $\operatorname{div}(a(x,\nabla u))$, a special case of which is the $p$-Laplacian. The reaction term is a nonlinearity function $f$ which exhibits $(p-1)$-subcritical growth. By using variational methods, we prove a multiplicity result on the existence of weak solutions for such problems. An explicit example of an application is also presented.

math.AP↗
arXiv2016

Nonlinear algebraic systems with discontinuous terms

Using a multiple critical points theorem for locally Lipschitz continuous functionals, we establish the existence of at least three distinct solutions for a parametric discrete differential inclusion problem involving a real symmetric and positive definite matrix. Applications to tridiagonal, fourth-order and partial difference inclusions are presented.

math.AP↗
arXiv2016

Higher nonlocal problems with bounded potential

The aim of this paper is to study a class of nonlocal fractional Laplacian equations depending on two real parameters. More precisely, by using an appropriate analytical context on fractional Sobolev spaces due to Servadei and Valdinoci, we establish the existence of three weak solutions for nonlocal fractional problems exploiting an abstract critical point result for smooth functionals. We emphasize that the dependence of the underlying equation from one of the real parameter is not necessarily of affine type.

math.AP↗
arXiv2016

On sequences of solutions for discrete anisotropic equations

Taking advantage of a recent critical point theorem, the existence of infinitely many solutions for an anisotropic problem with a parameter is established. More precisely, a concrete interval of positive parameters, for which the treated problem admits infinitely many solutions, is determined without symmetry assumptions on the nonlinear data. Our goal was achieved by requiring an appropriate behavior of the nonlinear terms at zero, without any additional conditions.

math.AP↗
arXiv2016

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.

math.AP↗
arXiv2016

Multiple solutions for elliptic equations involving a general operator in divergence form

In this paper, exploiting variational methods, the existence of three weak solutions for a class of elliptic equations involving a general operator in divergence form and with Dirichlet boundary condition is investigated. Several special cases are analyzed. In conclusion, for completeness, a concrete example of an application is presented by finding the existence of three nontrivial weak solutions for an uniformly elliptic second-order problem on a bounded Euclidean domain.

math.AP↗
arXiv2016

An eigenvalue problem involving a degenerate and singular elliptic operator

We study an eigenvalue problem involving a degenerate and singular elliptic operator on the whole space $\mathbb{R}^N$. We prove the existence of an unbounded and increasing sequence of eigenvalues. Our study generalizes to the case of degenerate and singular operators a result of A. Szulkin and M. Willem.

math.AP↗
arXiv2016

On a PDE involving the ${\cal A}_{p(\cdot)}$-Laplace operator

This paper establishes existence of solutions for a partial differential equation in which a differential operator involving variable exponent growth conditions is present. This operator represents a generalization of the $p(\cdot)$-Laplace operator, i.e. $Δ_{p(\cdot)}u={\rm div}(|\nabla u|^{p(\cdot)-2}\nabla u)$, where $p(\cdot)$ is a continuous function. The proof of the main result is based on Schauder's fixed point theorem combined with adequate variational arguments. The function space setting used here makes appeal to the variable exponent Lebesgue and Sobolev spaces.

math.AP↗