arXiv subjects

Dušan D. Repovš

Publications and source records attributed to Dušan D. Repovš.

At least 91 records · Page 5Linked to original sources

arXiv2019

Nonlocal Kirchhoff superlinear equations with indefinite nonlinearity and lack of compactness

We study the following Kirchhoff equation $$- \left(1 + b \int_{\mathbb{R}^3} |\nabla u|^2 dx \right) Δu + V(x) u = f(x,u), \ x \in \mathbb{R}^3.$$ A special feature of this paper is that the nonlinearity $f$ and the potential $V$ are indefinite, hence sign-changing. Under some appropriate assumptions on $V$ and $f$, we prove the existence of two different solutions of the equation via the Ekeland variational principle and Mountain Pass Theorem.

math.AP↗

arXiv2019

On a class of parametric $(p,2)$-equations

We consider parametric equations driven by the sum of a $p$-Laplacian and a Laplace operator (the so-called $(p,2)$-equations). We study the existence and multiplicity of solutions when the parameter $λ>0$ is near the principal eigenvalue $\hatλ_1(p)>0$ of $(-Δ_p,W^{1,p}_{0}(Ω))$. We prove multiplicity results with precise sign information when the near resonance occurs from above and from below of $\hatλ_1(p)>0$.

math.AP↗

arXiv2019

Robin problems with a general potential and a superlinear reaction

We consider semilinear Robin problems driven by the negative Laplacian plus an indefinite potential and with a superlinear reaction term which need not satisfy the Ambrosetti-Rabinowitz condition. We prove existence and multiplicity theorems (producing also an infinity of smooth solutions) using variational tools, truncation and perturbation techniques and Morse theory (critical groups).

math.AP↗

arXiv2019

Positive solutions for a class of singular Dirichlet problems

We consider a Dirichlet elliptic problem driven by the Laplacian with singular and superlinear nonlinearities. The singular term appears on the left-hand side while the superlinear perturbation is parametric with parameter $λ>0$ and it need not satisfy the AR-condition. Having as our starting point the work of Diaz-Morel-Oswald (1987), we show that there is a critical parameter value $λ_*$ such that for all $λ>λ_*$ the problem has two positive solutions, while for $λ<λ_*$ there are no positive solutions. What happens in the critical case $λ= λ_*$ is an interesting open problem.

math.AP↗

arXiv2019

Ambrosetti-Prodi problem with degenerate potential and Neumann boundary condition

We study the degenerate elliptic equation $-\mathop{\rm div}(|x|^α\nabla u) =f(u)+tϕ(x)+h(x)$ in a bounded open set $Ω$ with homogeneous Neumann boundary condition, where $α\in(0,2)$ and $f$ has a linear growth. The main result establishes the existence of real numbers $t_*$ and $t^*$ such that the problem has at least two solutions if $t\leq t_*$, there is at least one solution if $t_* t^*$. The proof combines a priori estimates with topological degree arguments.

math.AP↗

arXiv2019

Double-phase problems with reaction of arbitrary growth

We consider a parametric nonlinear nonhomogeneous elliptic equation, driven by the sum of two differential operators having different structure. The associated energy functional has unbalanced growth and we do not impose any global growth conditions to the reaction term, whose behavior is prescribed only near the origin. Using truncation and comparison techniques and Morse theory, we show that the problem has multiple solutions in the case of high perturbations. We also show that if a symmetry condition is imposed to the reaction term, then we can generate a sequence of distinct nodal solutions with smaller and smaller energies.

math.AP↗

arXiv2019

Some hemivariational inequalities in the Euclidean space

The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space $\mathbb{R}^d$ ($d\geq 3$). These hemivariational inequalities have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal group $O(d)$ and their actions on the Sobolev space $H^1(\mathbb{R}^d)$. Moreover, under an additional hypotheses on the dimension $d$ and in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schrödinger equations a concrete and meaningful example of an application is presented.

math.AP↗

arXiv2019

Double-phase problems and a discontinuity property of the spectrum

We consider a nonlinear eigenvalue problem driven by the sum of $p$ and $q$-Laplacian. We show that the problem has a continuous spectrum. Our result reveals a discontinuity property for the spectrum of a parametric ($p,q$)-differential operator as the parameter $β\rightarrow 1^-$.

math.AP↗

arXiv2019

Double phase transonic flow problems with variable growth: nonlinear patterns and stationary waves

In this paper we are concerned with a class of double phase energy functionals arising in the theory of transonic flows. Their main feature is that the associated Euler equation is driven by the Baouendi-Grushin operator with variable coefficient. This partial differential equation is of mixed type and possesses both elliptic and hyperbolic regions. After establishing a weighted inequality for the Baouendi-Grushin operator and a related compactness property, we establish the existence of stationary waves under arbitrary perturbations of the reaction.

math.AP↗

arXiv2019

Nonlinear nonhomogeneous boundary value problems with competition phenomena

We consider a nonlinear boundary value problem driven by a nonhomogeneous differential operator. The problem exhibits competing nonlinearities with a superlinear (convex) contribution coming from the reaction term and a sublinear (concave) contribution coming from the parametric boundary (source) term. We show that for all small parameter values $λ>0$, the problem has at least five nontrivial smooth solutions, four of constant sign and one nodal. We also produce extremal constant sign solutions and determine their monotonicity and continuity properties as the parameter $λ>0$ varies. In the semilinear case we produce a sixth nontrivial solution but without any sign information. Our approach uses variational methods together with truncation and perturbation techniques, and Morse theory.

math.AP↗

arXiv2019

Combinatorics of binary words and codimensions of identities in left nilpotent algebras

Numerical characteristics of polynomial identities of left nilpotent algebras are examined. Previously, we came up with a construction which, given an infinite binary word, allowed us to build a two-step left nilpotent algebra with specified properties of the codimension sequence. However, the class of the infinite words used was confined to periodic words and Sturm words. Here the previously proposed approach is generalized to a considerably more general case. It is proved that for any algebra constructed given a binary word with subexponential function of combinatorial complexity, there exists a PI-exponent, and its precise value is computed.

math.RA↗

arXiv2019

Positive solutions for nonlinear parametric singular Dirichlet problems

We consider a nonlinear parametric Dirichlet problem driven by the $p$-Laplace differential operator and a reaction which has the competing effects of a parametric singular term and of a Carathéodory perturbation which is ($p-1$)-linear near $+\infty$. The problem is uniformly nonresonant with respect to the principal eigenvalue of $(-Δ_p,W^{1,p}_0(Ω))$. We look for positive solutions and prove a bifurcation-type theorem describing in an exact way the dependence of the set of positive solutions on the parameter $λ>0$.

math.AP↗

arXiv2019

Periodic solutions for implicit evolution inclusions

We consider a nonlinear implicit evolution inclusion driven by a nonlinear, nonmonotone, time-varying set-valued map and defined in the framework of an evolution triple of Hilbert spaces. Using an approximation technique and a surjectivity result for parabolic operators of monotone type, we show the existence of a periodic solution.

math.AP↗

arXiv2019

Nonlinear Dirichlet problems with unilateral growth on the reaction

We consider a nonlinear Dirichlet problem driven by the $p$-Laplace differential operator with a reaction which has a subcritical growth restriction only from above. We prove two multiplicity theorems producing three nontrivial solutions, two of constant sign and the third nodal. The two multiplicity theorems differ on the geometry near the origin. In the semilinear case (that is, $p=2$), using Morse theory (critical groups), we produce a second nodal solution for a total of four nontrivial solutions. As an illustration, we show that our results incorporate and significantly extend the multiplicity results existing for a class of parametric, coercive Dirichlet problems.

math.AP↗

arXiv2019

Asymmetric Robin problems with indefinite potential and concave terms

We consider a parametric semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. In the reaction, we have the competing effects of a concave term appearing with a negative sign and of an asymmetric asymptotically linear term which is resonant in the negative direction. Using variational methods together with truncation and perturbation techniques and Morse theory (critical groups) we prove two multiplicity theorems producing four and five respectively nontrivial smooth solutions when the parameter $λ>0$ is small.

math.AP↗

arXiv2019

Nodal solutions for nonlinear nonhomogeneous Robin problems

We consider the nonlinear Robin problem driven by a nonhomogeneous differential operator plus an indefinite potential. The reaction term is a Carathéodory function satisfying certain conditions only near zero. Using suitable truncation, comparison, and cut-off techniques, we show that the problem has a sequence of nodal solutions converging to zero in the $C^1(\overlineΩ)$-norm.

math.AP↗

arXiv2018

Perturbations of nonlinear eigenvalue problems

We consider perturbations of nonlinear eigenvalue problems driven by a nonhomogeneous differential operator plus an indefinite potential. We consider both sublinear and superlinear perturbations and we determine how the set of positive solutions changes as the real parameter $λ$ varies. We also show that there exists a minimal positive solution $\overline{u}_λ$ and determine the monotonicity and continuity properties of the map $λ\mapsto\overline{u}_λ$. Special attention is given to the particular case of the $p$-Laplacian.

math.AP↗

arXiv2018

Double phase problems with variable growth

We consider a class of double phase variational integrals driven by nonhomogeneous potentials. We study the associated Euler equation and we highlight the existence of two different Rayleigh quotients. One of them is in relationship with the existence of an infinite interval of eigenvalues while the second one is associated with the nonexistence of eigenvalues. The notion of eigenvalue is understood in the sense of pairs of nonlinear operators, as introduced by Fučik, Nečas, Souček, and Souček. The analysis developed in this paper extends the abstract framework corresponding to some standard cases associated to the $p(x)$-Laplace operator, the generalized mean curvature operator, or the capillarity differential operator with variable exponent. The results contained in this paper complement the pioneering contributions of Marcellini, Mingione et al. in the field of variational integrals with unbalanced growth.

math.AP↗