The monograph emphasizes those basic abstract methods and theories that are useful in the study of nonlinear boundary value problems. The content is developed over six chapters, providing a thorough introduction to the techniques used in the variational and topological analysis of nonlinear boundary value problems described by stationary differential operators. The authors give a systematic treatment of the basic mathematical theory and constructive methods for these classes of nonlinear equations as well as their applications to various processes arising in the applied sciences. They show how these diverse topics are connected to other important parts of mathematics, including topology, functional analysis, mathematical physics, and potential theory. Throughout the book a balance is maintained between rigorous mathematics and physical applications. The primary readership includes graduate students and researchers in pure and applied nonlinear analysis. Remark: According to the Slovenian Research Agency (ARRS) criteria this book represents an outstanding research achievement, with the maximum possible score (A''=1), and it is the most cited book in Mathematical Reviews among all monographs published in the year 2019.

COBISS.SI-ID: 18583897

The Robin boundary condition is named after the French mathematician Victor Gustave Robin and is a combination of the values of a function and the values of its derivative on the boundary of the domain. This means that Robin boundary conditions are a weighted combination of Dirichlet boundary conditions and Neumann boundary conditions. This paper is concerned with the mathematical analysis of solutions of some semilinear Robin problems driven by the negative Laplacian plus an indefinite potential. A feature of this paper is the presence of a superlinear reaction term which need not satisfy the standard Ambrosetti-Rabinowitz growth condition. The main results establish existence and multiplicity properties of solutions. In the final part of this paper we generate an infinity of nontrivial smooth solutions by introducing symmetry on the reaction term. The proofs combine powerful variational, topological and analytic tools including truncation and perturbation techniques, Morse theory (critical groups) and the Lyapunov-Schmidt reduction method. Several examples illustrate the main abstract results of this paper. Remark: Journal of Differential Equations (published by Elsevier) is considered to be one of the leading journals in our field and our project group often publishes in this journal. By the Slovenian Research Agency criteria, this paper is a very high quality achievement, with the maximum possible score (A'=1).

COBISS.SI-ID: 18034521

The purpose of this paper was to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space R^d (d)2). 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(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. Remark: Advances in Nonlinear Analysis (published by De Gruyter) is ranked very high on the Science Citation list of journals (based on the Impact Factor) in categories “Mathematics” and “Applied Mathematics” and our project group has already several papers published in this journal. By the Slovenian Research Agency criteria, this paper is a very high quality achievement, with the maximum possible score (A'=1).

COBISS.SI-ID: 18703961

This paper is concerned with the qualitative analysis of a class of second-order nonlinear evolution inclusions. The feature of this paper is the presence of non-monotone and noncoercive viscosity term. The novelty of this paper is that develops a kind of parabolic regularization of the inclusion, initially introduced by Jacques-Louis Lions in the context of semilinear hyperbolic equations. In such a way, using the nonsmooth Clarke theory in combination with a priori estimates that allow to pass to the limit, we obtain a sufficient condition for the existence of solutions. The main abstract result of this paper is illustrated with a hyperbolic boundary value problem with forcing term. The methods introduced in this paper can be extended to other classes of nonlinear evolutionary inequality problems. This paper also extends in the framework of evolution inclusions some ideas developed in our monograph V. Radulescu, D. Repovš, Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis, Francis & Taylor, New York, 2015 [COBISS.SI-ID 17325401] in the framework of problems with variable exponent. Remark: Journal of Differential Equations (published by Elsevier) is considered to be one of the leading journals in our field and our project group often publishes in this journal. By the Slovenian Research Agency criteria, this paper is a very high quality achievement, with the maximum possible score (A'=1).

COBISS.SI-ID: 18207321

In this paper, we consider the following nonlinear Kirchhoff type problem: $$\begin{cases} - \Big (a+b \int_{\mathbb{R}^3} |\nabla u|^2 \Big) \Delta u + V(x)u = f(u), & \text{in} \quad \mathbb{R}^3 \; , \\ u \in H^1 (\mathbb{R}^3) \; , \end{cases}$$ where a,b ) 0 are constants, the nonlinearity f is superlinear at infinity with subcritical growth and V is continuous and coercive. For the case when f is odd in u we obtain infinitely many sign-changing solutions for the above problem by using a combination of invariant sets method and the Ljusternik-Schnirelman type minimax method. To the best of our knowledge, there are only few existence results for this problem. It is worth mentioning that the nonlinear term may not be 4-superlinear at infinity, in particular, it includes the power-type nonlinearity |u|^{p-2}u with p \in (2, 4]. Remark: Nonlinear Analysis: Theory, Methods and Applications (published by Elsevier) is considered to be one of the leading journals in our field and our project group often publishes in this journal. By the Slovenian Research Agency criteria, this paper is a very high quality achievement, with the maximum possible score (A'=1).

COBISS.SI-ID: 18506585