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Vicenţiu D. Rădulescu

Publications and source records attributed to Vicenţiu D. Rădulescu.

At least 55 records · Page 3Linked to original sources
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↗
arXiv2018

$(p,2)$-equations asymmetric at both zero and infinity

We consider a $(p,2)$-equation, that is, a nonlinear nonhomogeneous elliptic equation driven by the sum of a $p$-Laplacian and a Laplacian with $p>2$. The reaction term is $(p-1)$-linear but exhibits asymmetric behaviour at $\pm\infty$ and at $0^{\pm}$. Using variational tools, together with truncation and comparison techniques and Morse theory, we prove two multiplicity theorems, one of them providing sign information for all the solutions (positive, negative, nodal).

math.AP↗
arXiv2018

Nonlinear elliptic inclusions with unilateral constraint and dependence on the gradient

We consider a nonlinear Neumann elliptic inclusion with a source (reaction term) consisting of a convex subdifferential plus a multivalued term depending on the gradient. The convex subdifferential incorporates in our framework problems with unilateral constraints (variational inequalities). Using topological methods and the Moreau-Yosida approximations of the subdifferential term, we establish the existence of a smooth solution.

math.AP↗
arXiv2018

Positive solutions for nonvariational Robin problems

We study a nonlinear Robin problem driven by the $p$-Laplacian and with a reaction term depending on the gradient (the convection term). Using the theory of nonlinear operators of monotone-type and the asymptotic analysis of a suitable perturbation of the original equation, we show the existence of a positive smooth solution.

math.AP↗
arXiv2018

Periodic solutions for a class of evolution inclusions

We consider a periodic evolution inclusion defined on an evolution triple of spaces. The inclusion involves also a subdifferential term. We prove existence theorems for both the convex and the nonconvex problem, and we also produce extremal trajectories. Moreover, we show that every solution of the convex problem can be approximated uniformly by certain extremal trajectories (strong relaxation). We illustrate our results by examining a nonlinear parabolic control system.

math.AP↗
arXiv2018

Positive solutions for nonlinear nonhomogeneous parametric Robin problems

We study a parametric Robin problem driven by a nonlinear nonhomogeneous differential operator and with a superlinear Carathéodory reaction term. We prove a bifurcation-type theorem for small values of the parameter. Also, we show that as the parameter $λ>0$ approaches zero we can find positive solutions with arbitrarily big and arbitrarily small Sobolev norm. Finally we show that for every admissible parameter value there is a smallest positive solution $u^*_λ$ of the problem and we investigate the properties of the map $λ\mapsto u^*_λ$.

math.AP↗
arXiv2018

Nonhomogeneous Dirichlet problems without the Ambrosetti-Rabinowitz condition

We consider the existence of solutions of the following $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition: $-\mbox{div}(|\nabla u|^{p(x)-2}\nabla u)=f(x,u) \text{ in }Ω,$ and $u=0,\text{ on }\partial Ω.$ We give a new growth condition and we point out its importance for checking the Cerami compactness condition. We prove the existence of solutions of the above problem via the critical point theory, and also provide some multiplicity properties. Our results extend previous work by Q. Zhang and C. Zhao, Existence of strong solutions of a $p(x)$-Laplacian Dirichlet problem without the Ambrosetti-Rabinowitz condition, Comp. Math. Appl. 69 (2015), 1-12, and we establish the existence of solutions under weaker hypotheses on the nonlinear term.

math.AP↗
arXiv2018

Existence and multiplicity of solutions for resonant $(p,2)$-equations

We consider Dirichlet elliptic equations driven by the sum of a $p$-Laplacian $(2<p)$ and a Laplacian. The conditions on the reaction term imply that the problem is resonant at both $\pm\infty$ and at zero. We prove an existence theorem (producing one nontrivial smooth solution) and a multiplicity theorem (producing five nontrivial smooth solutions, four of constant sign and the fifth nodal; the solutions are ordered). Our approach uses variational methods and critical groups.

math.AP↗
arXiv2017

Positive solutions for superdiffusive mixed problems

We study a semilinear parametric elliptic equation with superdiffusive reaction and mixed boundary conditions. Using variational methods, together with suitable truncation techniques, we prove a bifurcation-type theorem describing the nonexistence, existence and multiplicity of positive solutions.

math.AP↗
arXiv2017

Nonhomogeneous Hemivariational Inequalities with Indefinite Potential and Robin Boundary Condition

We consider a nonlinear, nonhomogeneous Robin problem with an indefinite potential and a nonsmooth primitive in the reaction term. In fact, the right-hand side of the problem (reaction term) is the Clarke subdifferential of a locally Lipschitz integrand. We assume that asymptotically this term is resonant with respect the principal eigenvalue (from the left). We prove the existence of three nontrivial smooth solutions, two of constant sign and the third nodal. We also show the existence of extremal constant sign solutions. The tools come from nonsmooth critical point theory and from global optimization (direct method).

math.AP↗
arXiv2017

Resonant semilinear Robin problems with a general potential

We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. The reaction term is a Carathéodory function which is resonant with respect to any nonprincipal eigenvalue both at $\pm \infty$ and 0. Using a variant of the reduction method, we show that the problem has at least two nontrivial smooth solutions.

math.AP↗