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Groundstates of the planar Schrodinger-Poisson system with potential well and lack of symmetry
(Cambridge University Press, 2023-06-06) Liu, Zhisu; Radulescu, Vicentiu; Zhang, Jianjun
The Schrodinger-Poisson system describes standing waves for the nonlinear Schrodinger equation interacting with the electrostatic field. In this paper, we are concerned with the existence of positive ground states to the planar Schrodinger-Poisson system with a nonlinearity having either a subcritical or a critical exponential growth in the sense of Trudinger-Moser. A feature of this paper is that neither the finite steep potential nor the reaction satisfies any symmetry or periodicity hypotheses. The analysis developed in this paper seems to be the first attempt in the study of planar Schrodinger-Poisson systems with lack of symmetry.
Multiplicity of solutions for nonlinear coercive problems
(Elsevier, 2023-12-01) Diblík, Josef; Galewski, Marek; Radulescu, Vicentiu; Šmarda, Zdeněk
We are concerned in this paper with problems that involve nonlinear potential mappings satisfying condition (S) and whose potentials are coercive. We first provide mild sufficient conditions for the minimizing sequence in the Weierstrass-Tonelli theorem in order to have strongly convergent subsequences. Next, we establish a three critical point theorem which is based on the Pucci-Serrin type mountain pass lemma and which is an infinite dimensional counterpart of the Courant theorem. Ricceri-type three critical point results then follow. Some applications to Dirichlet boundary value problems driven by the perturbed Laplacian are given in the final part of this paper.
Positive supersolutions of non-autonomous quasilinear elliptic equations with mixed reaction
(Association des Annales de l'Institut Fourier, 2023-10-27) Aghajani, Asodallah; Radulescu, Vicentiu
We provide a simple method for obtaining new Liouville-type theorems for positive supersolutions of the We We provide a simple method for obtaining new Liouville-type theorems for positive supersolutions of the elliptic problem - Delta(p)u+ b(x)vertical bar del u vertical bar(pq/q+1) = c(x)u(q) in Omega, where Omega is an exterior domain in R-N with N >= p > 1 and q >= p - 1. In the case q not equal p - 1, we mainly deal with potentials of the type b(x) = vertical bar x vertical bar(a), c(x) = lambda vertical bar x vertical bar(sigma), where lambda > 0 and a, sigma is an element of R. We show that positive supersolutions do not exist in some ranges of the parameters p, q, a, sigma, which turn out to be optimal. When q = p - 1, we consider the above problem with general weights b(x) >= 0, c(x) > 0 and we assume that c(x)- b(p)(x)/p(p) > 0 for large vertical bar x vertical bar, but we also allow the case lim(vertical bar x vertical bar ->infinity)[c(x)- b(p)(x)/p(p)] = 0. The weights b and c are allowed to be unbounded. We prove that if this equation has a positive supersolution, then the potentials must satisfy a related differential inequality not depending on the supersolution. We also establish sufficient conditions for the nonexistence of positive supersolutions in relationship with the values of tau := lim sup(vertical bar x vertical bar ->infinity) vertical bar x vertical bar b(x) <= infinity. A key ingredient in the proofs is a generalized Hardy-type inequality associated to the p-Laplace operator.
Infinitely many smooth nodal solutions for Orlicz Robin problems
(Elsevier, 2023-08-17) Bahrouni, Anouar; Missaoui, Hlel; Radulescu, Vicentiu
In this note, we study a Robin problem driven by the Orlicz g-Laplace operator. In particular, by using a regularity result and Kajikiya's theorem, we prove that the problem has a whole sequence of distinct smooth nodal solutions converging to the trivial one. The analysis is developed in the most general abstract setting that corresponds to Orlicz-Sobolev function spaces.
Multiple and Nodal Solutions for Parametric Dirichlet Equations Driven by the Double Phase Differential Operator
(Springer Nature, 2023-07-04) Cai, Li; Papageorgiou, Nikolaos S.; Radulescu, Vicentiu
We consider a nonlinear parametric Dirichlet problem driven by the double phase differential operator. Using variational tools combined with critical groups, we show that for all small values of the parameter, the problem has at least three nontrivial bounded solutions which are ordered and we provide the sign information for all of them. Two solutions are of constant sign and the third one is nodal. Finally, we determine the asymptotic behavior of the nodal solution as the parameter converges to zero.

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