Nonhomogeneous Differential Operator Research Articles

Nonlinear multivalued Duffing systems

We consider a multivalued nonlinear Duffing system driven by a nonlinear nonhomogeneous differential operator. We prove existence theorems for both the convex and nonconvex problems (according to whether the multivalued perturbation is convex valued or not). Also, we show that the solutions of the nonconvex problem are dense in those of the convex (relaxation theorem). Our work extends the recent one by Kalita–Kowalski [7].

Nodal Solutions for Nonlinear Non-Homogeneous Robin Problems with an Indefinite Potential

AbstractWe consider a nonlinear Robin problem driven by a non-homogeneous differential operator plus an indefinite potential term. The reaction function is Carathéodory with arbitrary growth near±∞. We assume that it is odd and exhibits a concave term near zero. Using a variant of the symmetric mountain pass theorem, we establish the existence of a sequence of distinct nodal solutions which converge to zero.

Positive solutions to nonlinear nonhomogeneous inclusion problems with dependence on the gradient

The goal of the paper is to study a generalized elliptic inclusion problem driven by a nonhomogeneous partial differential operator with the Dirichlet boundary condition and a convection multivalued term. An existence theorem for positive solutions of the problem is established by applying the method of subsolution–supersolution, together with truncation and comparison techniques.

Nonlinear nonhomogeneous Robin problems with dependence on the gradient

We consider a nonlinear elliptic equation driven by a nonhomogeneous partial differential operator with Robin boundary condition and a convection term. Using a topological approach based on the Leray–Schauder alternative principle, together with truncation and comparison techniques, we show the existence of a smooth positive solution without imposing any global growth condition on the reaction term.

Multiple nodal solutions for nonlinear nonhomogeneous elliptic problems with a superlinear reaction

We consider nonlinear Dirichlet and Neumann problems driven by a nonlinear nonhomogeneous differential operator and with a (p−1)-superlinear Carathéodory reaction which does not satisfy the Ambrosetti–Rabinowitz condition. We prove two multiplicity theorems producing two nodal solutions, and answer two open questions posed by Aizicovici et al. (2013) and by Barletta and Papageorgiou (2014). Our approach uses variational methods together with suitable truncation techniques and flow invariance arguments.

Extremal solutions and strong relaxation for nonlinear multivalued systems with maximal monotone terms

We consider differential systems in RN driven by a nonlinear nonhomogeneous second order differential operator, a maximal monotone term and a multivalued perturbation F(t,u,u′). For periodic systems we prove the existence of extremal trajectories, that is solutions of the system in which F(t,u,u′) is replaced by extF(t,u,u′) (= the extreme points of F(t,u,u′)). For Dirichlet systems we show that the extremal trajectories approximate the solutions of the “convex” problem in the C1(T,RN)-norm (strong relaxation).

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 $\lambda>0$, 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 $\lambda>0$ 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.

Positive solutions for nonlinear nonhomogeneous parametric Robin problems

Abstract 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 {\lambda>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 * {u^{*}_{\lambda}} of the problem, and we investigate the properties of the map λ ↦ u λ * {\lambda\mapsto u^{*}_{\lambda}} .

Small perturbations of elliptic problems with variable growth

This paper deals with the study of a nonlinear eigenvalue problem driven by a new class of non-homogeneous differential operators with variable exponent and involving a nonlinear term with variable growth. The framework in the present paper corresponds to the case of small perturbations of the nonlinear term. Combining variational arguments with energy estimates, we establish the existence of eigenvalues in a neighborhood of the origin. Our abstract setting includes several models described by nonhomogeneous differential operators, including the case of the capillarity equation.

Nodal solutions for noncoercive nonlinear Neumann problems with indefinite potential

We consider a nonlinear Neumann problem driven by a nonhomogeneous differential operator and an indefinite potential. Using variational methods together with flow invariance arguments, we show that the problem has at least one nodal solution. The result presented in this paper gives an answer to the open question raised by Papageorgiou and Rădulescu (2016).

Multiplicity theorems for nonlinear nonhomogeneous Robin problems

We study a nonlinear Robin boundary value driven by a nonhomogeneous differential operator with a Carathéodory reaction and we look for multiple nontrivial solutions with sign information. We prove four such multiplicity theorems producing three nontrivial solutions, for resonant problems and for problems in which no global growth restriction is assumed on the reaction. Also, in the semilinear case, we show that we can have four nontrivial solutions, by producing a second nodal solution.

Non-autonomous Eigenvalue Problems with Variable (p 1,p 2)-Growth

Abstract We are concerned with the study of a class of non-autonomous eigenvalue problems driven by two non-homogeneous differential operators with variable ( p 1 , p 2 ) {(p_{1},p_{2})} -growth. The main result of this paper establishes the existence of a continuous spectrum consisting in an unbounded interval and the nonexistence of eigenvalues in a neighbourhood of the origin. The abstract results of this paper are described by two Rayleigh-type quotients and the proofs rely on variational arguments.

Existence, nonexistence and multiplicity of positive solutions for nonlinear, nonhomogeneous Neumann problems

We consider a nonlinear parametric Neumann problem driven by a nonhomogeneous differential operator and a strictly $$(p-1)$$ -sublinear reaction term. We prove a bifurcation-type result establishing the existence of a critical parameter value $$\lambda _*>0$$ such that for all $$\lambda >\lambda _*$$ the problem has at least two positive solutions, for $$\lambda =\lambda _*$$ it has at least one positive solution and for $$\lambda \in (0,\lambda _*)$$ there are no positive solutions. Also, for $$\lambda \ge \lambda _*$$ we show that the problem has a smallest positive solution $$\bar{u}_{\lambda }$$ and we investigate the continuity and monotonicity properties of the map $$\lambda \rightarrow \bar{u}_{\lambda }$$ .

Multiple and nodal solutions for parametric Neumann problems with nonhomogeneous differential operator and critical growth

We consider a parametric Neumann problem with nonhomogeneous differential operator and critical growth. Combining variational methods based on critical point theory, with suitable truncation techniques and flow invariance arguments, we show that for all large λ, the problem has at least three nontrivial smooth solutions, two of constant sign (one positive, the other negative) and the third nodal. We also study the asymptotic behavior of all solutions obtained when λ converges to infinity. The interesting point is that we do not impose any restrictions to the behavior of the nonlinear term f at infinity. Our work unifies and sharply improves several recent papers.

Nonlinear Nonhomogeneous Robin Problems with Superlinear Reaction Term

Abstract We consider a nonlinear Robin problem driven by a nonlinear, nonhomogeneous differential operator, and with a Carathéodory reaction term which is ( p - 1 ) ${{(p-1)}}$ -superlinear near ± ∞ ${\pm\infty}$ without satisfying the Ambrosetti–Rabinowitz condition and which does not have a standard subcritical polynomial growth. Using a combination of variational methods and Morse theoretic techniques, we prove a multiplicity theorem producing three nontrivial solutions (two of which have constant sign). In the process we establish some useful facts about the boundedness of the weak solutions of critical equations and the relation of Sobolev and Hölder local minimizers for functionals with a critical perturbation term.

The Brezis–Oswald Result for Quasilinear Robin Problems

Abstract In this paper we consider a nonlinear elliptic problem driven by a nonhomogeneous differential operator with Robin boundary conditions. We produce conditions on the reaction term near 0 + ${0^{+}}$ and near + ∞ ${+\infty}$ , which imply the existence and uniqueness of a positive solution. In the particular case of equations driven by the p-Laplacian, we show that these conditions are also necessary, extending in this way the semilinear Dirichlet work of Brezis and Oswald [5].

Nonlinear, nonhomogeneous parametric Neumann problems

We consider a parametric nonlinear Neumann problem driven by a nonlinear nonhomogeneous differential operator, with a Caratheodory reaction $f$ which is $p$-superlinear in the second variable, but not necessarily satisfying the usual in such cases Ambrosetti-Rabinowitz condition. We prove a bifurcation type result describing the dependence of positive solutions on the parameter $\lambda> 0$, show the existence of a smallest positive solution $\overline{u}_{\lambda}$ and investigate properties of the map $\lambda\mapsto\overline{u}_{\lambda}$. Finally, we show the existence of nodal solutions.

Infinitely Many Nodal Solutions for Nonlinear Nonhomogeneous Robin Problems

Abstract We consider a nonlinear Robin problem driven by a nonhomogeneous differential operator which incorporates the p-Laplacian as special case. The reaction f ⁢ ( z , x ) ${f(z,x)}$ is a Carathéodory function which need not satisfy a global growth condition and is only assumed to be odd near zero. Using variational tools, we show that the problem has a whole sequence of distinct nodal (that is, sign-changing) solutions.

Nodal solutions for nonlinear nonhomogeneous Neumann equations

We consider a nonlinear Neumann problem driven by a nonhomogeneous differential operator with a Caratheodory reaction which is $(p-1)$-sublinear near $\pm\infty$. Using variational tools we show that the problem has at least three nontrivial smooth solutions (one positive, one negative and a third nodal). Our formulation unifies problems driven by the $p$-Laplacian, the $(p,q) $ Laplacian and the $p$-generalized mean curvature operator.

Nonlinear nonhomogeneous periodic problems

We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator and a Caratheodory reaction. We show that it has at least three solutions, two of constant sign and the third nodal. In the particular case of the scalar \({p-}\)Laplacian and with a parametric reaction of equidiffusive type, we show that three solutions with precise sign exist if the parameter \({\lambda > \widehat{\lambda}_1(p)=}\) the first nonzero eigenvalue of the periodic scalar Laplacian. Finally, in the semilinear case \({(p=2),}\) we show that there is a second nodal solution, for a total of four nontrivial solutions all with sign information.