Arbitrarily small nodal solutions for nonhomogeneous Robin problems

In the present paper, we consider a nonlinear Robin problem driven by a nonhomogeneous differential operator and with a reaction which is only locally defined. Using cut-off techniques and variational tools, we show that the problem has a sequence of nodal solutions converging to zero in C 1 ( Ω ‾ ).

Existence and Relaxation Results for Second Order Multivalued Systems

We consider nonlinear systems driven by a general nonhomogeneous differential operator with various types of boundary conditions and with a reaction in which we have the combined effects of a maximal monotone term $A(x)$ A ( x ) and of a multivalued perturbation $F(t,x,y)$ F ( t , x , y ) which can be convex or nonconvex valued. We consider the cases where $D(A)\neq \mathbb{R}^{N}$ D ( A ) ≠ R N and $D(A)= \mathbb{R}^{N}$ D ( A ) = R N and prove existence and relaxation theorems. Applications to differential variational inequalities and control systems are discussed.

Nonlinear nonhomogeneous Dirichlet problems with singular and convection terms

We consider a nonlinear Dirichlet problem driven by a general nonhomogeneous differential operator and with a reaction exhibiting the combined effects of a parametric singular term plus a Carathéodory perturbation $f(z,x,y)$ f ( z , x , y ) which is only locally defined in $x \in {\mathbb {R}} $ x ∈ R . Using the frozen variable method, we prove the existence of a positive smooth solution, when the parameter is small.

CONSTANT SIGN AND NODAL SOLUTIONS FOR NONLINEAR ROBIN EQUATIONS WITH LOCALLY DEFINED SOURCE TERM

We consider a parametric Robin problem driven by a nonlinear, nonhomogeneous differential operator which includes as special cases the p-Laplacian and the (p,q)-Laplacian. The source term is parametric and only locally defined (that is, in a neighborhood of zero). Using suitable cut-off techniques together with variational tools and comparison principles, we show that for all big values of the parameter, the problem has at least three nontrivial smooth solutions, all with sign information (positive, negative and nodal).

Minimum action solutions of nonhomogeneous Schrödinger equations

In this paper, we are concerned with the qualitative analysis of solutions to a general class of nonlinear Schrödinger equations with lack of compactness. The problem is driven by a nonhomogeneous differential operator with unbalanced growth, which was introduced by Azzollini [1]. The reaction is the sum of a nonautonomous power-type nonlinearity with subcritical growth and an indefinite potential. Our main result establishes the existence of at least one nontrivial solution in the case of low perturbations. The proof combines variational methods, analytic tools, and energy estimates.

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

Positive Solutions for the Generalized Nonlinear Logistic Equations

We consider a nonlinear parametric elliptic equation driven by a nonhomogeneous differential operator with a logistic reaction of the superdiòusive type. Using variationalmethods coupled with suitable truncation and comparison techniques, we prove a bifurcation type result describing the set of positive solutions as the parameter varies.

Coercive and noncoercive nonlinear Neumann problems with indefinite potential

We consider nonlinear Neumann problems driven by a nonhomogeneous differential operator and an indefinite potential. In this paper we are concerned with two distinct cases. We first consider the case where the reaction is (