We consider a nonlinear logistic equation of superdiffusive type driven by a nonhomogeneous differential operator and a Robin boundary condition.We prove a multiplicity result for positive solutions which is global with respect to the parameter λ > 0 (bifurcation-type theorem).We also demonstrate the existence of a minimal positive solution u * λ and determine the monotonicity and continuity properties of the minimal solution map λ → u * λ .

We consider a general periodic system driven by a nonlinear, nonhomogeneous differential operator, with a maximal monotone term which is not defined everywhere. Using a topological approach based on Leray-Schauder alternative principle, we show the existence of a periodic solution.

We consider a nonlinear Neumann problem driven by the p-Laplacian and a reaction which consists of a singular term plus a (p-1) - linear perturbation which is resonant at +∞ with respect to the principal eigenvalue. Using variational methods together with suitable truncation, comparison and approximation techniques, we show that the problem admits two positive smooth solutions.

We consider a parametric nonlinear Robin problem driven by a non homogeneous differential operator. Using variational tools together with suitable truncation and perturbation techniques, we prove a bifurcation-type theorem describing the dependence of the set of positive solutions on the parameter.

We consider second order nonlinear Dirichlet systems driven by a nonlinear nonhomogeneous differential operator. The reaction term consists of a maximal monotone map A (.) plus a multivalued perturbation F depending also on the derivative. Using tools from multivalued analysis and from the theory of nonlinear operators of monotone type, we prove existence theorems both for the "convex" (F convex-valued) and the "nonconvex" (F nonconvex-valued) problems. We also present an example of a system with unilateral constraints.

We study a nonlinear Dirichlet problem driven by the sum of a p-Laplacian (p > 2) and a Laplacian and which at +/-infinity is resonant with respect to the spectrum of ( -Delta(p),W-0(1,P)(Omega)) and at zero is resonant with respect to the spectrum of (-Delta,H-0(1) (Omega)) (double resonance). We prove two multiplicity theorems providing three and four nontrivial solutions respectivelly, all with sign information. Our approach uses critical point theory together with truncation and comparison techniques and Morse theory.

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.

We consider a semilinear Neumann problem with an indefinite and unbounded potential, and a Caratheodory reaction term. Under asymptotic conditions on the reaction which make the energy functional coercive, we prove multiplicity theorems producing three or four solutions with sign information on them. Our approach combines variational methods based on the critical point theory with suitable perturbation and truncation techniques, and with Morse theory.

We consider a parametric semilinear Dirichlet problem driven by the Laplacian plus an indefinite unbounded potential and with a reaction of superdifissive type. Using variational and truncation techniques, we show that there exists a critical parameter value lambda(*) > 0 such that for all lambda > lambda(*) the problem has at least two positive solutions, for lambda = lambda(*) the problem has at least one positive solution, and no positive solutions exist when lambda is an element of (0, lambda(*)). Also, we show that for lambda >= lambda(*) the problem has a smallest positive solution.

We study a parametric nonlinear Dirichlet problem driven by a nonhomogeneous differential operator and with a reaction which is "concave" (i.e., (p - 1) - sublinear) near zero and "convex" (i.e., (p - 1) - superlinear) near +/-infinity. Using variational methods combined with truncation and comparison techniques, we show that for all small values of the parameter lambda > 0, the problem has at least five nontrivial smooth solutions (four of constant sign and the fifth nodal). In the Hilbert space case (p = 2), using Morse theory, we produce a sixth nontrivial smooth solution but we do not determine its sign.

We consider a nonlinear periodic problem driven by a nonhomogeneous differential operator plus an indefinite potential and a reaction having the competing effects of concave and convex terms. For the superlinear (concave) term we do not employ the usual in such cases Ambrosetti-Rabinowitz condition. Using variational methods together with truncation, perturbation and comparison techniques, we prove a bifurcation-type theorem describing the set of positive solutions as the parameter varies.

We consider nonlinear periodic equations driven by the scalar p-Laplacian and with a Caratheodory reaction which does not satisfy a global growth condition. Using truncation-perurbation techniques, variational methods and Morse theory, we prove a "three solutions theorem", providing sign information for all the solutions. In the semilinear case (p = 2), we produce a second nodal solution, for a total of four nontrivial solutions. We also cover problems which are resonant at zero.

We consider a parametric nonlinear Dirichlet problem driven by the p-Laplacian, with a singular term and a p-superlinear perturbation, which need not satisfy the usual Ambrosetti–Rabinowitz condition. Using variational methods together with truncation techniques, we prove a bifurcation-type theorem describing the behavior of the set of positive solutions as the parameter varies.

We consider a periodic problem driven by the scalar p-Laplacian and with a jumping (asymmetric) reaction. We prove two multiplicity theorems. The first concerns the nonlinear problem (1

We deal with an Ambrosetti–Prodi problem driven by the p-Laplace differential operator, with a “crossing” reaction which can be sublinear or superlinear (in the positive direction). Using variational methods based on the critical point theory, together with upper–lower solutions, truncation and comparison techniques and critical groups, we show the existence of a unique critical parameter value λ∗ such that for λ<λ∗ there are at least two nontrivial solutions, for λ=λ∗ there is at least one nontrivial solution, and for λ>λ∗ no solutions exist. We extend several recent results on this problem.

In this paper, we study a nonlinear elliptic equation driven by the sum of a p p -Laplacian and a Laplacian ( ( p , 2 ) \left ( p,2\right ) -equation), with a Carathéodory ( p − 1 ) \left ( p-1\right ) -(sub-)linear reaction. Using variational methods combined with Morse theory, we prove two multiplicity theorems providing precise sign information for all the solutions (constant sign and nodal solutions). In the process, we prove two auxiliary results of independent interest.

We consider a nonlinear Neumann problem driven by a nonhomogeneous differential operator with a Caratheodory reaction which is (p - 1)-sublinear near +/-infinity. 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.

We consider nonlinear, nonhomogeneous Dirichlet problems driven by the sum of a p-Laplacian (p > 2) and a Laplacian, with a reaction term which has space dependent zeros of constant sign. We prove three muliplicity theorems for such equations providing precise sign information for all solutions. In the first multiplicity theorem, we do not impose any growth condition on the reaction near +/-infinity. In the other two, we assume that the reaction is (p - 1) - linear and resonant with respect to principal eigenvalue of (-Delta(p), W-0(1,p) (Omega)). Our approach uses variational methods based on the critical point theory, together with suitable truncation and comparison techniques and Morse theory (critical groups).

We consider a nonlinear periodic problem drive driven by a nonhomogeneous differential operator which incorporates as a special case the scalar p-Laplacian, and a reaction which exhibits the competition of concave and convex terms. Using variational methods based on critical point theory, together with suitable truncation techniques and Morse theory (critical groups), we establish the existence of five nontrivial solutions, two positive, two negative and the fifth nodal (sign-changing). In the process, we also prove some auxiliary results of independent interest.