Search results for "nonlinear regularity theory"
showing 5 items of 5 documents
Solutions with sign information for nonlinear Robin problems with no growth restriction on reaction
2019
We consider a parametric nonlinear Robin problem driven by a nonhomogeneous differential operator. The reaction is a Carathéodory function which is only locally defined (that is, the hypotheses concern only its behaviour near zero). The conditions on the reaction are minimal. Using variational tools together with truncation, perturbation and comparison techniques and critical groups, we show that for all small values of the parameter λ > 0, the problem has at least three nontrivial smooth solutions, two of constant sign and the third nodal.
Positive and nodal solutions for nonlinear nonhomogeneous parametric neumann problems
2020
We consider a parametric Neumann problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential term. The reaction term is superlinear but does not satisfy the Ambrosetti-Rabinowitz condition. First we prove a bifurcation-type result describing in a precise way the dependence of the set of positive solutions on the parameter λ > 0. We also show the existence of a smallest positive solution. Similar results hold for the negative solutions and in this case we have a biggest negative solution. Finally using the extremal constant sign solutions we produce a smooth nodal solution.
(p,2)-equations resonant at any variational eigenvalue
2018
We consider nonlinear elliptic Dirichlet problems driven by the sum of a p-Laplacian and a Laplacian (a (p,2) -equation). The reaction term at ±∞ is resonant with respect to any variational eigenvalue of the p-Laplacian. We prove two multiplicity theorems for such equations.
Singular (p, q)-equations with superlinear reaction and concave boundary condition
2020
We consider a parametric nonlinear elliptic problem driven by the sum of a p-Laplacian and of a q-Laplacian (a (Formula presented.) -equation) with a singular and (Formula presented.) -superlinear reaction and a Robin boundary condition with (Formula presented.) -sublinear boundary term (Formula presented.). So, the problem has the combined effects of singular, concave and convex terms. We look for positive solutions and prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter varies.
Parameter dependence for the positive solutions of nonlinear, nonhomogeneous Robin problems
2020
We consider a parametric nonlinear Robin problem driven by a nonlinear nonhomogeneous differential operator plus an indefinite potential. The reaction term is $$(p-1)$$-superlinear but need not satisfy the usual Ambrosetti–Rabinowitz condition. We look for positive solutions and prove a bifurcation-type result for the set of positive solutions as the parameter $$\lambda >0$$ varies. Also we prove the existence of a minimal positive solution $$u_\lambda ^*$$ and determine the monotonicity and continuity properties of the map $$\lambda \rightarrow u_\lambda ^*$$.