Search results for "Nodal solution"
showing 10 items of 20 documents
Multiple Solutions with Sign Information for a Class of Coercive (p, 2)-Equations
2019
We consider a nonlinear Dirichlet equation driven by the sum of a p-Laplacian and of a Laplacian (a (p, 2)-equation). The hypotheses on the reaction f(z, x) are minimal and make the energy (Euler) functional of the problem coercive. We prove two multiplicity theorems producing three and four nontrivial smooth solutions, respectively, all with sign information. We apply our multiplicity results to the particular case of a class of parametric (p, 2)-equations.
Multiple solutions for nonlinear nonhomogeneous resonant coercive problems
2018
We consider a nonlinear, nonhomogeneous Dirichlet problem driven by the sum of a \begin{document}$p$\end{document} -Laplacian ( \begin{document}$2 ) and a Laplacian. The reaction term is a Caratheodory function \begin{document}$f(z,x)$\end{document} which is resonant with respect to the principal eigenvalue of ( \begin{document}$-\Delta_p,\, W^{1,p}_0(\Omega)$\end{document} ). Using variational methods combined with truncation and comparison techniques and Morse theory (critical groups) we prove the existence of three nontrivial smooth solutions all with sign information and under three different conditions concerning the behavior of \begin{document}$f(z,\cdot)$\end{document} near zero. By …
Multiple solutions with sign information for a (p,2)-equation with combined nonlinearities
2020
We consider a parametric nonlinear Dirichlet problem driven by the sum of a p-Laplacian and of a Laplacian (a (p,2)-equation) and with a reaction which has the competing effects of two distinct nonlinearities. A parametric term which is (p−1)-superlinear (convex term) and a perturbation which is (p−1)-sublinear (concave term). First we show that for all small values of the parameter the problem has at least five nontrivial smooth solutions, all with sign information. Then by strengthening the regularity of the two nonlinearities we produce two more nodal solutions, for a total of seven nontrivial smooth solutions all with sign informations. Our proofs use critical point theory, critical gro…
Three solutions for parametric problems with nonhomogeneous (a,2)-type differential operators and reaction terms sublinear at zero
2019
Abstract We consider parametric Dirichlet problems driven by the sum of a Laplacian and a nonhomogeneous differential operator ( ( a , 2 ) -type equation) and with a reaction term which exhibits arbitrary polynomial growth and a nonlinear dependence on the parameter. We prove the existence of three distinct nontrivial smooth solutions for small values of the parameter, providing sign information for them: one is positive, one is negative and the third one is nodal.
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.
Multiple solutions with sign information for semilinear Neumann problems with convection
2019
We consider a semilinear Neumann problem with convection. We assume that the drift coefficient is indefinite. Using the theory of nonlinear operators of monotone type, together with truncation and comparison techniques and flow invariance arguments, we prove a multiplicity theorem producing three nontrivial smooth solutions (positive, negative and 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.
Multiple solutions for (p,2)-equations at resonance
2019
We consider a nonlinear nonhomogeneous Dirichlet problem driven by the sum of a p-Laplacian and a Laplacian and a reaction term which is (p− 1)-linear near ±∞ and resonant with respect to any nonprincipal variational eigenvalue of (−∆p, W01,p(Ω)). Using variational tools together with truncation and comparison techniques and Morse Theory (critical groups), we establish the existence of six nontrivial smooth solutions. For five of them we provide sign information and order them.
Constant sign and nodal solutions for parametric anisotropic $(p, 2)$-equations
2021
We consider an anisotropic ▫$(p, 2)$▫-equation, with a parametric and superlinear reaction term.Weshow that for all small values of the parameter the problem has at least five nontrivial smooth solutions, four with constant sign and the fifth nodal (sign-changing). The proofs use tools from critical point theory, truncation and comparison techniques, and critical groups. Spletna objava: 9. 9. 2021. Abstract. Bibliografija: str. 1076.
Least energy solutions to the Dirichlet problem for the equation −D(u) = f (x, u)
2017
Let be a bounded smooth domain in RN. We prove a general existence result of least energy solutions and least energy nodal ones for the problem −u = f(x, u) in u = 0 on ∂ (P) where f is a Carathéodory function. Our result includes some previous results related to special cases of f . Finally, we propose some open questions concerning the global minima of the restriction on the Nehari manifold of the energy functional associated with (P) when the nonlinearity is of the type f(x, u) = λ|u| s−2u − μ|u| r−2u, with s, r ∈ (1, 2) and λ,μ > 0.