Search results for "Nodal solution"
showing 10 items of 20 documents
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 …
A multiplicity theorem for parametric superlinear (p,q)-equations
2020
We consider a parametric nonlinear Robin problem driven by the sum of a \(p\)-Laplacian and of a \(q\)-Laplacian (\((p,q)\)-equation). The reaction term is \((p-1)\)-superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. Using variational tools, together with truncation and comparison techniques and critical groups, we show that for all small values of the parameter, the problem has at least five nontrivial smooth solutions, all with sign information.
Multiple nodal solutions for semilinear robin problems with indefinite linear part and concave terms
2017
We consider a semilinear Robin problem driven by Laplacian plus an indefinite and unbounded potential. The reaction function contains a concave term and a perturbation of arbitrary growth. Using a variant of the symmetric mountain pass theorem, we show the existence of smooth nodal solutions which converge to zero in $C^1(\overline{\Omega})$. If the coefficient of the concave term is sign changing, then again we produce a sequence of smooth solutions converging to zero in $C^1(\overline{\Omega})$, but we cannot claim that they are nodal.
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…
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.
Nonlinear Nonhomogeneous Elliptic Problems
2019
We consider nonlinear elliptic equations driven by a nonhomogeneous differential operator plus an indefinite potential. The boundary condition is either Dirichlet or Robin (including as a special case the Neumann problem). First we present the corresponding regularity theory (up to the boundary). Then we develop the nonlinear maximum principle and present some important nonlinear strong comparison principles. Subsequently we see how these results together with variational methods, truncation and perturbation techniques, and Morse theory (critical groups) can be used to analyze different classes of elliptic equations. Special attention is given to (p, 2)-equations (these are equations driven…
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.
Multiple solutions for semilinear Robin problems with superlinear reaction and no symmetries
2021
We study a semilinear Robin problem driven by the Laplacian with a parametric superlinear reaction. Using variational tools from the critical point theory with truncation and comparison techniques, critical groups and flow invariance arguments, we show the existence of seven nontrivial smooth solutions, all with sign information and ordered.
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.
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.