In this work, we study the following resonant boundary value problem (1)−u′′(x)=g(x,u(x))+f(x);x∈(0,2π)u(0)=u(2π)u′(0)=u′(2π),where g:0,2π×R→R is continuous and bounded, and f∈L2(0,2π) is assumed to ...have mean-value zero. Using a variational approach and infinite-dimensional Morse theory, we prove existence and multiplicity of periodic solutions of the boundary problem (1), where the nonlinearity g satisfies an Ahmad–Lazer–Paul condition.
In this article, we study the existence and multiplicity of solutions to the problem {−Δu=g(x,u), in Ω;u=0, on ∂Ω, where Ω is a bounded domain in RN(N⩾2) with smooth boundary, and g:Ω¯×R→R is a ...differentiable function. We will assume that g(x,s) has a resonant behavior for large negative values of s and that a Landesman–Lazer type condition is satisfied. We also assume that g(x,s) is superlinear, but subcritical, for large positive values of s. We prove the existence and multiplicity of solutions for problem (1.1) by using minimax methods and infinite-dimensional Morse theory.
We consider a nonlinear Dirichlet problem driven by the sum of a
-Laplace and a Laplacian (a
-equation). The reaction exhibits the competing effects of a parametric concave term plus a Caratheodory ...perturbation which is resonant with respect to the principle eigenvalue of the Dirichlet
-Laplacian. Using variational methods together with truncation and comparison techniques and Morse theory (critical groups), we show that for all small values of the parameter, the problem has as least six nontrivial smooth solutions all with sign information (two positive, two negative and two nodal (sign changing)).
In this article, we study the semilinear elliptic boundary value problem $$\displaylines{ -\Delta u = -\lambda_1u^{-}+g(x,u),\quad \text{in }\Omega;\cr u =0,\quad \text{on }\partial\Omega, }$$ where ...\(u^{-}\) denotes the negative part of \(u:\Omega\to \mathbb{R}\); \(\lambda_1\) is the first eigenvalue of the N-dimensional Laplacian with Dirichlet boundary conditions in a connected, open, bounded set \(\Omega\subset\mathbb{R}^N\), \(N\geq 2\); and \(g: \overline{\Omega}\times\mathbb{R}\to\mathbb{R}\) is a continuous function. Assuming a one-sided Ahmad-Lazer-Paul condition, we establish conditions for existence and multiplicity of solutions by using variational methods and infinite-dimensional Morse Theory
For more information see https://ejde.math.txstate.edu/special/01/r2/abstr.html
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
In this paper, we are interested in the existence of multiple nontrivial $ T $-periodic solutions of the nonlinear second ordinary differential equation $ \ddot{x}+V_x(t, x) = 0 $ in $ N(\geq 1) $ ...dimensions. Using homological linking and morse theory, we get at least two critical points of the functional corresponding to our problem. And, we also prove that two critical points are different by critical groups. Then, we obtain there are at least two nontrivial $ T $-periodic solutions of the problem.
Nodal solutions for (p,2)-equations AIZICOVICI, SERGIU; PAPAGEORGIOU, NIKOLAOS S.; STAICU, VASILE
Transactions of the American Mathematical Society,
10/2015, Letnik:
367, Številka:
10
Journal Article
Recenzirano
Odprti dostop
-(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.>
In this article we study the existence of solutions to the problem $$\displylines{ -\Delta u = g(x,u) \quad \text{in } \Omega; \cr u = 0 \quad\text{on } \partial\Omega, }$$ where $\Omega$ is a smooth ...bounded domain in $\mathbb{R}^N$ $(N\geq 2)$ and $g:\overline{\Omega}\times\mathbb{R}\to\mathbb{R}$ is a differentiable function with g(x,0)=0 for all $x\in\Omega$. By using minimax methods and Morse theory, we prove the existence of at least three nontrivial solutions for the case in which an asymmetric condition on the nonlinearity g is assumed. The first two nontrivial solutions are obtained by employing a cutoff technique used by Chang et al in 9. For the existence of the third nontrivial solution, first we compute the critical group at infinity of the associated functional by using a technique used by Liu and Shaoping in 19. The final result is obtained by using a standard argument involving the Morse relation.
Celotno besedilo
Dostopno za:
DOBA, IZUM, KILJ, NUK, PILJ, PNG, SAZU, UILJ, UKNU, UL, UM, UPUK
In this paper, we consider the p(x)-Laplacian equations on the bounded domain. The nonlinearity is superlinear but does not satisfy the usual Ambrosetti–Rabinowitz condition near infinity, or its ...dual version near zero. Existence and multiplicity results are obtained via Morse theory and modified functional methods. In a sense, we expand a recent result of Gasiński and Papageorgiou L. Gasiński, N.S. Papageorgiou, Anisotropic nonlinear Neumann problems, Calc. Var. Partial Differential Equations 42 (2011) 323–354.
We consider a Dirichlet problem driven by the anisotropic (
p
,
q
)-Laplacian (double phase problem) and with a reaction term which exhibits asymmetric behavior as
x
→
±
∞
. Using variational tools, ...truncation, and comparison techniques and critical groups, we prove a multiplicity theorem producing four nontrivial solutions all with sign information and ordered.
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