Results 51 to 60 of about 3,762 (154)

A mountain pass theorem without Palais–Smale condition

Comptes Rendus. Mathématique, 2005
Given a Hilbert space (H,〈⋅,⋅〉), Λ an interval of R and J∈C2(H,R) whose gradient ∇J:H→H is a compact mapping, we consider a family of functionals of the type: I(λ,u)=〈u,u〉−λJ(u),(λ,u)∈Λ×H. Without further compactness assumptions, we present a deformation lemma to detect critical points. In particular, if I(λ¯,⋅) has a ‘mountain pass structure’ for some
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A note on Palais-Smale condition and coercivity

Differential and Integral Equations, 1990
It has been observed [the second author, An existence theorem on multiple critical points and its applications in nonlinear P.D.E., in Differential geometry and differential equations, Proc. Symp., Changchun/China 1982, 479-483 (1986); the third author, An introduction to critical point theory (1988)] that, for a \(C^ 1\) function \(\varphi\) bounded ...
Čaklović, L., Li, Shu Jie, Willem, M.
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Planar Choquard equations with critical exponential reaction and Neumann boundary condition

Mathematische Nachrichten, Volume 297, Issue 10, Page 3847-3869, October 2024.
Abstract We study the existence of positive weak solutions for the following problem: −Δu+α(x)u=∫ΩF(y,u)|x−y|μ1dyf(x,u)inΩ,∂u∂η+βu=∫∂ΩG(y,u)|x−y|μ2dνg(x,u)on∂Ω,$$\begin{equation*} \begin{aligned} \hspace*{65pt}-\Delta u + \alpha (x) u &= {\left(\int \limits _{\Omega }\frac{F(y,u)}{|x-y|^{{\mu _1}}}\;dy\right)}f(x,u) \;\;\text{in} \; \Omega,\\ \hspace ...
Sushmita Rawat, Vicenţiu D. Rădulescu, K. Sreenadh   +2 more
wiley   +1 more source

An estimate on the relative Morse index for strongly indefinite functionals

Electronic Journal of Differential Equations, 2001
We extend the Benci and Rabinowitz linking theorem to strongly indefinite functionals satisfying the Palais-Smale condition. More precisely, we show an upper estimate for a relative Morse index of critical points.
A. Abbondandolo, P. Felmer, J. Molina
doaj  

Multiplicity Results of Solutions to the Fractional p-Laplacian Problems of the Kirchhoff–Schrödinger–Hardy Type

Mathematics
This paper is devoted to establishing multiplicity results of nontrivial weak solutions to the fractional p-Laplacian equations of the Kirchhoff–Schrödinger type with Hardy potentials.
Yun-Ho Kim
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Multiple solutions for nonlinear discontinuous strongly resonant elliptic problems

Abstract and Applied Analysis, 2000
We consider quasilinear strongly resonant problems with discontinuous right-hand side. To develop an existence theory we pass to a multivalued problem by, roughly speaking, filling in the gaps at the discontinuity points.
Nikolaos C. Kourogenis, Nikolaos S. Papageorgiou   +1 more
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Least energy solutions for a class of (p1,p2)$(p_{1}, p_{2})$‐Kirchhoff‐type problems in RN$\mathbb {R}^{N}$ with general nonlinearities

Journal of the London Mathematical Society, Volume 110, Issue 4, October 2024.
Abstract We examine the following (p1,p2)$(p_{1}, p_{2})$‐Kirchhoff‐type problem: −M1∥∇u∥Lp1(RN)p1Δp1u−M2∥∇u∥Lp2(RN)p2Δp2u=g(u)inRN,u∈W1,p1(RN)∩W1,p2(RN),$$\begin{equation*} {\left\lbrace \def\eqcellsep{&}\begin{array}{ll}-M_{1}\left(\Vert \nabla u\Vert ^{p_{1}}_{L^{p_{1}}(\mathbb {R}^{N})}\right)\Delta _{p_{1}}u-M_{2}\left(\Vert \nabla u\Vert ^{p_{2 ...
Vincenzo Ambrosio
wiley   +1 more source

Existence and multiplicity of solutions for Dirichlet problems involving the p(x)-Laplace operator

Electronic Journal of Differential Equations, 2013
In this article, we study superlinear Dirichlet problems involving the p(x)-Laplace operator without using the Ambrosetti-Rabinowitz's superquadraticity condition.
Mustafa Avci
doaj  

Generalized noncooperative Schrödinger–Kirchhoff–type systems in RN$\mathbb {R}^N$

Mathematische Nachrichten, Volume 297, Issue 6, Page 2092-2121, June 2024.
Abstract We consider a class of noncooperative Schrödinger–Kirchhof–type system, which involves a general variable exponent elliptic operator with critical growth. Under certain suitable conditions on the nonlinearities, we establish the existence of infinitely many solutions for the problem by using the limit index theory, a version of concentration ...
Nabil Chems Eddine, Dušan D. Repovš
wiley   +1 more source

Multiple critical points theorems without the Palais–Smale condition

Journal of Mathematical Analysis and Applications, 2004
zbMATH Open Web Interface contents unavailable due to conflicting licenses.
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