Journal ArticleDOI
Seven solutions with sign information for sublinear equations with unbounded and indefinite potential and no symmetries
TLDR
In this paper, a semilinear Dirichlet problem with an unbounded and indefinite potential and a superlinear reaction which need not satisfy the usual Ambrosetti-Rabinowitz condition was considered.Abstract:
We consider a semilinear Dirichlet problem with an unbounded and indefinite potential and a superlinear reaction which need not satisfy the usual, in such cases, Ambrosetti-Rabinowitz condition. Using a combination of variational methods (critical point theory) with truncation and comparison techniques, with Morse theory and with flow invariance arguments, we show that the problem has at least seven nontrivial smooth solutions and provide sign information for all of them.read more
Citations
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Multiplicity of solutions for resonant Neumann problems with an indefinite and unbounded potential
TL;DR: In this paper, the authors examined semilinear Neumann problems driven by the Laplacian plus an unbounded and indefinite potential and showed that the reaction is a Caratheodory function which exhibits linear growth near ± ∞.
Journal ArticleDOI
Multiple solutions to a Robin problem with indefinite weight and asymmetric reaction
TL;DR: The existence of two nontrivial smooth solutions to a semilinear Robin problem with indefinite unbounded potential and asymmetric nonlinearity f is established in this article, and a third nonzero solution exists provided f is C 1.
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Multiple solutions to a Robin problem with indefinite weight and asymmetric reaction
TL;DR: The existence of two nontrivial smooth solutions to a semilinear Robin problem with indefinite unbounded potential and asymmetric nonlinearity was established in this article, where crossing and resonance are allowed.
Journal ArticleDOI
Noncoercive resonant (p,2)-equations with concave terms
TL;DR: In this article, a nonlinear Dirichlet problem driven by the sum of a p-Laplace and a Laplacian (a (p, 2 ) {(p, 2)} -equation) was considered and it was shown that for all small values of the parameter, the problem has as least six nontrivial smooth solutions all with sign information.
Journal ArticleDOI
Robin problems with indefinite, unbounded potential and reaction of arbitrary growth
TL;DR: In this paper, an elliptic Robin problem driven by the negative Laplacian plus an indefinite and unbounded potential and with a reaction of arbitrary growth which exhibits z-dependent zeros of constant sign was studied.
References
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Book
Critical Point Theory and Hamiltonian Systems
Jean Mawhin,Michel Willem +1 more
TL;DR: The direct method of the Calculus of Variations, Fenchel Transform and duality, Minimax Theorems for Indefinite Functional, Borsuk-Ulam Theorem and Index Theories, Lusternik-Schnirelman Theory and Multiple Periodic Solution with Fixed Energy, Morse-Ekeland Index, Morse Theory, and Morse Theory for Second Order Systems as discussed by the authors.
Book
Variational methods: Applications to nonlinear partial differential equations and Hamiltonian systems
TL;DR: Variational problems are part of our classical cultural heritage as discussed by the authors, and variational methods have been extensively studied in the literature, including lower semi-continuity results, the compensated compactness method, the concentration compactness methods, Ekeland's variational principle, and duality methods or minimax methods including the mountain pass theorems, index theory, perturbation theory, linking and extensions of these techniques to non-differentiable functionals and functionals defined on convex sets.
Journal ArticleDOI
A Strong Maximum Principle for some quasilinear elliptic equations
TL;DR: In this paper, it was shown that the Strong Maximum Principle is true for weak solutions of − Δu + β(u) = f with β a non-negative superharmonic continuous function in a domain Ω ⊂ ℝ� n�,n ⁽ 1,n ↽ 1.