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where Ω = ℝN or Ω = B1, N ⩾ 3, p > 1 and . Using a suitable map we transform problem (1) into another one without the singularity 1/|x|2. Then we obtain some bifurcation results from the radial solutions corresponding to some explicit values of λ.
We discuss a Conley index calculation which is of importance in population models with large interaction. In particular, we prove that a certain Conley index is trivial.
For systems of elliptic equations of Fitzhugh–Nagumo type on bounded domains and with small diffusion in one equation, we construct solutions with multiple sharp peaks close to each other and close to, but not on, the boundary. This is a striking contrast to results for scalar equations.
For some symmetric domains, we also construct similar multipeak solutions except that here the peaks are not close to each other.
A counterexample has been constructed to show that a conjectured global solution structure for bifurcation of non-trivial solutions from a simple eigenvalue of the linearization at zero really can occur. In addition, new results and counterexamples have been obtained for bifurcation from an eigenvalue of geometric multiplicity 1 and odd algebraic multiplicity.
Some new global results are given about solutions to the boundary value problem for the
Euler–Lagrange equations for the Ginzburg–Landau model of a one-dimensional superconductor.
The main advance is a proof that in some parameter range there is a branch of
asymmetric solutions connecting the branch of symmetric solutions to the normal state. Also,
simplified proofs are presented for some local bifurcation results of Bolley and Helffer. These
proofs require no detailed asymptotics for solution of the linear equations. Finally, an error
in Odeh's work on this problem is discussed.
In this paper, we prove the uniqueness of the decaying positive solution on all of Rn for certain second order non linear elliptic equations. This improves earlier work of a number of authors. These problems occur in the theory of peak solutions. In particular, our results apply to a number of non-smooth nonlinearities which occur as limiting equations in population problems.
We consider a competition–diffusion system and study its singular limit as the interspecific
competition rate tend to infinity. We prove the convergence to a Stefan problem with zero
latent heat.
In this note we discuss the radial positive solutions of
formula here
where D is the annulus {s∈Rn
[ratio ]b<|s|<1}. Here
0<b<1, n[ges ]2, and
g[ratio ]R→R is a suitable C1
function with g(0)[ges ]0 but with g changing sign. We answer
a question
on page 223 of the original Gidas–Ni–Nirenberg paper
[8], by showing that the
maximum of these solutions occurs at a point sε, where
|sε|→b as ε→0.
This also
shows that the non-negativity condition in the result on page 223 of
[8] is necessary.
We discuss the existence of positive solutions of some singularity perturbed elliptic equations on convex domains with nonlinearity changing sign. In particular, we obtain solutions with both a boundary layer and a sharp interior peak.
We study the existence of changing sign solutions of an elliptic semilinear boundary value problem, which arises as a limiting equation of the two species Lotka–Volterra competing equations system. Using variational methods and a result of D'Aujourd'hui, we find conditions which are both sufficient and necessary for this existence problem.
In this paper, we study the perturbation of zeros of maps of Banach spaces where the maps are invariant under continuous groups of symmetries. In some cases, we allow the perturbed maps partially to break the symmetries. Our results improve earlier results of the author by removing smoothness conditions on the group action. The key new idea is a regularity theorem for the zeros of invariant Fredholm maps.
We study the existence of zeros of a perturbed nonlinear operator near a zero of the unperturbed operator in the case where both operators are invariant under a symmetry group. To do this, we first correct some work of Rubinsztein on the G-homotopy groups of spheres.
Our basic theorem is a version of the implicit function theorem in the case of continuous groups of symmetries. The result is sufficiently general to cover a great many applications. It generalizes some earlier work of the author and corrects and improves some work of Vanderbauwhede. We also consider the breaking of symmetries problem and the variational case. Finally, we apply our results to study the periodic solutions of an ordinary differential equation.
The proof of Lemma 6 (and thus of Theorem 9) has a gap in it. While m(B) → 0 as r → ∞ for each fixed h in Ni, it is not clear (and probably false) that this holds uniformly for h in . However Lemma 6 (and thus Theorem 9) holds with only trivial modifications of the given proof if one of the following holds: (i) ygi(y) → ∞ as |y| → ∞; (ii) m{x∈Ω: h(x) = 0} = 0 for every h in (iii) Niis one dimensional; (iv) there is a subset A of Ω such that h(x) = 0 if x ∈ A and h ∈ and m{x∈Ω\A: h(x) = 0} = 0 for every h ∈ . Assumption (ii) holds under very weak conditions. For example, the methods in [1] and regularity theorems imply that (ii) holds if there is a closed subset T of Ω of measure zero such that either (a) Ω\T is connected and aij (i, j = 1, …, n) are locally Lipschitz continuous on Ω\T or (b) for each component A of Ω\T, the aij have Lipschitz extensions to Ā and T is a “nice” set. (For example, it suffices to assume that T is a smooth submanifold of Ω though much weaker conditions would suffice.) Remember that we are assuming Ω is connected.
We study bifurcation problems in the presence of continuous groups of symmetries and obtain theorems on the existence and uniqueness of solutions. We also briefly consider some applications.
We study the existence of solutions of the Dirichlet problem for weakly nonlinear elliptic partial differential equations. We only consider cases where the nonlinearities do not depend on any partial derivatives. For these cases, we prove the existence of solutions for a wide variety of nonlinearities.
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