1,720,982 research outputs found
Positive eigenvectors and simple nonlinear maps
Full text linkFor linear operators L,T and nonlinear maps P, we describe classes of simple maps F = I −PT, F = L−P between Banach and Hilbert spaces, for which no point has more than two preimages. The classes encompass known examples (homeomorphisms, global folds) and the weaker, geometric, hypotheses suggest new ones. The operator L may be the Laplacian with various boundary conditions, as in the original Ambrosetti-Prodi the- orem, or the operators associated with the quantum harmonic oscillator, the hydrogen atom, a spectral fractional Laplacian, elliptic operators in non-divergent form. The maps P include the Nemitskii map P(u) = f(u) but may be non-local, even non-variational. For self-adjoint operators L, we employ familiar results on the nondegeneracy of the ground state. On Banach spaces, we use a variation of the Krein-Rutman theorem
On the number of closed solutions for polynomial ODE's and a special case of Hilbert's 16th problem
No full textOn the count and the classification of periodic solutions to forced pendulum-type equations
No full textRadial and non radial solutions for Hardy-Hénon type elliptic systems
No full textWe discuss existence and non-existence of positive solutions for the following system of Hardy and Hénon type:
where is a bounded domain in , N ≥ 3, p, q > 1, and α, β > −N. We also study symmetry breaking for ground states when Ω is the unit ball in
Non-radial maximizers for functionals with exponential non-linearity in R-2
No full textWe consider the functional F:H-0(1)(B(0,1))-> R
F(u)=integral(B(0,1)) vertical bar x vertical bar(alpha)(e(p vertical bar u vertical bar gamma)-1-p vertical bar u vertical bar(gamma))dx
where alpha>0, p>0, 1<= 2, and B(0,1) is the unit ball in R-2. We prove that for any p>0, 1<2 and 0<4 pi, gamma=2 no maximizer of F(u) on the unit ball in H-0(1) is radially symmetric provided that alpha is large enough. This extends a result of Smets, Su and Willem concerning the existence of non-radial ground state solutions for the Rayleigh quotient related to the Henon equation with Dirichlet boundary conditions
Qualitative Properties of Solutions of Semilinear Elliptic Systems
Full text linkThe article explores the qualitative properties of solutions to ellip- tic equations and systems, focusing particularly on whether solutions retain the symmetry of their domains. According to the well-known Gidas-Ni-Nirenberg theorem, positive solutions to certain autonomous elliptic equations in radial domains are radial themselves. However, this symmetry can be broken in equations with power weight terms. The article also examines related results for systems of these weighted equations
On Trudinger-Moser type inequalities with logarithmic weights
No full textTrudinger–Moser type inequalities for radial Sobolev spaces with logarithmic weights are considered. The precise Trudinger–Moser growths in dependence on the logarithmic terms, and the corresponding sharp Moser type exponents are determined. In a particular case a critical Trudinger–Moser growth of double exponential type is found
Eigenvalues and bifurcation for Neumann problems with indefinite weights
Full text linkWe consider eigenvalue problems and bifurcation of positive solutions for elliptic equations with indefinite weights and with Neumann boundary conditions. We give complete results concerning the existence and non- existence of positive solutions for the superlinear coercive and non-coercive problems, showing a surprising complementarity of the respective results
Bifurcation beyond the principal eigenvalues for Neumann problems with indefinite weights
Full text linkThis paper is devoted to the study of the effects of indefinite weights on some following nonlinear Neumann problems. Our results establish a relation between the position of a parameter and the number of nontrivial classical solutions of these problems. The proof combines spectral analysis tools, variational methods and the Clark multiplicity theorem
Trudinger–Moser type inequalities with logarithmic weights in dimension N
No full textWe consider borderline embeddings of Trudinger–Moser type for weighted Sobolev spaces in bounded domains in RN . The embeddings go into Orlicz spaces with exponential growth functions. It turns out that the most interesting weights are powers of the logarithm, for which an explicit dependence of the maximal growth functions can be established. Corresponding Moser type results are also proved, with explicit sharp exponents. In the particular case of a logarithmic weight with the limiting exponent N − 1, a maximal growth of double exponential type is obtained, while for any larger exponent the embedding goes into L∞
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