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Non-radial solutions for some semilinear elliptic equations on the disk. (English) Zbl 1404.35198
Summary: Starting with approximate solutions of the equation \(-\varDelta u = w u^3\) on the disk, with zero boundary conditions, we prove that there exist true solutions nearby. One of the challenges here lies in the fact that we need simultaneous and accurate control of both the (inverse) Dirichlet Laplacian and nonlinearities. We achieve this with the aid of a computer, using a Banach algebra of real analytic functions, based on Zernike polynomials. Besides proving existence, and symmetry properties, we also determine the Morse index of the solutions.

35J91 Semilinear elliptic equations with Laplacian, bi-Laplacian or poly-Laplacian
Ada95; MPFR
Full Text: DOI
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