Nonlinear Eigenvalue Problem for p‐Laplacian in IR^N

Abstract

The nonlinear eigenvalue problem for p-Laplacianis considered. We assume that 1 < p < N and that the functionfis of subcritical growth with respect to the variable u. The existence and C'."-regularity of the weak solution is proved. 9*

Solving the $p$-Laplacian on manifolds

“…License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use From (6) and (7) one deduces that J (u * ) ≤ lim i→∞ inf J (u i ) = m; hence J (u * ) = m. By the usual arguments from variational calculus, one deduces that u * is a weak solution to (1). The C 1,α regularity follows from Theorem 1 in [5].…”

“…If M = R n with 1 < p < n and h ≥ 0, then equation (1) (and in fact a more general eigenvalue problem) is solved in [1].…”

L ∞-Estimates for nonlinear elliptic Neumann boundary value problems

“…To prove the L ∞ −regularity of u, we will use the Moser iteration technique (see e.g. [5][6][7][8][9]). It suffices to consider the proof in case 1 ≤ p ≤ N , otherwise we would be done.…”

MULTIPLE SOLUTIONS FOR A p-LAPLACIAN SYSTEM WITH NONLINEAR BOUNDARY CONDITIONS

“…In fact, the study of scalar equations had really started in the middle of 80s by M.Ôtani [34] in one dimension and then in dimension N by F. de Thélin [19]. Later, the results are generalized to other kinds of equations or systems involving p-Laplacian in R N or bounded open set Ω ⊂ R N (see, e.g., [1,3,4,5,6,7,8,13,18,20,23,24,27,29,30,33,35] and the references therein).…”