Multiplicity of solutions for mixed local-nonlocal elliptic equations with singular nonlinearity

We will prove multiplicity results for the mixed local-nonlocal elliptic equation of the form eqnarray split -ₚu+ (-) ₚˢ u split eqnarray where equation* (-) ₚˢ u (x) = c₍, ₒP. V. ₑ䂞|u (x) -u (y) |^p-2 (u (x) -u (y) ) |x-y|^{n+sp} d y, equation* and -ₚ is the usual p-Laplace operator. Under the assumptions that is a bounded domain in R^n with regular enough boundary, p>1, n> p, s (0, 1), >0 and r (p-1, p^*-1) where p^* is the critical Sobolev exponent, we will show there exist at least two weak solutions to our problem for 00, assuming strict convexity of, for p=2 and s (0, 1/2), we will show the existence of at least two positive weak solutions to the problem, for small values of, extending the result of garaingeometric. Here c₍, ₒ is a suitable normalization constant, and P. V. stands for Cauchy Principal Value.