Compactness and existence results for the p-Laplace equation

Abstract

Given 1 < p < N and two measurable functions V (r) ≥ 0 and K (r) > 0, r > 0, we define the weighted spacesand study the compact embeddings of the radial subspace ofas a particular case. We consider exponents q1, q2, q that can be greater or smaller than p. Our results do not require any compatibility between how the potentials V and K behave at the origin and at infinity, and essentially rely on power type estimates of their relative growth, not of the potentials separately. We then apply these results to the investigation of existence and multiplicity of finite energy solutions to nonlinear p-Laplace equations of the formwhere V and g (|·| , u) with u fixed may be vanishing or unbounded at zero or at infinity. Both the cases of g super and sub p-linear in u are studied and, in the sub p-linear case, nonlinearities with g (|·| , 0) = 0 are also considered. MSC (2010):Primary 35J92; Secondary 35J20, 46E35, 46E30

Radial Nonlinear Elliptic Problems with Singular or Vanishing Potentials

“…Moreover, multiplicity results can also be obtained. We leave the details to interested reader, which we refer to [3,4] for similar results and related arguments.…”

Radial quasilinear elliptic problems with singular or vanishing potentials

“…Introduction. In this paper, we pursue the work we made in [2,5,4,3,6,8,10] and, in particular, we complete the results we proved in [6]. In all these works, we have obtained embedding and compactness results for weighted Sobolev spaces, which, by variational methods, yielded existence and multiplicity results for nonlinear elliptic equations in R N .…”

“…These results are not available anymore for potentials which may be singular or vanishing, at zero or at infinity, and new embedding theorems are needed. Much work has been done in recent years in this direction, and we refer the reader for example to the following papers and the references therein: [5,4,10,13] for the case A = 1, p = 2 of the usual laplacian operator, [1,11,9,12,14,16,17,7,3] for the p-laplacian case A = 1, 1 < p < N (see also [2] for equations involving the bi-laplacian operators), [8] for the case A = 1, p = 2 of the weighted laplacian, and the already mentioned paper [15], which inspired the present work, for the general case A = 1, 1 < p < N with a potential on the derivatives. The main novelty of our approach with respect to [15] (here and in [6]) is two-fold.…”

Ground state solution of critical Schrödinger equation with singular potential

“…It also arises in many other branches of mathematical physics, such as nonlinear dynamics, plasma physics, condensed matter physics and cosmology, see [9,28]. For singular potential V satisfies conditions (V 1 ) and (V 2 ), the equations have been investigated by many researchers, [2,3,5,14,20] for equation with the Laplace operator, [6,7,23,21] for equation with the p-Laplace operator, [8,13,15] for equation with the biharmonic operator.…”

“…They attained the radial inequalities with respect to the parameters a, a 0 , b, b 0 , then established main results on continuous and compact embeddings and the existence of solution to equation (GQ). Badiale-Guida-Rolando [6] generalized the embedding results under different conditions on V and Q, and explored the existence of solution to equation (GQ) with the sub-critical and super-critical growth.…”