Abstract
In this paper, we prove some continuous and compact embedding theorems for weighted Sobolev spaces, and consider both a general framework and spaces of radially symmetric functions. In particular, we obtain some a priori Strauss-type decay estimates. Based on these embedding results, we prove the existence of ground state solutions for a class of quasilinear elliptic problems with potentials unbounded, decaying and vanishing.
Highlights
In this paper, we consider the following quasilinear elliptic problems: ⎧⎨– pu + V (x)|u|p– u = K (x)|u|q– u, x ∈ RN, ⎩|u(x)| → as |x| → ∞, ( . )where N >, < p ≤ N, – pu = – div(|∇u|p– ∇u), V (x) and K(x) are nonnegative measurable functions, and may be unbounded, decaying and vanishing.Recently, these type elliptic equations have been widely studied
Based on variational methods and some compact embedding results, we obtain the existence of ground and bounded state solutions for problem ( . )
It is worth pointing out that we provide here a unified approach what conditions the potentials V (x) and K(x) should satisfy so that problem ( . ) and problem ( . ) have ground and bound state solutions, respectively
Summary
Rabinowitz [ ] proved the existence of a ground state solution for problem [ ], Ambrosetti, Malchiodi and Ruiz [ ] obtained the ground and bound state solutions for problem In , as the potentials V (x) and K(x) are radially symmetric, Su, Wang and Willem [ ] obtained the existence of a ground state solution for problem
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