Compactly supported solutions for a semilinear elliptic problem in ℝ<sup><i>n</i></sup> with sign-changing function and non-Lipschitz nonlinearity
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Abstract
For a sign-changing function a ( x ) we consider the solutions of the following semilinear elliptic problem in ℝ n with n ≥ 3: where γ > 0 and 0 < q < 1 < p < ( n + 2)/( n − 2). Under an appropriate growth assumption on a − at infinity, we show that all solutions are compactly supported. When Ω + = { x ∈ ℝ n | a ( x ) > 0} has several connected components, we prove that there exists an interval on γ in which the solutions exist. In particular, if a ( x ) = a (| x |), by applying the mountain-pass theorem there are at least two solutions with radial symmetry that are positive in Ω + .
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| Category | Codex | Gemma |
|---|---|---|
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