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| EXISTENCE OF DISCONTINUOUS SOLUTIONS FOR A DOUBLY DEGENERATE ELLIPTIC EQUATIONS ON $R^N$ |
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Citation： |
Wang Junyu.EXISTENCE OF DISCONTINUOUS SOLUTIONS FOR A DOUBLY DEGENERATE ELLIPTIC EQUATIONS ON $R^N$[J].Chinese Annals of Mathematics B,1993,14(2):175~182 |
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Net amount： 804 |
Authors： |
Wang Junyu; |
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| Abstract： |
It is demonstrated that under the hypotheses I—III the problem
$\[\left\{ {\begin{array}{*{20}{c}}
{div((k(U) + \varepsilon )|DU{|^{M - 1}}DU) = f(|x|,U) + \varepsilon U{\text{ }}in{\text{ }}{R^N},N > 1,{\text{ (1}}{\text{.1}}{{\text{)}}_\varepsilon }} \ {U(0) > 0,U(x) \geqslant 0{\text{ on }}{R^N},U(x) \to 0{\text{ as }}|x| \to + \infty {\text{ }}(1.2)}
\end{array}} \right.\]$
for each fixed $\epsilon >0$ has infinitely many distinct radially symmetric solutions $U_\epsilon=V_\epsilon(|x|)$ such that $V_\epsilon(s),s^{N-1}(k(V_\epsilon(s))+\epsilon)|V'(s)|^{M-1}V'_\epsilon(s)\in C[0,+\infinity)\capC^1(0,+\infinity)$,
$\[\left\{ {\begin{array}{*{20}{c}}
{({s^{N - 1}}(k({V_\varepsilon }(s)) + \varepsilon )|V'(s){|^{M - 1}}V'(s)) = {\varepsilon ^{N - 1}}(f(s,{V_\varepsilon }(s)) + \varepsilon {V_\varepsilon }(s))for{\text{ }}s > 0,{{(1.3)}_\varepsilon }} \ {{V_\varepsilon }(0) = B > 0,{V_\varepsilon }(s) \geqslant 0{\text{ for }}s > 0,and{\text{ }}{V_\varepsilon }( + \infty ) = 0,(1.4)}
\end{array}} \right.\]$
where B is a positive number chosen arbitrarily, which extends the result in [3]. In particular, the author proves that $U_0(x)=V_0(|x|)$ is a weak solution of the problem $(l.l)_0-(1.2)$. |
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