Existence of constant sign solutions for the p-Laplacian problems in the resonant case with respect to Fučík spectrum

Mieko Tanaka

doi:10.55937/sut/1266408981

June 2009 Existence of constant sign solutions for the

p

-Laplacian problems in the resonant case with respect to Fučík spectrum

Mieko Tanaka

Author Affiliations +

SUT J. Math. 45(2): 149-166 (June 2009). DOI: 10.55937/sut/1266408981

Abstract

We consider the following the $p$ -Laplacian equation in a bounded domain $\rm{\Omega}$ :

$\left\{ {\matrix{ { - \Delta _p u = f(x,u)} \hfill & {{\rm{in }}\Omega }, \hfill \cr {u = 0} \hfill & {{\rm{on}}\partial \Omega }. \hfill \cr } } \right.$

We treat the case of nonlinearity term $f$ satisfying the following conditions

$f(x,t) = \{ \matrix{ {{a_0}t_ + ^{p - 1} - {b_0}t_ - ^{p - 1} + o(|t{|^{p - 1}})} & {{\rm{at\;0,}}} \cr {at_ + ^{p - 1} - bt_ - ^{p - 1} + o(|t{|^{p - 1}})} & {{\rm{at}} \infty {\rm{,}}} \cr }$

for constants ${a_0},{b_0},a$ and $b$ . We prove the existence of a positive solution or a negative solution in the case of $({a_0} - {\lambda _1})(a - {\lambda _1}) = 0$ or $({b_0} - {\lambda _1})(b - {\lambda _1}) = 0$ respectively, where ${\lambda _1}$ is the first eigenvalue of $- {\rm{\Delta }}_p$ .

Acknowledgements

The author would like to express her sincere thanks to Professor Shizuo Miyajima for helpful comments and encouragement.

Citation

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Mieko Tanaka. "Existence of constant sign solutions for the $p$ -Laplacian problems in the resonant case with respect to Fučík spectrum." SUT J. Math. 45 (2) 149 - 166, June 2009. https://doi.org/10.55937/sut/1266408981