On a Semilinear Elliptic Equation in R2 When the Exponent Approaches Infinity

Ren, Xiufang; Wei, Jinlong

doi:10.1006/jmaa.1995.1011

Cited by 6 publications

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“…On the other hand, by Lemma 3.3 of [13], there exists a constant C > 0 independent of p such that pc p > C. Hence, we can obtain pE p (u ± p ) ≥ C. Using Lemma 2.1, we have C ≤ p R 2 |u ± p (x)| p+1 dx. Next we prove the second inequality.…”

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confidence: 83%

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Asymptotic behavior of sign-changing radial solutions of a semilinear elliptic equation in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> when exponent approaches <inline-formula><tex-math id="M2">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula>

Ao¹,

Liu²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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In this paper, we consider a semi-linear elliptic equation in R 2 with the nonlinear exponent approaching infinity. We study asymptotic behavior of sign-changing once radial solutions obtained by . Assuming up(0) > 0, we prove that a suitable rescaling of the positive part u + p converges to the unique regular solution of Liouville equation in R 2 , while a suitable rescaling of the negative part u − p converges to a solution of a singular Liouville equation in R 2 . We also obtain the asymptotic value of the L ∞ -norms of u − p and u + p . Moreover, we show that pup blow up at the origin and pup convergence to the fundamental solution of −∆ + 1 in R 2 (up to a multiplier).

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confidence: 83%

“…First let us recall some known results for radial solutions. The asymptotic behavior of the ground state solution of (1.1) was studied by Ren and Wei in [13]. They show that the L ∞ (R 2 ) norm of solutions remain bounded and the normalized solution converges to the fundamental solution of −∆ + 1 in R 2 .…”

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confidence: 99%

Asymptotic behavior of sign-changing radial solutions of a semilinear elliptic equation in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^2 $\end{document}</tex-math></inline-formula> when exponent approaches <inline-formula><tex-math id="M2">\begin{document}$ +\infty $\end{document}</tex-math></inline-formula>

Ao¹,

Liu²

2020

Discrete &Amp; Continuous Dynamical Systems - A

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“…We state a lemma from our earlier work [11] with a brief proof. where C does not depend on anything and |Q| is the Lebesgue measure of Q. Therefore…”

Section: Some Estimatesmentioning

confidence: 99%

On a Two-Dimensional Elliptic Problem with Large Exponent in Nonlinearity

Ren¹,

Wei²

1994

Transactions of the American Mathematical Society

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Abstract. A semilinear elliptic equation on a bounded domain in R2 with large exponent in the nonlinear term is studied in this paper. We investigate positive solutions obtained by the variational method. It turns put that the constrained minimizing problem possesses nice asymptotic behavior as the nonlinear exponent, serving as a parameter, gets large. We shall prove that cp , the minimum of energy functional with the nonlinear exponent equal to p , is like (&Ke)lf2p~^2 as p tends to infinity.Using this result, we shall prove that the variational solutions remain bounded uniformly in p . As p tends to infinity, the solutions develop one or two peaks. Precisely the solutions approach zero except at one or two points where they stay away from zero and bounded from above.Then we consider the problem on a special class of domains. It turns out that the solutions then develop only one peak. For these domains, the solutions enlarged by a suitable quantity behave like a Green's function of -A. In this case we shall also prove that the peaks must appear at a critical point of the Robin function of the domain.

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Asymptotic behavior of least energy nodal solutions for biharmonic Lane–Emden problems in dimension four

Chen,

Cheng,

Zhao

2023

Calc. Var.

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On a Semilinear Elliptic Equation in R2 When the Exponent Approaches Infinity

Cited by 6 publications

References 0 publications

On a Two-Dimensional Elliptic Problem with Large Exponent in Nonlinearity

Asymptotic behavior of least energy nodal solutions for biharmonic Lane–Emden problems in dimension four

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