Positive and sign-changing solutions for nonlinear equations with rapid growing weights

ABSTRACT

We prove the existence of a positive ground state and a sign-changing solution with exactly two nodal domains for the equation $-\mathrm{div}(K(x)\mathrm{\nabla }u)=\lambda K(x)|x{|}^{\beta }|u{|}^{q-2}u+\gamma K(x)|u{|}^{2^{\ast }-2}u$ in ${\mathbb{R}}^N$, where $N\ge 3$, $2<q<2N/(N-2)$, $\lambda >0$ is a parameter, $K(x)=\exp (\frac{|x{|}^{\alpha }}{4})$, $\alpha \ge 2$ and $\beta =(\alpha -2)\frac{(2^{\ast }-q)}{(2^{\ast }-2)}$. The equation is subcritical if $\gamma =0$ and critical if $\gamma =1$.

KEYWORDS:

• Subcritical and critical equation
• exponential weights
• self-similar solutions
• ground state
• positive solution
• nodal domain
• sign-changing solution
• weak solutions

2020 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary 35J60
• Secondary 35B08
• 35B33
• 35C06
• 35J20
• 49J45

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

G. Figueiredo has been partially supported by CNPq and FAPDF and M. Montenegro has been partially supported by CNPq and FAPESP.