We prove existence and bifurcation results for a semilinear eigenvalue problem in ${ℝ}^N$ $(N\ge 2)$, where the linearization — $$ has no eigenvalues. In particular, we show that under rather weak assumptions on the coefficients $\lambda =0$ is a bifurcation point for this problem in $H^1,H^2$ and $L^p$ $(2\le p\le \infty )$.

Let ${\lambda }_1\left(Q\right)$ be the first eigenvalue of the Sturm-Liouville problem $$y^{\text{'}\text{'}}-Q\left(x\right)y+\lambda y=0,y\left(0\right)=y\left(1\right)=0,0<x<1.$$ We give some estimates for $m_{\alpha ,\beta ,\gamma }={inf}_{Q\in T_{\alpha ,\beta ,\gamma }}{\lambda }_1\left(Q\right)$ and $M_{\alpha ,\beta ,\gamma }={sup}_{Q\in T_{\alpha ,\beta ,\gamma }}{\lambda }_1\left(Q\right)$, where $T_{\alpha ,\beta ,\gamma }$ is the set of real-valued measurable on $\left[0,1\right]$ $x^{\alpha }{(1-x)}^{\beta }$-weighted $L_{\gamma }$-functions $Q$ with non-negative values such that ${\int }_0^1{x}^{\alpha }{(1-x)}^{\beta }{Q}^{\gamma }\left(x\right)\mathrm{d}x=1$ $(\alpha ,\beta ,\gamma \in ℝ,\gamma \ne 0)$.

We study the boundary value problem $-{\mathrm{d}iv\left(\right(\left|\nabla u\right|}^{p_1\left(x\right)-2}+{\left|\nabla u\right|}^{p_2\left(x\right)-2}\left)\nabla u\right)=f(x,u)$ in $\Omega$, $u=0$ on $\partial \Omega$, where $\Omega$ is a smooth bounded domain in ${ℝ}^N$. Our attention is focused on two cases when $f(x,u)=\pm (-\lambda |u{|}^{m\left(x\right)-2}u+\left|u{|}^{q\left(x\right)-2}u\right)$, where $m\left(x\right)=max\{p_1\left(x\right),p_2\left(x\right)\}$ for any $x\in \overline{\Omega }$ or $m\left(x\right)<q\left(x\right)<\frac{N·m\left(x\right)}{(N-m(x\left)\right)}$ for any $x\in \overline{\Omega }$. In the former case we show the existence of infinitely many weak solutions for any $\lambda >0$. In the latter we prove that if $\lambda$ is large enough then there exists a nontrivial weak solution. Our approach relies on the variable exponent theory of generalized Lebesgue-Sobolev spaces, combined with a ${\mathbb{Z}}_2$-symmetric version for even...