Positive solutions of nonlinear elliptic equations—existence and nonexistence of solutions with radial symmetry in $L_{p}(\textbf {R}^{N})$

Abstract:

It is shown that when $r$ is nonincreasing, radially symmetric, continuous and bounded below by a positive constant, the solution set of the nonlinear elliptic eigenvalue problem \[ - \Delta u = \lambda u + r{u^{1 + \sigma }},\qquad u > 0\qquad {\text {on}} {\mathbf {R}^N},\qquad u \to 0\qquad {\text {as}} {\text {|x|}} \to \infty \], contains a continuum $\mathcal {C}$ of nontrivial solutions which is unbounded in $\mathbf {R} \times {L_p}({\mathbf {R}^N})$ for all $p \geq 1$. Various estimates of the ${L_p}$ norm of $u$ are obtained which depend on the relative values of $\sigma$ and $p$, and the Pohozaev and Sobolev embedding constants.