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

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