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

Abstract

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 \[ − Δ u = λ u + r u 1 + σ , u > 0 on R N , u → 0 as |x| → ∞ - \Delta u = \lambda u + r{u^{1 + \sigma }},\qquad u > 0\qquad {\text {on}}\,{\mathbf {R}^N},\qquad u \to 0\qquad {\text {as}}\,{\text {|x|}} \to \infty \] , contains a continuum 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.